SPIRIT OF MATHEMATICS

NATIONAL MATHEMATICAL YEAR 2O12

2012-01-17T22:07:00.000+05:30

PM declares 2012 as 'National Mathematical Year'

Chennai: Declaring 2012 as the 'National Mathematical year' as a tribute to maths wizard Srinivasa Ramanujan, Prime Minister Manmohan Singh on Monday voiced concern over the "badly inadequate" number of competent mathematicians in the country.

He also said that the perception that pursuit of mathematics does not lead to attractive career possibilities "must change."

"It is a matter of concern that for a country of our size, the number of competent mathematicians that we have is badly inadequate", he said at a function to here mark the 125th birth anniversary of Ramanujan.

Singh also declared December 22, the birthday of Ramanujan, as 'National Mathematics Day.'

Students have not pursued mathematics at advanced levels over more than three decades, which has resulted in a decline in quality of mathematics teachers at schools and colleges, Singh who is on a two-day visit to the state, told a galaxy of academics at Madras University.

"There is a general perception in our society that the pursuit of mathematics does not lead to attractive career possibilities. This perception must change. This perception may have been valid some years ago, but today there are many new career opportunities available to mathematics and the teaching perception itself has become much more attractive in recent years", Singh said.

The Prime Minister said the mathematical community has a duty to find out "ways and means" to address the shortage of top quality mathematicians and reach out to the public, especially in the modern context, where mathematics has tremendous influence on every kind of human endeavour.

Noting that the Central government has pursued a policy of encouraging scientific activities of diverse kinds, the Prime Minister said, "Given our traditions, we naturally attach special importance to mathematics...in many ways, mathematics can be regarded as the mother science".

He said Ramanujan overcame formidable difficulties to reach the pinnacle of greatness, illustrating the inadequacy of University evaluation system in the early decades of the last century, while at the same time showing the system displayed enough flexibility to take care of mavericks like him.

"There have been many reforms since those days but there would still be talent which would elude proper evaluation. Our institutions of higher learning must be sensitive to this problem."

"A genius like Ramanujan would shine bright even in the most adverse of circumstances, but we should be geared to encourage and nurture good talent which may not be of the same calibre as that of Ramanujan", Singh said.

Honouring Professor Robert Kanigel, who has written a biography of Ramanujan, Singh said this book has made Ramanujan well known to the public at large all over the world.

He said the country was proud of Ramanujan and Tamil Nadu has a special claim on him for he was a Tamilian.

"Along with CV Raman and Subramanyam Chandrashekhar (both Nobel laureates), he is among the three great men of science and mathematics that Tamil Nadu and India have given to the world of modern times", he said.

AN INTERESTING FACT IN FEBRAURY 2010

2010-01-12T13:00:00.005+05:30

An interesting fact about Febraury 2010.
4 Sundays
4 Mondays
4 Tuesday
4 Wednesday
4 Thursday
4 Friday
4 Saturday

REALLY AMAGING

A WEDDING PROPOSAL BY Mr. ALGEBRA TO Mr. CALCULUS

2009-12-16T14:22:00.005+05:30

Dear Mr Calculus
I am asking your opinion about the marriage of my sons " Mr Zero" and "Mr Infinity" with your daughters Miss Differentiation and Miss Integration. I have consulted 'Mr Vector' who told me that this Marriage is strictly according to 3rd law of marriage " To every husband, there is equal and opposite wife" He further continued that the two pairs will lead a happy life. I have also come to know that your daughters are in love with my sons.

Of My sons, Zero is very popular among the students and possess a high name and fame. As regard to 2nd son infinity any thing added or subtracted from him he remains unchanged. After consulting the formula and Pundit of Logarithm, you may fix up the date for marriage. I am confident that you will accept the proposal. You can also consult Aunt dynamics, Alpha, Beta, Gamma, Delta and Statistics. Sigma will accompany us in marriage party. Please convey my best wishes to sister Geometry and her daughter "Co ordinate geometry"

Your sincerely
Mr ALGEBRA
MATHEMATICS LANE
MATHS HOUSE
MATHS CITY

SALARY THEOREM

2009-11-16T20:45:00.003+05:30

MATHEMATICS-LANGUAGE OF THE UNIVERSE

2009-10-15T19:53:00.002+05:30

Do you know which langauge shared by all human beings regardless of the culture, religion, or gender?
It is Mathematics , the only language shared by all human beings and language of symbols.
Pi is still 3.14159, regardless of what country you are in . Adding up the cost of a basket full of groceries involves the same math process regardless of whether the total is expressed in dollars, yen or rupees. With this universal language, all of us, no matter what our unit of exchange, are likely to arrive at math results the same way.
All of us possess the ability to be " literate" in the shared language of math. The math literacy is called numeracy, and it is this shared language of numbers that connects us with people across continents and through time. It is what links ancient scholars and medieval merchants, astronauts and artists peasants and presidents
With this language we can explain
the mysteries of the universe or secrets of DNA.
We can understand the forces of planetary motion
Discover cures for catastrophic diseases
calculate the distance from any city of America to any city of India
We can build computers and transfer information across the globe.
We can save money for retirement
So math is not just for calculus majors. It is for all of us.
It is not just calculating difficult equations. It is about making better decisions and , hopefully leading richer, fuller lives.
Mathematics is Universal by the following quotes

"I bow to that glorious Lord of the Jainas, who as the shining lamp of the know-ledge of numbers made to shine whole of the universe", said Mahaviracharya in Ganita Sara Sangraha.

The laws of nature are but the mathematical thoughts of God. ~Euclid

I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe - because like Spinoza's God, it won't love us in return. ~Bertrand Russell, 1912

The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization. ~John Kemeny

Mathematics is the gateway and the key to other sciences- Roger Bacon

INTERSTING NUMBER 37

2009-09-16T21:43:00.000+05:30

NUMBER 37 IS AN INTERESTING NUMBER:
SEE THE PATTERN:
111/(1+1+1)=37
222/(2+2+2)=37
333/(3+3+3)=37
444/(4+4+4)=37
555/5+5+5=37
666/(6+6+6)=37
777/(7+7+7)=37
888/(8+8+8)=37
999/(9+9+9)=37

YOUR AGE BY CHOCOLATE MATHS

2009-09-13T00:24:00.000+05:30

Hi Everybody,

Have a sweet morning.. or .afternoon.. or .evening !!!

YOUR AGE BY CHOCOLATE MATHS

Don't tell me your age , the Hershey Man will know!

YOUR AGE BY CHOCOLATE MATHS This is pretty neat.

DON'T CHEAT BY SCROLLING DOWN FIRST!
It takes less than a minute .
Work this out as you read .

Don't read the bottom until you've worked it out!

1. Pick the number of times a week you'd like to have chocolate
(more than once but less than 10)

2. Multiply this number by 2 (just to be bold)

3.. Add 5

4. Multiply it by 50 -- I'll wait while you get the calculator
(or use the one listed under 'accessories' on your computer)

5. If you already had your birthday this year add 1759 ...
If you haven't, add 1758.

6. Now subtract the four digit year that you were born

You should have a three digit number

The first digit of this was your original number
(i.e., how many times you want to have chocolate each week)

The next two numbers are

YOUR AGE! (Oh YES, it is!!)

2009 IS THE ONLY YEAR IT WILL WORK, SO SPREAD IT AROUND WHILE IT LASTS.
Chocolate
Calculator

MATH FUN

2009-09-13T00:22:00.000+05:30

Let me take only a minute

Just do the following multiplication :

13837 x Your Age x 73 = ? ? ?

You get very interesting resut, let me know.

You get the same result if you multiply 10001 * your age * 101

How is that?

FIND OUT MISTAKE

2009-09-11T23:33:00.004+05:30

Find out the mistake in this :

Division by Zero

Everyone knows that0/2=0 , the problem is that far too many people also say that

or 2 = 2 ! Remember that division by zero is undefined! You simply cannot divide by

zero so don’t do it!

Here is a very good example of the kinds of havoc that can arise when you divide by zero. See if you can find the mistake that I made in the work below.

1. a = b We’ll start assuming this to be true.

2. ab = a ^2 Multiply both sides by a.

3. ab - b^2 = a^ 2 - b^2 Subtract b^2 from both sides.

4. b ( a - b) = ( a + b ) ( a - b )Factor both sides.

5. b = a + b Divide both sides by a - b .

6. b = 2b Recall we started off assuming a = b .

7. 1 = 2 Divide both sides by b.

So, we’ve managed to prove that 1 = 2! Now, we know that’s not true so clearly we

made a mistake somewhere. Can you see where the mistake was made?

The mistake was in step 5. Recall that we started out with the assumption a = b .

However, if this is true then we have a - b = 0 ! So, in step 5 we are really dividing by zero!

That simple mistake led us to something that we knew wasn’t true, however, in most

cases your answer will not obviously be wrong. It will not always be clear that you are dividing by zero, as was the case in this example. You need to be on the lookout for this kind of thing.

Remember that you CAN’T divide by zero!

APPLICATION OF DIFFERENTIATION

2009-06-27T23:13:00.001+05:30

Uses of Differentiation

Increasing and Decreasing Functions

An increasing function is a function where: if x₁ > x₂, then f(x₁) > f(x₂) , so as x increases, f(x) increases. A decreasing function is a function which decreases as x increases. Of course, a function may be increasing in some places and decreasing in others. A point where a function changes from an increasing to a decreasing function or visa-versa is known as a turning point. A turning point is a type of stationary point (see below).

We can use differentiation to determine if a function is increasing or decreasing:

A function is increasing if its derivative is always positive. A function is decreasing if its derivative is always negative.

Examples

y = -x has derivative -1 which is always negative and so -x is decreasing.

y = x² has derivative 2x, which is negative when x is less than zero and positive when x is greater than zero. Hence x² is decreasing for x<0>0 .

Stationary Points

Stationary points are points on a graph where the gradient is zero. There are three types of stationary points: maximums, minimums and points of inflection (/inflexion). The three are illustrated here:

Example

Find the coordinates of the stationary points on the graph y = x² .
We know that at stationary points, dy/dx = 0 (since the gradient is zero at stationary points).

By differentiating, we get: dy/dx = 2x. Therefore the stationary points on this graph occur when 2x = 0, which is when x = 0.

When x = 0, y = 0, therefore the coordinates of the stationary point are (0,0). In this case, this is the only stationary point. If you think about the graph of y = x², you should know that it is "U" shaped, with its lowest point at the origin. This is what we have just found.

Maximum, Minimum or Point of Inflection?

At all the stationary points, the gradient is the same (= zero) but it is often necessary to know whether you have found a maximum point, a minimum point or a point of inflection. Therefore the gradient at either side of the stationary point needs to be looked at (alternatively, we can use the second derivative).
At maximum points, the gradient is positive just before the maximum, it is zero at the maximum and it is negative just after the maximum. At minimum points, the gradient is negative, zero then positive. Finally at points of inflexion, the gradient can be positive, zero, positive or negative, zero, negative. This is illustrated here:

Example

Find the stationary points on the graph of y = 2x² + 4x³ and state their nature (i.e. whether they are maxima, minima or points of inflexion).
dy/dx = 4x + 12x²
At stationary points, dy/dx = 0
Therefore 4x + 12x² = 0 at stationary points
Therefore 4x( 1 + 3x ) = 0
Therefore either 4x = 0 or 3x = -1
Therefore x = 0 or -1/3

When x = 0, y = 0
When x = -1/3, y = 2x² + 4x³ = 2(-1/3)² + 4(-1/3)³ = 2/9 - 4/27 = 2/27

Looking at the gradient either side of x = 0:
When x = -0.0001, dy/dx = negative
When x = 0, dy/dx = zero
When x = 0.0001, dy/dx = positive
So the gradient goes -ve, zero, +ve, which shows a minimum point.

Looking at the gradient either side of x = -1/3 .
When x = -0.3334, dy/dx = +ve
When x = -0.3333..., dy/dx = zero
When x = -0.3332, dy/dx = -ve
So the gradient goes +ve, zero, -ve, which shows a maximum point.

Therefore there is a maximum point at (-1/3 , 2/27) and a minimum point at (0,0).

Solving Practical Problems

This method of finding maxima and minima is very useful and can be used to find the maximum and minimum values of all sorts of things.

Example

Find the least area of metal required to make a closed cylindrical container from thin sheet metal in order that it might have a capacity of 2000p cm³.

The total surface area of the cylinder, S, is 2pr² + 2prh
The volume = pr²h = 2000p
Therefore pr²h = 2000p.
Therefore h = 2000/r²

Therefore S = 2pr² + 2pr( 2000/r² )
= 2pr² + 4000p
r

So we have an expression for the surface area. To find when the surface area is a minimum, we need to find dS/dr .
dS = 4pr - 4000p
dr r²
When dS/dr = 0:
4pr - (4000p)/r² = 0
Therefore 4pr = 4000p
r²
So 4pr³ = 4000p
So r³ = 1000
So r = 10

You should then check that this is indeed a minimum using the technique above.
So the minimum area occurs when r = 10. This minimum area is found by substituting into the equation for the area the value of r = 10.

S = 2pr² + 4000p
r
= 2p(10)² + 4000p
10
= 200p + 400p
= 600p

Therefore the minimum amount of metal required is 600p cm²

2008-12-05T14:37:00.002+05:30

STORY OF PI

Did u know that Archimedes was the first mathematician to discover the value of pi up to 10000 digits!!!

Who was the first mathematician to give the approximate value of “pi” which is commonly accepted today? YES IT IS INDIAN Aryabhata(he gave the value of pi as 3.1416)

Notes on Pi: Pi is the most famous ratio in mathematics, and is one of the most ancient numbers known to humanity. Pi is approximately 3.14 – the number of times that a circle’s diameter will fit around the circle. Pi goes on forever, and can’t be calculated to perfect precision: 3.1415926535897932384626433832795028841971693993751…. This is known as the decimal expansion of pi. No apparent pattern emerges in the succession of digits – a predestined yet unfathomable code. They do not repeat periodically, seemingly to pop up by blind chance, lacking any perceivable order, rule, reason, or design – “random” integers, ad infinitum.

In 1991, the Chudnovsky brothers in New York, using their computer, m zero, calculated pi to two billion two hundred sixty million three hundred twenty one thousand three hundred sixty three digits (2,260,321,363). They halted the program that summer.

Pi has had various names through the ages, and all of them are either words or abstract symbols, since pi is a number that can’t be shown completely and exactly in any finite form of representation. Pi is a transcendental number. A transcendental number is a number but can’t be expressed in any finite series of either arithmetical or algebraic operations. Pi slips away from all rational methods to locate it. It is indescribable and can’t be found. Ferdinand Lindemann, a German mathematician, proved the transcendence of pi in 1882.

Pi possibly first entered human consciousness in Egypt. The earliest known reference to pi occurs in a Middle Kingdom papyrus scroll, written around 1650 BCE by a scribe named Ahmes. He began scroll with the words: “The Entrance Into the Knowledge of All Existing Things” and remarks in passing that he composed the scroll “in likeness to writings made of old.” Towards the end of the scroll, which is composed of various mathematical problems and their solutions, the area of a circle is found using a rough sort of pi.

Around 200 BCE, Archimedes of Syracuse found that pi is somewhere about 3.14 (in fractions, Greeks did not have decimals). Knowledge of pi then bogged down until the 17th century. Pi was then called the Ludolphian number, after Ludolph van Ceulen, a German mathematician. The first person to use the Greek letter for the number was William Jones, an English mathematician, who coined it in 1706.

Physicists have noted the ubiquity of pi in nature. Pi is obvious in the disks of the moon and the sun. The double helix of DNA revolves around pi. Pi hides in the rainbow, and sits in the pupil of the eye, and when a raindrop falls into water pi emerges in the spreading rings. Pi can be found in waves and ripples and spectra of all kinds, and therefore pi occurs in colours and music. Pi has lately turned up in superstrings.

Pi occurs naturally in tables of death, in what is known as a Gaussian distribution of deaths in a population; that is, when a person dies, the event “feels” pi. It is one of the great mysteries why nature seems to know mathematics.

(NOTE: The above information was gleaned from an article in The New Yorker magazine, March 2, 1992, called “Profiles: The Mountains of Pi”)

REGIONAL MATHEMATICS OLYMPIAD QUESTION PAPER 2008

2008-11-12T12:17:00.001+05:30

CLICK ON THE FOLLOWING LINK FOR
REGIONAL MATHEMATICS OLYMPIAD QUESTION PAPER 2008
http://khvmathematics.files.wordpress.com/2008/11/oly1.jpg

A GOOD BLOG FOR IIT/AIEEE ASPIRANTS

2008-08-19T16:29:00.004+05:30

HERE IS THE BLOG THAT CONTAINS INFORMATIONS RELATED TO AIEEE/IIT, VIEWS OF IIT TOPPERS, AND FOR E BOOKS DOWNLOAD. HERE YOU CAN DOWNLOAD NCERT BOOKS. IRDOV PHYSICS, RESNICK HOLLIDAY FUNDAMENTALS OF PHYSICS AND MANY MORE VISIT: http://iit-dreams.blogspot.com/

What is the best way to study mathematics

2008-06-22T00:07:00.000+05:30

Once open a time a lady wanted to hire a cook for her restaurant. She interviewed many candidates and finally she selected the one who impressed her in the interview. Lady was very impressed with his knowledge of cooking. He was like a cooking encyclopedia.

On his first day of the job, lady thought of taking his trial. She asked him to cook “Kadai Chicken“. She started eagerly waiting for the food. Food was served and she had her first bite. It was pathetic in taste. She could not swallow even a single bite of it. The lady was shocked and asked the man “It is so horrible in taste. Are you sure you can cook food?” He replied, “Madam, sorry for the food. Actually I have never done cooking before. I just had taken lessons on cooking from experts and during my training classes, I saw the instructor cooking. Also, I have read and learned all the recipes of making excellent food.”

Moral of the story is, there is a difference in knowing how to do things and actually being able to do things. By merely just knowing how to do things does not make you expert on actually doing it yourself.

Learning Mathematics is like learning the art of solving problems (actually doing) and not just knowing formulae and concepts (acquiring knowledge). So if you want to improve your mathematics, you need to focus more and more on problem solving instead of just reading theories, formulae, and solutions.

Following are some instructions/tips that would definitely help you learn mathematics better:

Always study Mathematics sitting on a study table with paper and pen to use: More you write, better you remember. Even if you are reading concepts and learning formulae, write it and learn it. Mathematics needs a higher level of concentration. Whether you are solving a problem or reading mathematical steps of a solution you need better concentration and focus. So my suggestion would be to sit on a table chair with no disturbance around. If your room is noisy, you can put cotton balls in your ears.

Spend more time on solve problem instead of reading solutions/theories/formulae: More you practice, better you would learn. It is very important that you solve problems to learn topics in mathematics. Just understanding concepts and learning formulae would not be sufficient to be able to solve questions in exam. In mathematics more than 50% of the knowledge comes through tricks/methods involved in solving problems. If you don’t practice questions, you don’t acquire this knowledge. In fact learning in Mathematics starts the day you start solving problems with pen and paper.

Step by Step learning:Learn theory and formulae first. Practice them in written. You should start reading solved-examples only after learning the concepts and formulae. This is must for easy understanding of the solved-examples as in every questions you use multiple formulae. If you don’t remember formulae well, you will take more time to understand the solution. After finishing examples, you need to solve level-1 (easy-to-average level) problems.

How to decide level? If you are not able to solve, go through solution. If you can understand the solution by just glancing it (as a hint), then it is level-1 (easy-to-difficult) problem. If you have to go through complete solution step by step and then finally you can understand the solution, then it is a level-2 (average-to-difficult) problem. If you find it hard to understand solution, it means it is level-3 (difficult-to-very difficult) problem. These levels are relative as every student has his own potential.

Once you have solved 30-40 level-1 problems and have thoroughly revised them to a level that you remember the ideas of most of them, you can then move to level-2 problems. Practice at least 30-40 level-2 problems. Don’t solve level-3 problems. They are not important and you can confidently leave them. Trying to solve them can be negative as they can break your confidence in the topic.

Revision and Re-Learning: Generally when you are not able to solve problems, you see their solutions. But you do nothing after that. In 1-2 weeks time you forget the solution. I am sure if you face that question again, you would not be able to solve it. So what is the point spending time on the question at first stage.

I suggest after reading the solution, you try to solve it yourself with paper and pen. Don’t worry if you know the solution now as you have read the solution. Mind will retain only if you do it with your hands. Then, mark the level of the question for future revision. After few week, try all questions again which are level -1 and Level-2. Do them like a test. Shortlist 50 such questions and take a 2-hours test. Even before exam, when you are confused what to revise, take out Level-1 and Level-2 (or just level-2) problems and revise them. If you don’t mark them, you cannot revise them.

Don’t refer solutions without trying problems: With all the books and study material around, most students have a tendency to read a question and immediately jump to see the solution. This is totally wrong and if you continue this for a long time, you will become dependent on solutions and you develop a bad habit of surrendering. Where as in mathematics, you need a fighting attitude. Try hard to crack the trick. I know you don’t have too much time to spend on each question but at least in each 60-70% questions you should attempt yourself first (spend 10-15 minutes average time on each question) and then refer solution. It is very important to try first as your brain develops only when you put stress on it.

Generate your Interest to perform better: No doubt, People who like mathematics perform better than others. As it involves applying tricks (like in puzzles and games) to solve problems, you perform better if you are liking what you are doing. You need to do problem solving when you are willing to do it. Feel proud if you are able to solve a question, feel thrilled rather than feeling frustrated when you take help of solutions (or help of others) to solve the problems. As I suggested in “Revision and Re-Learning”, those questions which you are able to solve through solutions, solve them again. When you are able to solve them again, you will feel good and that will help in generating interest.

Make Flash Cards for better learning: To learn formulae and even tricks involved in problem solving, make paper based card (paper sheets) and keep them with you. You can memorize them even when you are not at your desk, may be when you in a car/bus, in school, while walking, etc. This helps in building your knowledge, generates interest and above all you are utilizing your non productive time.

Help others if you get chance: If somebody needs your help in solving a problem and you know how to solve it, never miss the opportunity to help her. This generates confidence in you as well as your interest would also go up. More confident you are, better you can think.

-Manmohan Gupta
(HOD Mathematics, VMC)
Mathematics is Fun

A review for Mathematics site

2008-05-15T11:12:00.003+05:30

For about many days I am back again . After valuation work for PU students and many more reasons Now I am giving certain ideas for mathematics.
ವಿಸಿಟ್ ದಿ ಸೈಟ್
i. e. http://mathsisinteresting.blogspot.com
(. The site is really helpful and Every mathematics students must read the articles. The author has given many applications of mathematics in real life and in other fields. His article"(Solving Maths Develops Plan Foward Capability)" will provoke to know the importance of mathematics. http://www.limeehai.com/478/how-to-study-math-wisely/ How to study maths wisely and many more articles are good. So visit the said website for increasing mathematical knowledge.

To prepare for IIT and AIEEE visit: goiit.com

JARASANDHA NUMBERS

2008-02-23T18:00:00.003+05:30

In our Indian epic Mahabharatha, we come across a demonaic figure named 'JARASANDHA' . He had a boon that if he was split into two parts and thrown apart, the parts would rejoin and return to life. In fact, he was given life by the two halves of his body.
In the field of Mathematics, we have numbers exhibiting the same property as Jarasandha.

Consider a number of the form XC . This may split as two numbers X and C and if these numbers are added and squared we get the same number XC again i.e
XC ----(X+C)^2 = 10^n X +C =XC

ILLUSTRATIONS:
81=(8+1)^2 = (10^1 x8)+1=81
9801=(98+01)^2 =99^2 =9801 =10^2x98+01=9801
3025=(30+25)^2 =55^2= 3025
2025=(20+25)^2 =45^2 =2025
88209=(88+209)^2=297^2 =88209

Can You find such numbers , please give such numbers

HOW TO FIND THE LAST DIGIT OF ANY NUMBER

2008-01-20T22:18:00.002+05:30

How to find the last digit of a number?

It is very difficult to find the last digit of a number by usual traditional method. In number theory finding the last digit or finding the remainder when any number is divided by 10 is having more importance and also one of interesting problem.

However if u remember and used the following rule, we can easily find the last digit.

Last digit of the number of the form mⁿ
m n=1 n=2 n=3 n=4 n=5
0 0 0 0 0 0
1 1 1 1 1 1
2 2 4 8 6 2
3 3 9 7 1 3
4 4 6 4 6 4
5 5 5 5 5 5
6 6 6 6 6 6
7 7 9 3 1 7
8 8 4 2 6 8
9 9 1 9 1 9

Last digit of mⁿ is changing upto n=4 when n=5, the last digit is repeated. Hence the last digit is periodic with 4

METHOD: To find the last digit of number mⁿ Divide n by 4, If the remainder is r, then the last digit of the number is (the last digit of given number)^r. otherwise if remainder is zero, then the last digit of number is last digit of m^4 i.e (last digit of given number)^4.

ILLUSTRATIONS:
1) The last digit of the number
N=(2003)⁵⁰⁰³ +(6007)⁹⁰⁵ –(209)⁶⁰⁸ =3³ +7¹ -9⁴ =14-1=13, the last digit is 3.
2)Find the remainder when

(608)¹⁸²⁹ is divided by 10

Sol: Finding the remainder when any number is divided by 10 is same as finding the last digit of the number , since when any number is divisible by 10 the remainder is the last digit of the given number. Last digit of 608 is 8 and when 1829 is divided by 4 , remainder is 1

Hence the last digit is 8¹ (by rule )
3)Find the last digit of 4^1328
Here when 1328 is divided by 4 , Remainder is 0 , so the last digit of 4^1328 is last digit of 4^4 i.e 6
FINDING THE LAST DIGIT ANY NUMBER OF THE FORM m^n ENDING WITH 0,1,2,3,4,5,6,7,8,9
a)The last digit of 5^1 is 5
5^2 is 5
5^3 is 5
Hence the last digit of 5^any number is 5
b)The last digit 6^1 is 6
6^2 is 6
6^3 is 6
and so on.
Hence the last digit of 6^any number is 6
c)The last digit of
11^1 is 1
11^2 is 1
11^3 is 1
11^4 is 1 and so on
The last digit of (any number ending with 1)^any number is 1
d)Similarly we can show that the last digit of (any number any number ending with 0)^any number is 0

e)Last digit of 2^1 is 2
2^2 is 4
2^3 is 8
2^4 is 16 ie 6
2^5 is 32 ie 2
2^6 is 64 i.e 4
2^7 is 128 i.e 8
2^8 is 256 i.e 6
In general we can see that the last digit of (any number ending with 2)^any number is 2,4,8,6
when power n is odd , the last digit is 2 and 8; when the power is even, the last digit is 4 and 6
2²ⁿ ≡ 6(mod 10) when n is even

≡ 4(mod 10) when n is odd

f)3²ⁿ ≡ 1(mod 10) when n is even .

≡9(mod 10) when n is odd

g) We can also see that last digit of (any number ending with 7)^any number is 1 when n is odd and 9 when n is even
7²ⁿ ≡1(mod 10) when n is even.

≡ 9(mod 10) when n is odd

h)The last digit of (any number ending with 8)^any number is is 8, 6 or 4

8²ⁿ ≡ 6(mod 10) when n is even.

≡ 4(mod 10) when n is odd.

j)9ⁿ ≡ 1(mod 10) when n is even

≡ 9(mod 10) when n is odd

To find the unit or last digit remember the following:(Try to prove)

a)5ⁿ≡5(mod 10) b)6ⁿ =6(mod 10)

c)4ⁿ ≡ 6(mod 10) when n is even .

≡ 4(mod 10) when n is odd

d)9ⁿ ≡ 1(mod 10) when n is even

≡ 9(mod 10) when n is odd

e)3²ⁿ ≡ 1(mod 10) when n is even .

≡ 9(mod 10) when n is odd

f)2²ⁿ ≡ 6(mod 10) when n is even

≡ 4(mod 10) when n is odd

g)8²ⁿ ≡ 6(mod 10) when n is even.

≡ 4(mod 10) when n is odd.

h) 7²ⁿ ≡1(mod 10) when n is even.

≡ 9(mod 10) when n is odd i.e

When n is positive integer ,

the last digit of 5ⁿ is 5 ; the last digit in 6ⁿ is 6.

The last digit in 3⁴ⁿ is 1; The last digit in 9ⁿ is 1 or 9 according as n is odd or even. The last digit of 11ⁿ is 1, 6ⁿ is 6 , 4ⁿ is 4 or 6 according as n is odd or even.

examples:

Unit digit of 17¹⁸⁹ =unit digit of 7¹⁸⁹ =7

Unit digit of 28²⁰⁰ =unit digit of 8²⁰⁰ =6

Unit digit of 43²⁰ =unit digit of 3²⁰ =1

Unit digit of 79²⁷ =unit digit of 9²⁷ =9

Unit digit of 84³⁷ =unit digit of 4³⁷ =4

STORY OF ZERO

2008-01-15T08:42:00.000+05:30

STORY OF ZERO

Perhaps the most fundamental contribution of ancient India to the progress of civilisation is the decimal system of numeration including the invention of the number zero. This system uses 9 digits and a symbol for zero to denote all integral numbers, by assigning a place value to the digits. This system was used in Vedas and Valmiki Ramayana. Mohanjodaro and Harappa civilisations (3000 B.C.) also used this system.

If zero merely signified a magniutude or a direction seperatorI(i.e. separting those above the zero level from those below the zero level), the Egyptian zero, nfr, dating back atleast four thousand years, amply served these purposes. The ancient Egyptians (5000 B.C.) had a system based on 10, but they didn't use positional notation. Thus to represent 673, they would draw six snares, seven heel bones and three vertical strokes.

If zero was merely a place holder symbol, indicating the absence of a magnitude at a specified place position (such as, for example, the zero in 10 indicated the absence of any 'tens' in one hundred and one), then such a zero was already present in the babylonian number system long before the first recorded occurence of the Indian zero. Babylonians in Mesopotamia (3000 B.C.) had a sexagesimal system using base 60. Greeks and Romans had a cumbersome system (try to write 2376 in Roman numerals).

If zero was represented by just an empty space within a well defined postional number system, such zero was present chinese mathematics a few centuries before the Indian zero.

Many civilisations had some concept of "zero" as nothing - for example, if you have two cows and they both die, you are left with nothing.

However, the Indians were the first to see that zero can be used for something beyond nothing - at different places in a number, it adds different values. For example, 76 is different from 706, 7006, 760 etc.

Indian zero alluded to in the question was a multi faceted mathematical object: a symbol, a number, a magnitude, a direction seprator and a place holder, all in one operating with a fully established positional number system. Such a zero occured only twice in history- the indian zero which is now the universal zero and the Mayan zero which occupied in solitary isolation in central america around the beginning of commaon Era.

Brahmgupta (598 AD - 660 AD) was the first to give the rules of operation of zero.

A + 0 = A, where A is any quantity.
A - 0 = A,
A x 0 = 0,
A / 0 = 0

He was wrong regarding the last formula. This mistake was corrected by Bhaskara (1114 AD - 1185 AD), who in his famous book Leelavati, claimed that division of a quantity by zero is an infinite quantity or immutable God.

The ancient Indians represented zero as a circle with a dot inside. The word 'zero' comes from the Arabic "al-sifr". Sifr in turn is a transiliteration of the Sanskrit word "soonya" meaning of void or empty, which became later the term for zero. This and the decimal number system fascinated Arab scholars who came to India. Arab mathematician Al-Khowarizmi (790 AD - 850 AD) wrote Hisab-al-Jabr wa-al-Muqabala (Calculation of Integration and Equation) which made Indian numbers popular. "Soonya" became "al-sifr" or "sifr". The impact of this book can be judged by the fact that "al-jabr" became "Algebra" of today.

An Italian Leonardo Fibonacci (1170 AD - 1230 AD) took this number system to Europe. The Arabic "sifr" was called "zephirum" in Latin, and acquired many local names in Europe including "cypher". In the beginning, the merchants used to Roman numbers found the decimal system a new idea, and referred to these numbers as "infidel numbers", as the Arabs were called infidels because they had invaded the holy land of Palestine.

However, nowadays this system is called Hindu-Arabic System. This positional system of representing integers revolutionised the mathematical calculations and also helped in Astronomy and accurate navigation. The use of positional system to indicate fractions was introduced around 1579 AD by Francois Viete. The dot for a decimal point came to be used a few years later, but did not become popular until its use by Napier.

CALCULUS JOKES

2008-01-02T22:56:00.000+05:30

A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying "I differentiate you!"

One day he met a new patient; and true to form he stared at him and said "I differentiate you!", but for once, his victim's expression didn't change.

Surprised, the mathematician marshalled his energies, stared fiercely at the new patient and said loudly "I differentiate you!", but still the other man had no reaction. Finally, in frustration, the mathematician screamed out "I DIFFERENTIATE YOU!"

The new patient calmly looked up and said, "You can differentiate me all you like: I'm e to the x." (CALCULUS)

The functions are sitting in a bar, chatting (how fast they go to zero at infinity etc.). Suddenly, one cries "Beware! Derivation is coming!"
All immediately hide themselves under the tables, only the exponential sits calmly on the chair.

The derivation comes in, sees a function and says "Hey, you don't fear me?"
"No, I'am e to x", says the exponential self-confidently.
"Well" replies the derivation "but who says I differentiate along x?" (CALCULUS)

Big party; every possible function is having fun, chatting and drinking

this evening.

In an n-dimensional corner e^x stands bitter and alone.  Near the lonely

one there's a small group of exponential functions, and 2^x within them

turns to see e^x on it's corner.

- Hey, e^x, come-on, integrate yourself - Said 2^x pointing to the group.

- What for - whispers e^x - it makes no difference

Math and Alcohol don't mix, so...

PLEASE DON'T DRINK AND DERIVE

Then there's every parent's scream when their child walks into the

room dazed and staggering:

OH NO...YOU'VE BEEN TAKING DERIVATIVES!!

The limit as n goes to infinity of sin(x)/n is 6.

Proof: cancel the n in the numerator and denominator.

In a dark, narrow alley, a function and a differential operator meet:
"Get out of my way - or I'll differentiate you till you're zero!"
"Try it - I'm e^x..."
Same alley, same function, but a different operator:
"Get out of my way - or I'll differentiate you till you're zero!"
"Try it - I'm e^x..."
"Too bad... I'm d/dy."

What is interesting about the number 2008? and mathematical horoscope for 2008

2007-12-31T20:09:00.000+05:30

What is interesting about the number 2008?

I WISH ALL MY BLOG READERS
HAPPY NEW YEAR 2008
I hope may 2008 will bring success to all our ways of life in future

Mathematically , I am thinking of what specail about number 2008, coming new year,

1.2008 is a Kaprekar constant in base 3(see my article about kaprekar constant in the blog)

2. 2008 is the prime number 251 multiplied by the sum its own digits.

But 2008 is a boring number(only numerically speaking I hope) and to make things worse my attic is still untidy.
Find something interesting about this number and let me know in the comments. If it passes my own personal test of ‘interestingness’ (loosely defined as anything that makes me say “thats cool”) then I will update this page with the results.

IF YOU WANT MORE FACTS ABOUT OTHER NUMBER FROM 0,1, ...............9999 YOU CAN VISIT THE WEB: http://www.stetson.edu/~efriedma/numbers.html

Mathematical Horoscope for 1-1-2008

AREAS (Mar 21-Apr 19) Today a close friend who has let down her boundaries will have a dilemma. Though you have several axes to grind with her, you should not let this be a factor. If she can count on you, you will get closure.	E-BAR (Sep 23-Oct 22) Today you will intersect a radical from class, a tan gent of average height. He will be a cute guy, but obtuse. Help him by reading his abstract, so he can get his degree and reach his proper rank.
TORUS (Apr 20-May 20) Today you will be in your prime. A clique of friends will give you a ring and hold an enjoyable function. You will feel connected to this circle of friends, and will be the center of attention.	SCORE-PI-O (Oct 23 - Nov 21) You have reached a critical point in your life. You are on edge and have about reached your limit. This minor identity crisis has kept you in knots for some time. Seek a group of solid friends to set things right.
M and I (May 21-Jun 21) Today you will get a visit from an ex, though you won't know why. The jerk is still square, a real number, quite a character. He will feed you a line about being a pair again, and this will cause an argument, making you tensor and tensor.	SLANTED-AREAS (Nov 22-Dec 21) Today you should stay in bed and catch some Z's. Stay horizontal and sleep like a log. This complete lack of activity will mean a minimum of stress, but it is also a product of the power you have.
ANSWER (Jun 22-Jul 22) Don't commute today. If you go by plane, the floor will drop out. If your mode is by car, you will hit a field of trees in the median. Even walking on your own digits will result in a funeral plot.	CAPRI-CHORD (Dec 22-Jan 19) This period of your life will be similar to one year ago, when your life was simple and ordered. It is integral that you be careful, lest you do commit an improper error. Look for a sign, but stick to your roots.
E-O (Jul 23-Aug 22) Today life will throw you a curve. You and a loved one will diverge, and the distance will feel empty. Though you will see her less, her existence proves your perfect friendship is in a separate category. Don't let it make a difference.	A-SQUARY-PLUS (Jan 20-Feb 18) Keep an open mind today. Stop projecting and admit the magnitude of your problems for a second. If you are analytic, the origin of your exact difficulty will soon be clear, and your troubles will decrease by half.
ERGO (Aug 23-Sep 22) On the surface, today will seem like an irrational day, continuously full of problems. The constant chaos will be far from ideal. But the day will be less negative and more normal than it first appears. Stay to your regular routine and minimize variation.	PI-CEES (Feb 19-Mar 20) Today will be an odd day. You will face many complex problems, all equally frustrating. But you will maximize your joy by finding value in all of them, and it will turn out to be a fine day.

collections taken from : http://www.stetson.edu

HOW TO STUDY MATHEMATICS

2007-12-25T08:15:00.000+05:30

For the benefit of maths students Here I am giving link and article about How to study mathematics written by Paul Dawkins

How To Study Mathematics

How to Study Mathematics - Written by Paul Dawkins

Before I get into the tips for how to study math let me first say that everyone studies differently and there is no one right way to study for a math class. There are a lot of tips in this document and there is a pretty good chance that you will not agree with all of them or find that you can’t do all of them due to time constraints. There is nothing wrong with that. We all study differently and all that anyone can ask of us is that we do the best that we can. It is my intent with these tips to help you do the best that you can given the time that you’ve got to work with.
Now, I figure that there are two groups of people here reading this document, those that are happy with their grade, but are interested in what I’ve got to say and those that are not happy with their grade and want some ideas on how to improve. Here are a couple of quick comments for each of these groups.

If you have a study routine that you are happy with and you are getting the grade you want from your math class you may find this an interesting read. There is, of course, no reason to change your study habits if you’ve been successful with them in the past. However, you might benefit from a comparison of your study habits to the tips presented here.

If you are not happy with your grade in your math class and you are looking for ways to improve your grade there are a couple of general comments that I need to get out of the way before proceeding with the tips. Most people who are doing poorly in a math class fall into three main categories.

The first category consists of the largest group of students and these are students that just do not have good study habits and/or don’t really understand how to study for a math class. Students in this category should find these tips helpful and while you may not be able to follow all of them hopefully you will be able to follow enough of them to improve your study skills.

The next category is the people who spend hours each day studying and still don’t do well. Most of the people in this category suffer from inefficient study habits and hopefully this set of notes will help you to study more efficiently and not waste time.

The final category is those people who simply aren’t spending enough time studying. Students are in this category for a variety of reasons. Some students have job and/or family commitments that prevent them from spending the time needed to be successful in a math class. To be honest there isn’t a whole lot that I can do for you if that is your case other than hopefully you will become a more efficient in your studies after you are through reading this. The vast majority of the students in this category unfortunately, don’t realize that they are in this category. Many don’t realize how much time you need to spend on studying in order to be successful in a math class. Hopefully reading this document will help you to realize that you do need to study more. Many simply aren’t willing to make the time to study as there are other things in their lives that are more important to them. While that is a decision that you will have to make, realize that eventually you will have to take the time if you want to pass your math course.

Now, with all of that out of the way let’s get into the tips. I’ve tried to break down the hints and advice here into specific areas such as general study tips, doing homework,, studying for exams, etc. However, there are three broad, general areas that all of these tips will fall into.

Math is Not a Spectator Sport

You cannot learn mathematics by just going to class and watching the instructor lecture and work problems. In order to learn mathematics you must be actively involved in the learning process. You’ve got to attend class and pay attention while in class. You’ve got to take a good set of notes. You’ve got to work homework problems, even if the instructor doesn’t assign any. You’ve got to study on a regular schedule, not just the night before exams. In other words you need to be involved in the learning process.

The reality is that most people really need to work to pass a math class, and in general they need to work harder at math classes than they do with their other classes. If all that you’re willing to do is spend a couple of hours studying before each exam then you will find that passing most math classes will be very difficult.

If you aren’t willing to be actively involved in the process of learning mathematics, both inside and outside of the class room, then you will have trouble passing any math class.

Work to Understand the Principles
You can pass a history class by simply memorizing a set of dates, names and events. You will find, however, that in order to pass a math class you will need to do more than just memorize a set of formulas. While there is certainly a fair amount of memorization of formulas in a math class you need to do more. You need to understand how to USE the formulas and that is often far different from just memorizing them.

Some formulas have restrictions on them that you need to know in order to correctly use them. For instance, in order to use the quadratic formula you must have the quadratic in standard form first. You need to remember this or you will often get the wrong answer!

Other formulas are very general and require you to identify the parts in the problem that correspond to parts in the formula. If you don’t understand how the formula works and the principle behind it, it can often be very difficult to use the formula. For example, in a calculus course it’s not terribly difficult to memorize the formula for integration by parts for integrals. However, if you don’t understand how to actually use the formula and identify the appropriate parts of the integral you will find the memorized formula worthless.

Mathematics is Cumulative:
You’ve always got to remember that mathematics courses are cumulative. Almost everything you do in a math class will depend on subjects that you’ve previously learned. This goes beyond just knowing the previous sections in your current class to needing to remember material from previous classes.

You will find a college algebra class to be very difficult without the knowledge that you learned in your high school algebra class. You can’t do a calculus class without first taking (and understanding) an Algebra and a Trigonometry class.
So, with these three main ideas in mind let’s proceed with some more specific tips to studying for a math class. Note as well that several of the tips show up in multiple sections since they are either super important tips or simply can fall under several general topics.

General Tips for Studying Mathematics

These are some general tips that where either important enough to single out or just didn’t seem to fit into any of the other sections.

• Go To Class. Remember that math is cumulative. If you don’t go to class you will miss important material that will be used in later sections and/or important announcements.
• Get to Class On Time. Sometime important announcements are only given during the first few minutes of a class.
• LISTEN During Class. In order to get something out of the class you need to listen while in class. Often this can be difficult to do but it is very important. Sometimes important ideas will not be written down on the board, but instead just spoken by the instructor. Watch for things the instructor emphasizes, even if just in words. This often means the instructor thinks it’s important. The more important that an instructor thinks a topic is, the more likely that it will show up on the exam!
• Take Good Notes. Try to write down everything that instructor puts on board. It may seem easy when watching the instructor, but it often is not so easy when it comes time for you to do it. A good set of notes will help remind you how to do these problems. For some instructors writing down everything may be difficult. In these cases you should try to write down as much as possible. Note as well that this seems to contradict the previous tip. It is often hard to both listen and take a good set of notes. This is something that one often only gains with practice. You need to be able to listen while you are writing down the important parts of the lecture.

• If you find that are having trouble both listening and taking good notes then you might consider bringing in a tape recorder and record the lecture. Note however that prior to doing this you should first speak with your instructor. There are a few instructors out there in the world that do not like to have their lectures recorded.

• Ask Questions. If you don’t understand something then ask your instructor. Chances are you are not the only one who doesn’t understand.
• Listen When Others Ask Questions. When other students ask questions make sure you listen to both the question and the answer. It may be that the student asking the question thought of something that you didn’t think of.

• Review Notes After Class. After each class you should review your notes. Note the topics that you found confusing and formulate questions that you can ask your instructor or tutor to help you understand the topic.

• Make a Set of Index Cards. Make a set of index cards with important formulas and concepts on them. You can carry these around with you to look over when you’ve got a few spare minutes. Use them to help you memorize the important formulas and concepts.

• Learn The (Proper) Notation. Instructors will assume you know it so you’ll need to and many instructors will take points off for bad notation.

• Get Into A Study Group. It is often helpful to study in groups. People often look at things differently so someone else may see how to solve a problem that can’t do or understand a topic that you find confusing.

• Note Due Dates. Write down the due dates for homework and dates for exams someplace you’ll see them so you don’t forget about them.

• Budget Adequate Time For Studying/Homework. It often takes more time studying mathematics to learn the subject than you may require in other classes. Homework will often take more time than you had originally thought it would. Keep this in mind as you budget time.

• Do Homework After Each Class. At the end of each class budget some time to look over the homework from that days lecture and attempt to do it Doing this will allow you time to really work at understanding the concepts covered that day. Do not wait until the last minute to do the homework as this often results in an incomplete homework set and/or an incomplete understanding of the concept.

• Do Homework Without Notes and Book. After the first few homework problems, put your notes and book up and try to do the remaining problems without referring to your notes and/or book. In most cases you will not have these during your exams so get used to doing problems without them.

• Do More Homework. Do not limit yourself to just the homework that your instructor assigns. The more problems that you work the better off you’ll be.
• Practice, Practice, Practice. Practice as much as possible. The only way to really learn how to do problems is work lots of them. The more you work, the better prepared you will be come exam time.
• Persevere. You will not just instantly get every topic that is covered in a math class. There will be some topics that you will have to work at before you completely understand. The only way to really grasp some topics is to go home and think about it and work some problems. You will often find that after a little work a topic that initially baffled you will all of a sudden make sense.
• Keep Old Homework and Exams. Do not throw away homework and exams once you get them back. The homework is a good source of study material for exams and both the homework and exams is a good source of study material for comprehensive final exams (if you’ve got one).
• Don’t Forget Your Textbook. If you get stuck on a topic that was discussed in class do not forget that you do have a textbook. Often the text book will contain examples not worked in class and/or a different approach to a problem.
• Seek Help If You Need It. If you are having trouble with your math class you have many options open to you and you should take advantage of them. You can go to your instructor’s office hours, go to the tutoring room or hire a tutor to get help.
• Have the Proper Attitude. Always do the best that you can. Do not do try to do just enough to get by. Doing this can lead to major problems if you aren’t careful. If you are trying to do just enough to get by then all it takes is one bad exam and you are now failing the course. You should always do the best that you can and strive for the best grade that you can possible get.
Taking Notes
Here are a couple of tips for taking notes in the class.
• Listen in Class. Do not just write down what you see on the board. No instructor is going to write down every word they say and sometimes the important ideas won’t get written down.
• Write Down Explanatory Remarks. Make sure you write down any explanatory remarks the instructor makes. These often won’t get written down by the instructor, but can tell you how to work a particular kind of problem or why the instructor used one formula/method over another for a given problem.
• Note Important Formulas/Concepts. If an instructor emphasizes a particular formula or concept then make note of it. This probably means the instructor feels that it’s important and important formulas and concepts are much more likely to show up on an exam.
• Question Your Instructor. If you are unclear on something ask questions.
• Note Topics You Don’t Understand. If you are having trouble understanding something being presented note that in the margin and at least write down the key words. Leave yourself a couple of lines so you can fill in the missing details later once you’ve gotten help to understand the concept.
• Review/Edit Your Notes. As soon you can after class go back over your notes. Look for any errors and/or omissions. Fill in any information you didn’t have time to write down in class.
• Review Regularly. At regular intervals sit down and review your notes so that you can learn and retain the information. Remember, that this information will probably be required down the road so it’s best to learn it as soon as possible.
Getting Help
Getting help when you are in trouble is one of the most important things that you can do in a math class. Here are a couple of things that you can do the get help.
• Get Help When You Need It. Do not wait until the last minute to get help. When you start running into problems it is time to get help. Remember that math is cumulative. If you don’t get help right away you will only be making it all that more difficult to understand future material.
• Ask Questions in Class. This way you get the help you need AND stay actively involved in the class.
• Visit the Instructor’s Office Hours. The instructor has office hours for a reason, so use them!
• Form a Study Group. Many people find it convenient to study in a group. Different people will see things differently and may see a way to work a problem that you don’t know how to do.
• Go to the Tutor Lab. If your school has a free math tutoring lab (many do and many don’t) then make use of it! That’s what it’s there for. Here at Lamar University the free math tutoring lab located in Lucas 209.
• Get a Private Tutor. You can always hire a private tutor for some help. In almost every school you will find people how are willing to tutor you for a fee. Most math departments keep a list of tutors or can tell you where you might be able to find one. If you are here at Lamar University you can access a list of current tutors at http://www.math.lamar.edu/turorlist.asp or stop by the Math Departments office in Lucas 200 for a list.
• Ask Good Questions. Saying “I don’t understand this section.” or “I don’t get it.” is not the best way to seek help. It just doesn’t imply what you’re having trouble with and so will probably not get your questions answered. Be specific with your questions. What exactly is it about this section don’t you understand? Many instructors won’t help you unless you do have specific questions and/or can articulate exactly what it is that you don’t understand and/or “get”. If you really don’t understand a given section that is okay, but be prepared to tell your instructor just what is causing the problems. Do you not understand the concept? Are you having trouble getting the problems started? Are you having trouble identifying that the problems use the ideas in the given section?

• Have Attempted Work With You. When you get help make sure and bring the attempts that you’ve made on the problem. This will help the person helping you to understand just where you’re having problem. Again, many instructors won’t help you unless you’ve got something to show that you’ve at least attempted a problem. If you really don’t understand how to even start the problem then bring in work for other problems in that section that you have done. This accomplishes two things. First, it shows the instructor that you have been working and second, it gives the instructor something to look at to see just where you’re having trouble understanding the problem you’re trying to work.
Homework
Note that this section contains some general tips on making the most out of your homework. The next section contains tips on actually working homework problems.
• Understand the Purpose of Homework. Instructors do not give you homework assignments to make your life miserable (well some might, but most don’t!). Homework assignments are given to help you to learn the material in the class and to develop good reasoning and problems solving skills. Mathematics is just not a subject that most people will instantly understand everything single topic after hearing the instructors lecture. Most people need to work on some problems in order to really start to understand the topic. That is the point of the homework. It gives you a set of problems that will help you to understand the topics. Remember that it almost always seems easier to watch and instructor doing problems on the board than it actually is. You won’t know if you truly understand the material and can do the problem if you don’t attempt the homework.
• Have The Actual Assignment. This may seem like a silly tip, but make sure that you accurately write down the assignment and due date. This is one of the more common mistakes that students make with homework.
• Do the Assignment Promptly. You should always do the assignment as soon after the lecture as possible while the lecture is still fresh in your mind. Do not wait until the last minute to do the whole assignment when comments made by the instructor are still fresh in your mind. Often these little asides that an instructor makes won’t seem important at the time, but when it comes time to doing the homework the reason for making the comment will become clear.
• Be Organized. When you start working on homework make sure that you’ve got all the materials that you’ll need to do the homework such as notes and textbook.
• Review. Go back over the lecture for each section and review any examples that the instructor worked to make sure that you understand the ideas from that section. Make note of any common errors that your instructor may have mentioned. Do the same with the text book. Read the section and note examples worked and common errors mentioned in the text book.

One of the biggest roadblocks in doing homework that I’ve seen in many students is that they completely ignore the notes and/or text. They look at a problem and if they can’t see how to do it they give up and go to the next problem. Often there will be a similar problem in the notes and/or text that can help you to get started!
• Read/Follow the Directions. Make sure that you read and follow all the directions for both the homework set and the individual problems.
• Be Neat. Make sure that you write neatly. This will help the instructor as he/she is grading the assignment and you when you are going over the assignment in preparation for an exam.
• Show All Work. Make sure that you show all of your work. Do not just give the answer. Many instructors will not accept homework that consists only of answers and no work. Also, do not skip large chunks of the work. Instructors aren’t mind readers and so they won’t know how you got from Step 1 to Step 3 unless you also show Step 2. This is also important if you made a mistake in Step 2. If the instructor can’t determine how you got from Step 1 to Step 3 you’re liable to lose far more points than you would have lost for the error in Step 2 had you shown it. Showing your work will also help you when you are reviewing for the exam.
• Check Your Work. Always go back over your work and make sure that you’ve not made any simple arithmetic/sign errors.
Problem Solving
In the previous section there were some general tips in regards to homework sets as a whole. Here are some tips to help you actually work the problems. Note that some of the ideas were important enough that they are actually in both sections.
• Read the Problem. Read the problem to get an idea of what you’re being asked to do. This one of the biggest sources of point loss that I’ve seen in grading. Too many students just skim the problem and assume they know what’s going on and what they are being asked to do.
• Read the Problem Again. Now that you know what you’re being asked to do, read the problem again. This time around make note of what you are given and what you need to find. Also make sure that you understand just what you’re being asked to do.
• Clearly Note What You Are Asked to Find. Clearly write down somewhere what you’re being asked to find.
• Clearly Note What You Know. Clearly write down somewhere all the information you’ve been given.
• Draw a Diagram. If appropriate draw a diagram and label what you know and what you need to find. Often diagrams will suggest the solution technique, so it’s a good idea to get into the habit of drawing them.
• Devise a Plan. Try to figure out what you’re going to need to work the problem. Identify formulas that may help you. See if there are any intermediate steps/answers that will be needed in order to arrive at the final answer.

• Work a Similar Problem. If you can’t figure out how to work the problem find a similar problem that is simpler. Work this then go back and compare what you did in the simpler problem to the problem you’re asked to do.
• Work the Plan. Once you’ve got the plan, work it out to get the answer.
• Check Your Solution. Is the answer in proper form? Does your answer make sense? If possible, plug your answer back into the original problem as a check.
• Go Back Over the Problem. Once you’re satisfied that you’ve got the correct answer go back over the problem. Identify concepts/methods/formulas that were used for the problem. Try to understand why these concepts/methods/formulas were used on this problem. Look for identifying characteristics that will help you identify this kind of problem in the future.
Studying for Exams
Here are some tips on studying for exams.
• Start on Day One. You should always be studying for the next exam. Do a little each day, or at the very least start studying 2 – 3 days before the exam. Do NOT start studying the night before the exam. Cramming, while a time honored college tradition, just doesn’t work as well as spending time each day studying, especially with a math class.
• Get a Good Nights Sleep. Get a good nights sleep the night before the exam. It is important to be well rested and mentally sharp when you take the exam.
• Make a List of Important Concepts/Formulas. Review your notes and make a concise list of important concepts and formulas. Make sure you know these formulas and more importantly how to use them!
• Rework Homework Problems. Do not just read over the homework problems. Actually rework them. Writing down the steps will help you to remember them. Make sure that you try to do the problems without looking at the solutions.
• Rework Book/Notes Examples. Cover up the solutions to book or note examples and try to rework them. When looking for problems from the book don’t forget that most books have a review section at the end of each chapter that usually contains more problems.
• Look for Identifying Characteristics in Problems. While doing your homework you knew which section it came out of. This provided some clues as to the solution process. During an exam you won’t have this to help you. So, while reviewing your homework look for identifying characteristics that will give you clues on how to identify that kind of problem.
• Take a Practice Exam. Find some problems and treat them a practice test. Give yourself a time limit and don’t use your notes or book.
Taking an Exam
Taking exams is probably one of the most important things that you’ll do in a math class and so it’s important to do the best that you can. Here are some ideas that will help you while you’re taking an exam.

• RELAX!!! This is the first step to successfully taking an exam. Unfortunately, it’s also one of the hardest things to do. The more worked up and nervous you are during the exam the more likely you are to forget something or blank out. The worst thing that you can do in an exam is panic.
• Be Smart. By this I mean be smart as you take the exam. You should go over the exam three times. First, go over the exam and work all the problems that you KNOW you can do. Second, work all the problems that you think you can do, but aren’t sure. Last, go back and work the remaining problems. In this way you will get all the points that you know you can get.
• Be Time Efficient. Watch the clock. Don’t spend a huge amount of time trying to get the points for one problem! If you spend too much time trying to get the points for one problem, you may not be able to finish the exam and lose more points than you gained by taking all that time for that one problem.
• If You’re Stuck Move On. If you find that you’re stuck on a problem, move on to a different problem and come back later to finish the problem. Don’t waste time trying to get 10 points and then not be able to finish and miss 20 points because you ran out of time. Note that is really the same as the previous tip, but it was important enough to make again and in a different way.
• Show All Your Work. Make it as easy as possible for the instructor to see how much you do know. Try to write a well-reasoned solution. If your answer is incorrect, the instructor will assign partial credit based on the work you show. Don’t leave it to the instructor to read your mind trying to figure out if you knew what you were doing or not.
• Never Leave a Problem Blank. You should never leave a problem blank. Even if you don’t know how to finish the problem, write down as much as you do know. The instructor can only give partial credit if there is something written down for the problem. Writing something down is not a guarantee of partial credit, but not writing anything down is a guarantee of no partial credit!
• Read the Problem. Make sure that you read the questions carefully and completely before you answer the question. If the instructor asks for the answer to be written in a specific form then make sure that you do that.
• Does Your Answer Make Sense? Make sure you’ve gotten an answer that makes sense.
• Recheck Your Work. If time permits go back over the exam and check every problem. This means completely rework the problem from scratch.
Learn From Your Errors
This is probably one of the more important sections here and also one of the most over looked. Learning from your mistakes can only help you.
• Review Homework. When you get your homework back review it looking for errors that you made.
• Review Exams. Do the same thing with exams.

• Understand the Error. When you find an error in your homework or exams try to understand what the error is and just what you did wrong. Look for something about the error that you can remember to help you to avoid making it again.
• Get Help. If you can find the error and/or don’t understand why it was an error then get help. Ask the instructor, your tutor, or a classmate who got the problem correct.
• Rushed Errors. If you find yourself continually making silly arithmetic or notational errors then slow down when you are working the problems. Most of these types of errors happen because students get in a hurry and don’t pay attention to what they are doing.
• Repeated Errors. If you find yourself continually making errors on one particular type of problem then you probably don’t have a really good grasp of the concept behind that type of problem. Go back and find more examples and really try to understand just what you are doing wrong or don’t understand.
• Keep a List of Errors. Put errors that you keep making in a “list of errors”. With each error write down the correct method/solution. Review the list after you complete a problem and see if you’ve made any of your “common” errors.
© Paul Dawkins 2006 11 http://tutorial.math.lamar.edu/terms.aspx

A MATHS STUDENT LOVE LETTER

2007-12-21T17:06:00.000+05:30

My Dear Love,

Yesterday, I was passing by your rectangular house in
trigonometric lane. There I saw you with your cute circular
face, conical nose and spherical eyes, standing in your
triangular garden.

Before seeing you, my heart was a null set, but when a
vector of magnitude (likeness) from your eyes at a
deviation of theta radians made a tangent to my heart,
it differentiated.

My love for you is a quadratic equation with real roots,
which only you can solve by making good binary relation
with me. The cosine of my love for you extends to infinity.
I promise that I should not resolve you into partial functions
but if I do so, you can integrate me by applying the limits
from zero to infinity.

You are as essential to me as an element to a set. The
geometry of my life revolves around your acute personality.
My love, if you do n ot meet me at parabola restaurant on
date 10 at sunset, when the sun is making an angle of
160 degrees, my heart would be like a solved polynomial
of degree 10. With love from your higher order derivatives
of maxima and minima, of an unknown function.

Yours ever loving,
Pythagoras
De-Morgan's Law,
opposite to Guass house.
Binomial Avenue, MATHSCITY,
MATHEMATICA-
PIN : αβγδ∞θ
MOBILE: 1234567890

SOME QUOTATIONS ON MATHS(COLLECTIONS)

2007-12-20T19:02:00.000+05:30

God exists since mathematics is consistent, and the Devil exists since

we cannot prove it.    -- Andre Weil

"God made the whole (natural) numbers. The others, were man-made" -Leopold Kronecker [German mathematician, 1823-1891]:
______________________________________________________________________

If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.

P. Turan, "The Work of Alfred Renyi", Matematikai Lapok 21, 1970, pp 199-210

Life is good for only two things, discovering mathematics and teaching

mathematics.

        --Siméon Poisson

Young man, in mathematics you don't understand things, you just get used to

them.

 -- John von Neumann (1903-1957) [Hungarian/US mathematician and

scientist]

Anyone who cannot cope with mathematics is not fully human.  At best he

is a tolerable subhuman who has learned to wear shoes, bathe and not

make messes in the house. -- Lazarus Long, "Time Enough for Love", (Robert

A. Heinlein (1907-19??) [US science-fiction writer] )

Mathematicians are like lovers. Grant a mathematician the least

principle, and he will draw from it a consequence which you must also

grant him, and from this consequence another.

-- Bernard Le Bovier Fontelle

Math is like love -- a simple idea but it can get complicated.

"A mathematician is a blind man in a dark room looking for a black cat

which isn't there" - Charles R. Darwin (1809-1882) [English biologist]

Copernicus, Nicholaus  (1473-1543)

Mathematics is written for mathematicians
______________________________________________________________________________________

"A person who can, within a year, solve x^2 - 92y^2 = 1 is a mathematician."

   -- Brahmagupta

Bell, Eric Temple (1883-1960)

"Obvious" is the most dangerous word in mathematics.

The traditional mathematics professor of the popular legend is

absentminded. He usually appears in public with a lost umbrella in each

hand. He prefers to face the blackboard and to turn his back to the

class. He writes a, he says b, he means c; but it should be d. Some of his

sayings are handed down from generation to generation.

"In order to solve this differential equation you look at it till a

solution occurs to you."

"This principle is so perfectly general that no particular application of

it is possible."

"Geometry is the science of correct reasoning on incorrect figures."

"My method to overcome a difficulty is to go round it."

"What is the difference between method and device? A method is a device

which you used twice."

How to Solve It. Princeton: Princeton University Press. 1945.

"No human investigation can be called real science if it cannot be demonstrated mathematically." Leonardo da Vinci
---------------------------------------------------------------------------------------------------
I was x years old in the year x^2. (when asked his age.)
More funny Augustus de Morgan quotes
____________________________________________________________________

Infinity

"Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. –- Albert Einstein

"The notion of infinity is our greatest friend; it is also the greatest enemy of our peace of mind." -- James Pierpont

"The infinite! No other question has ever moved so profoundly the spirit of man." -- David Hilbert

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited;" -- Galileo Galilei

"The infinite in mathematics is always unruly unless it is properly treated." -- Edward Kasner and James Newman

"God created infinity, and man, unable to understand infinity, had to invent finite sets." -- Gian Carlo Rota

___________________________________________________________________

Life is …

‘Life is complex: it has both real and imaginary components.”

"Life is too short for long division."

"A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction." – Tolstoy

"The arithmetic of life does not always have a logical answer."-- Inshirah Abdur-Rauf

"A few honest men are better than numbers." -- Oliver Cromwell

"I must study politics and war that my sons may have liberty to study mathematics and philosophy." -- John Adams

"Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio."-- Thomas Robert Malthus

"Life is good for only two things: discovering mathematics and teaching mathematics." -- Simeon Poisson

__________________________________________________________________

Mathematician and Politician-an anology

2007-12-14T21:36:00.000+05:30

Consider the situation we know that the sum of the angles of a traingle is 180 degree. This theorem is from Eucledean plane geometry in which sides of traingle are straight line segments.

But our earth being spherical we can not draw straight line segment any where on the surface of the earth. So there exist other geometries known as Non Euclidean Geometries to deal with such situations. Important among these goemetries are 1)Riemannian geometry 2)Bolyai-Labachovsky geometry.

In one of thes geometries the sum of the angles of traingle is less than 180 degree and in other it is greater than 180 degree . Wonder is How mathematics being an exact subject can give different results about the same property of traingle. why this happens?

But this will focus that there is nothing like exact or absolute truth in mathematics, but validity of any result depends on the postulate (i.e axioms) chosen while starting the study of branch of Mathematics in which that result lies. In other words, any result or propositoon in mathematics has to be consistent with the axioms on which it is based.

Based on this concept we have following good humourous story

There were two freinds. One was a mathematican and the other a politician. They were fast friends from their childhool and they had maintained their friendship thorughout, even though they belonged to different professions.

Once the politician told his mathematical frined: "we are birds of the same feather. We both talk nonsencse. The day before yesterday I came to meet you in your school. you were teaching in a class. I did not want to disturb you; therefore I did not call you outside. But your voice being loud, I could hear from outside what you were teaching. You were telling your students that the 'sum of the angles of a traingle is 180 degree. We did study some thing like this, but I do not remember exactly what it was. So I took it that what you were teaching in the class was correct. Yesterday also I came to meet you and that time also you were teaching. But his time I heard you telling your students that the ' sum of the angles of a traingle is less than 180 degree, and in the other class you tell that it is less than 180 degree. . Mathematics being an exact subject, only one of these two statements can and must be true. So your case is like only. We politicians also tell one thing one platform and exactly opposite thing on the other platform. So I say that we both are birds of the same feather; we both talk nonsense".

To this mathematician replied: " Yes, my dear friend, we both talk nonsense. But there is a difference. You talk inconsistent nonsense, while we talk consistent nonsense. Our statements, though they may look contradictory, have to consistent with the axioms with which we start our subject.

WHY NOBLE PRIZE IS NOT GIVEN FOR MATHEMATICS?

2007-12-08T22:12:00.000+05:30

WHY NOBEL PRIZE IS NOT FOR MATHEMATICS?

Though, Mathematics is called the “Queen of all Sciences”, ironically, there is no Nobel prize in Mathematics. The reason mentioned by Nobel was that mathematics is not a practical science. Though there is no Nobel prize, many mathematicians have won the same. The British philosopher –logician Bertrand Russell won the Nobel prize for literature in 1950.

Here I have given different opinions and views about why nobel prize is not given for mathematics?

Actually, nobel prize has as its underlying policy of rewarding such monumental works which affect the lives of people practically.
even in the field of physics or chemistry if some piece of work has got no practical significance it is not rewarded..like einstein got his prize 4 xplaining photoelectric effect n not 4 theory of relativity which was undoubtedly his greatest work.mathmeticians are also awarded nobel prize if their theory is of supreme practical significane.4 example,john nash was given the prize in economics 4 his work on the game theory
Fields Prize - The Nobel Prize of Mathematics
Fields Medal is the top international scientific prize regarded as the Nobel Prize of mathematics. Founded at the behest of Canadian mathematician John Charles Fields, the medal was first awarded in 1936 . The International Mathematical Union (IMU) awards between two and four medals every four years during the International Congress of Mathematicians (ICM). They are awarded every four years by the International Mathematical Union to one or more outstanding researchers.The Fields Medal is widely viewed as the top honor a mathematician can receive. It comes with a monetary award, which in 2006 was US$13,400. Also, the monetary award is much lower than the roughly US$1.3 million given with each Nobel prize.
Nobel prizes were created in the will of the Swedish chemist Alfred Nobel. In fact, his will speaks of prizes for those "inventions or discoveries" of greatest practical benefit to mankind. Though its true that Literature falls under a similar catagory(unlike mathematics which has no direct implication) BUT with a BIG DIFFERENCE. i.e., it can phychologically effect human behaviour and attitude towards various aspects of mankind. Ex: a prose/song witten by a poet can make someone relax! But not mathematics "to the general public". But ofcourse i do agree(sometimes) upon a noble prize in mathematics... but adding such a prize may be "against" the will of Alfred Nobel... because he was the one to frame the prize and not me or you... further, I do believe it has MAJOR "indirect applications" which effects works in Physics, chem., Arch., and other Major sciences... So, the best i think that can be done is to increase the standards(i.e. selection criteria) and the prize amount of Fields prize at par to that of Nobel prize. I hope this explains it!! what do u say...?
true story of Alfred Nobel
The story why Nobel Prize is not given for mathematics goes this way: Alfred nobel and swedish mathematician Gustav Mittag Leffler loved the same girl. Before the invention of dynamite, Nobel was not wealthy, while Leffler was a Professor of Mathematics in the Swedish University. Nobel lost his beloved to the mathematician. This bitterness remained in him and surfaced hen he made his will to use his wealth to award exellence in science and literature. He left out mathematics saying mathematics doesnot directy improve human life (but the trur reason was because he lost in love to a mathematician)
dear friends! let me make the thing more clear... i can agree that its virtually difficult for anyone to stand in this vast ocean of mathematics! but... what i mean to say precisely, is that "Nobel prize was started with the will of Alfred Nobel"... and NOT a committee/org./etc... and the rules were set by he himself... which we cannot change!!!
But if its so badly required... then a saperate award for mathematics can be started with the same name i.e. "Nobel prize for Mathematics"... BUT without the logo of Alfred Nobel in the token of honour i.e. the one present in the medal. finally, i do hope that the issue is more clear... and that the suggestions are well understood... What's more important is the appreciation of various work done by researchers and on top mathematics by itself!! thanks a lot.
sorry if i was rude or something! but i mean... anyone would know that mathematics is not all that! i mean... literature makes a person relax? isnt that a very relative and a thing that is sooo dependent on the situation?
and without mathematics there absolutely cannot be a strong base for any other accurate sciences....
acc to me.. i think the reason has to do something with nobels personal affairs.... i have read it in more than dozen sources.
and i mean seriously, the prize money does not count when u do kind of a brilliant research and when u do it for the love!

· No Nobel for Maths because...

Number of theories have been put forward to explain the omission of mathematics from Nobel's original list. The most colorful suggestion is that Nobel was miffed at mathematicians after discovering that his wife had had an affair with the Swedish mathematician Magnus Mittag-Leffler. Of all the theories, this is the easiest to dismiss, for the simple reason that Nobel never had a wife. Another oft-repeated suggestion is that Nobel hated mathematics after doing poorly in it at school. It may or may not be true that Nobel wasn't good at math, but there is no evidence to suggest that a negative high school experience in the math class led to a desire to get back at the mathematicians later in life by not giving them one of his prizes.

By far the most likely explanation, I think, is that he viewed mathematics as merely a tool used in the sciences and in engineering, not as a body of human intellectual achievement in its own right. He also did not single out biology, possibly likewise regarding it as just a tool for medicine, a not unreasonable view to have in the late 19th century.

Left to your opinion what do u say ?

MATHEMATICAL LINKS FOR OLYMPIAD

2007-11-24T19:05:00.000+05:30


	Winning Maths - the South African Mathematical Talent Search
	UCT Mathematics Competition - a schools competition, from Grade 8 to 12
	South Africa and the International Mathematical Olympiad and Pan African Mathematics Olympiad
	South African Mathematical Olympiad
	Dynamic Math Learning - by Michael de Villiers
	Rubik's Cube - interactive

MATH JOKES

2007-11-24T18:44:00.000+05:30

SALARY THEOREM
The less you know, the more you make.
Proof:

Postulate 1: Knowledge is Power.
Postulate 2: Time is Money.As every engineer knows: Power = Work / Time
And since Knowledge = Power and Time = Money
It is therefore true that Knowledge = Work / Money .
Solving for Money, we get:
Money = Work / Knowledge
Thus, as Knowledge approaches zero, Money approaches infinity, regardless of the amount of Work done.(CALCULUS, INDERTERMINATE FORMS)

YOU MIGHT BE A MATHEMATICIAN if:

You Might Be a Mathematician if...

you are fascinated by the equation e^Πi +1
you know by heart the first fifty digits of pi.

you have tried to prove Fermat's Last Theorem.

you know ten ways to prove Pythagoras' Theorem.

your telephone number is the sum of two prime numbers.

you have calculated that the World Series actually diverges.

you are sure that differential equations are a very useful tool.

you comment to your wife that her straight hair is nice and parallel.

when you say to a car dealer "I'll take the red car or the blue one", you must add "but not both of them."

MATHEMATICAL OLYMPIAD

2007-11-20T18:06:00.002+05:30

Here I am giving List of websites which is immensly helpful for preparing for mathematical olympiad.

http://www.kvntpcbadarpur.com/Olympiad.htm: which gives notes (downloadable) regarding inequalities , algebra, Number theory etc
http://imo.math.ca/: International Mathematical olympiad problems and solutions
http://cage.rug.ac.be/~hvernaev/olympiad.html: A collection of large no of olympiad problems
www.cmi.ac.in/~vipul/olymp_resources/ : This is the home page of Mr vipul naik who represented India for the International mathematical olympiad and this gives good ideas and articles related to olympiad
BEST OF LUCK

MATH JOKES

2007-11-19T22:04:00.000+05:30

HOW TO PUT AN ELEPHANT INTO A REFRIGERATOR:

Analysis:

1) Differentiate it and put into the refrig.

Then integrate it in the refrig.

2) Redefine the measure on the referigerator (or the elephant).

3) Apply the Banach-Tarsky theorem.

Number theory:

1) First factorize, second multiply.

2) Use induction. You can always squeeze a bit more in.

Algebra:

1) Step 1. Show that the parts of it can be put into the refrig.

Step 2. Show that the refrig. is closed under the addition.

2) Take the appropriate universal refrigerator and get

a surjection from refrigerator to elephant.

Topology:

1) Have it swallow the refrig. and turn inside out.

2) Make a refrig. with the Klein bottle.

3) The elephant is homeomorphic to a smaller elephant.

4) The elephant is compact, so it can be put into a finite collection

of refrigerators. That's usually good enough.

5) The property of being inside the referigerator

is hereditary. So, take the elephant's mother,

cremate it, and show that the ashes fit inside the refrigerator.

6) For those who object to method 3 because it's cruel to animals.

Put the elephant's BABY in the refrigerator.

Algebraic topology:

Replace the interior of the refrigerator by its

universal cover, R^3.

Linear algebra:

1) Put just its basis and span it in the refrig.

2) Show that 1% of the elephant will fit inside the refrigerator.

By linearity, x% will fit for any x.

Affine geometry:

There is an affine transformation putting the

elephant into the refrigerator.

Set theory:

1) It's very easy!

refrigerator = { elephant } 2) The elephant and the interior of the

refrigerator both have cardinality c.

Geometry:

Declare the following:

Axiom 1. An elephant can be put into a refrigerator.

Complex analysis:

Put the refrig. at the origin

and the elephant outside the unit circle.

Then get the image under the inversion.

Numerical analysis:

1) Put just its trunk and refer the rest to the error term.

2) Work it out using the Pentium.

Statistics:

1) bright statistician.

Put its tail as a sample and say "Done."

2) dull statistician.

Repeat the experiment pushing the elephant to the refrig.

3) Our NEW study shows that you CAN'T put the elephant

in the refrigerator.


JESUS AND DISCIPLES:
Jesus and his disciples were walking around one day, when Jesus said, "The

Kingdom of Heaven is like 3x squared plus 8x minus 9."  The disciples

looked very puzzled, and finally asked Peter, "What on earth does Jesus

mean - the Kingdom of Heaven is like 3x squared plus 8x minus 9?  Peter

said, "Don't worry.  It's just another one of his parabolas."
source: Special Category:   Afterlife

COW-CALCULUS

(__)

(oo)

/---------\/

/ | x=a(b)||

* ||------||

^^ ^^

Mathematical Cow

(developer of

cow-culus)

RANKING OF ENGINEERING COLLEGES

2007-11-17T22:38:00.000+05:30

For the benefit of students of CET aspiranst of Karnataka, Here I am giving the Ranking of Engineering colleges based on the admissions of 2007-08
Ranking of Colleges *

Colleges throughout Karnataka

1. RV College of Engineering, Bangalore
2. MS Ramiah Institute of Tech, Bangalore
3. B.M.S College of Engineering, Bull Temple Road, Bangalore
4. Shri Jayachamarajendra College of Engineering, Mysore
5. P.E.S. Institute of Technology, Bangalore
6. Bangalore Institute of Technology, Bangalore
7. RNS Institute of Technology, Bangalore
8. JSS Academy of Technical Education, Bangalore
9. Siddaganga Institute of Technology, Tumkur
10. National Institute of Engineering, Mysore
11. BVB College of Engg & Tech, Hubli
12. Sir M Visvewsvaraya Institute of Technology, Bangalore
13. Shri Dharmasthala Manjunatheswara College of Engineering, Dharwad
14. Dayanand Sagar College of Engineering, Bangalore
15. NMAM Institute of Tech, Nitte, Karkala
16. Dr. Ambedkar Institute of Technology, Bangalore
17. Gogte Inst of Tech, Belgaum
18. Vidya Vardhaka College of Engg
19. BNMIT, Bangalore
20. BMS Int of Tech, Yelahanka, Bangalore
21. Jawaharlal Nehru National College of Engg, Shimoga
22. Sapthagiri College of Engg, Bangalore
23. Bapuji Inst of Engg & Tech, Davengere
24. CMR Inst of Tech, Bangalore
25. KS Inst of Tech, Bangalore
26. Global Academy of Tech, Bangalore
27. PES College of Engg, Mandya
28. KLES College of Engg and Tech, Belgaum
29. Don Bosco Inst of Tech, Bangalore
30. Atria Institute of Technology, Bangalore
31. Vemana Institute of Technology, Bangalore
32. Sri Siddhartha Institute of Technology, Tumkur
33. Poojya Doddappa Appa College of Engineering, Gulbarga
34. Basaveshwar Engineering College, Bagalkot
35. East west insitute of Technology, Bangalore
36. HKBK College of Engineering, Bangalore
37. Golden Valley Institute of Technology, Kolar
38. GM Institute of Technology, Davangere
39. New Horizon College of Engineering, Bangalore
40. Sri Venkateshwara College of Engineering, Bangalore
41. Acharya Institute of Technology, Bangalore

* This ranking of some top engineering colleges is based on their performance and demand for seats in the previous years. This ranking is solely the view of this website, and is not endorsed by the CET Cell or any college.

2007-11-16T23:21:00.000+05:30

"A mathematician is a blind man in a dark room looking for a black cat

which isn't there" - Charles R. Darwin (1809-1882) [English biologist]

Copernicus, Nicholaus  (1473-1543)

Mathematics is written for mathematicians.

#NoSpam.Q: Math is like love -- a simple idea but it can get complicated

Life is good for only two things, discovering mathematics and teaching

mathematics.

        --Siméon Poisson

If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.

P. Turan, "The Work of Alfred Renyi", Matematikai Lapok 21, 1970, pp 199-210

KAPREKAR CONSTANT

2007-11-13T22:48:00.000+05:30

Take a 4 digit no.(not all digit same)
arrange it's digit in asscending & descending order substract the no.
keep this process on
u always have no 6174
ex:-
let no is 4000
4000-0004=3996
9963-3699=6264
6642-2466=4176
7641-1467=6174
it's always true
It's actually Kaprekar constant, named after Dattatreya Ramchandra Kaprekar. He was a gifted school teacher. It is said that he came across this number when he was sitting on one of the benches at Dombivli railway station (Dombivli is a city in Maharashtra, near Mumbai)and by looking at the number 6174 on the waggon carrying the goods that was passing by before him.

Now people are doing extensive research in this. They've found Kaprekar routine and also Kaprekar constants in other bases(6174 is the Kaprekar constant in base 10). More about the inventor can be read at:

http://drkaprekar.quickseek.com/

Timeline of mathematics

2007-11-08T14:19:00.000+05:30

Timeline of mathematics

A timeline of Pure and Applied Mathematics

2450 BC - Egypt, first systematic method for the approximative calculation of the circle on the basis of the Sacred Triangle 3-4-5,
1650 BC - Rhind Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents first known aproximate value of π at 3.16 and first attempt at squaring the circle.
530 BC - Pythagoras studies propositional geometry and vibrating lyre strings; his group discovers the irrationality of the square root of two,
370 BC - Eudoxus states the method of exhaustion for area determination,
350 BC - Aristotle discusses logical reasoning in Organon,
300 BC - Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in Catoptrics,
260 BC - Archimedes computes &pi to two decimal places using inscribed and circumscribed polygons and computes the area under a parabolic segment,
225 BC - Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola,
200 BC ?240 BC - Eratosthenes uses his sieve algorithm to isolate prime numbers and finds the number of primes is infinite,
140 BC - Hipparchus develops the bases of trigonometry,
250 - Diophantus uses symbols for unknown numbers in terms of the syncopated algebra,
250 - Diophantus writes Arithmetica the first systematic treatise on algebra,
450 - Tsu Ch'ung-Chih and Tsu Kꮧ-Chih compute π to six decimal places,
550 - Hindu mathematicians give zero a numeral representation in a positional notation system,
628 - Brahmagupta writes Brahma- sphuta- siddhanta,
750 - Al-Khawarizmi - Considered father of modern algebra. First mathematician to work on the details of 'Arithmetic and Algebra of inheritance' besides the systematisation of the theory of linear and quadratic equations.
895 - Thabit ibn Qurra - The only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations.
975 - Al-Batani - Extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tan gent, secant and their reciprocals. Derived the formula: sin α = tan α / (1+tan α) and cos α = 1 / (1 + tan α).
1020 - Abul Wafa - Gave this famous formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the parabola and the volume of the paraboloid.
1030 - Ali Ahmed Nasawi - Develops the division of days into 24 hours, hours into 60 minutes and minutes into 60 seconds.
1070 - Omar Khayyam begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations. Invented the second and third degree of quadratic equations.
1202 - Leonardo Fibonacci demonstrates the utility of Arabic numerals in his Book of the Abacus,
1424 - Ghiyath al-Kashi - computes π to sixteen decimal places using inscribed and circumscribed polygons,
1520 - Scipione dal Ferro develops a method for solving cubic equations,
1535 - Niccolo Tartaglia develops a method for solving cubic equations,
1540 - Lodovico Ferrari solves the quartic equation,
1596 - Ludolf van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons,
1614 - John Napier discusses Napierian logarithms in Mirifici Logarithmorum Canonis Descriptio,
1617 - Henry Briggs discusses decimal logarithms in Logarithmorum Chilias Prima,
1619 - Ren頄escartes discovers analytic geometry,
1629 - Pierre de Fermat develops a rudimentary differential calculus,
1634 - Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle,
1637 - Pierre de Fermat claims to have proven Fermat's last theorem in his copy of Diophantus' Arithmetica,
1654 - Blaise Pascal and Pierre de Fermat create the theory of probability,
1655 - John Wallis writes Arithmetica Infinitorum,
1658 - Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle,
1665 - Isaac Newton invents his calculus,
1668 - Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbolic segment,
1671 - James Gregory discovers the series expansion for the inverse-tangent function,
1673 - Gottfried Leibniz invents his calculus,
1675 - Isaac Newton invents an algorithm for the computation of functional roots,
1691 - Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations,
1693 - Edmund Halley prepares the first mortality tables statistically relating death rate to age,
1696 - Guillaume de L'Hੴal states his rule for the computation of certain limits,
1696 - Jakob Bernoulli and Johann Bernoulli solve brachistochrone problem, the first result in the calculus of variations,
1706 - John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places,
1712 - Brook Taylor develops Taylor series,
1722 - Abraham De Moivre states De Moivre's theorem connecting trigonometric functions and complex numbers,
1724 - Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in Annuities on Lives,
1730 - James Stirling publishes The Differential Method,
1733 - Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false,
1733 - Abraham de Moivre introduces the normal distribution to approximate the binomial distribution in probability,
1734 - Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations,
1736 - Leonhard Euler solves the problem of the Seven bridges of Ksberg, in effect creating graph theory,
1739 - Leonhard Euler solves the general homogenous linear ordinary differential equation with constant coefficients,
1742 - Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach's conjecture,
1748 - Maria Gaetana Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana,
1761 - Thomas Bayes proves Bayes' theorem,
1762 - Joseph Louis Lagrange discovers the divergence theorem,
1789 - Jurij Vega improves Machin's formula and computes π to 140 decimal places,
1794 - Jurij Vega publishes Thesaurus Logarithmorum Completus,
1796 - Carl Friedrich Gauss presents a method for constructing a heptadecagon using only a compass and straightedge and also shows that only polygons with certain numbers of sides can be constructed,
1796 - Adrien-Marie Legendre conjectures the prime number theorem,
1797 - Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms,
1799 - Carl Friedrich Gauss proves that every polynomial equation has a solution among the complex numbers,
1805 - Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set of observations,
1807 - Joseph Fourier announces his discoveries about the trigonometric decomposition of functions,
1811 - Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration,
1815 - Sim鯮-Denis Poisson carries out integrations along paths in the complex plane,
1817 - Bernard Bolzano presents the intermediate value theorem---a continuous function which is negative at one point and positive at another point must be zero for at least one point in between,
1822 - Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane,
1824 - Niels Henrik Abel partially proves that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots,
1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths -- he assumes the function being integrated has a continuous derivative,
1825 - Augustin-Louis Cauchy introduces the theory of residues in complex analysis,
1825 - Johann Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's last theorem for n = 5,
1825 - Andr魍arie Amp貥 discovers Stokes' theorem,
1828 - George Green proves Green's theorem,
1829 - Nikolai Ivanovich Lobachevsky publishes his work on hyperbolic non-Euclidean geometry,
1831 - Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green,
1832 - ɶariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory,
1832 - Peter Dirichlet proves Fermat's last theorem for n = 14,
1835 - Peter Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions,
1837 - Pierre Wantsel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge,
1841 - Karl Weierstrass discovers but does not publish the Laurent expansion theorem,
1843 - Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem,
1843 - William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative,
1847 - George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what are now called Boolean algebras,
1849 - George Gabriel Stokes shows that solitary wavess can arise from a combination of periodic waves,
1850 - Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points,
1850 - George Gabriel Stokes rediscovers and proves Stokes' theorem,
1854 - Bernhard Riemann introduces Riemannian geometry,
1854 - Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space,
1858 - August Ferdinand M⩵s invents the M⩵s strip,
1859 - Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers,
1870 - Felix Klein constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate,
1873 - Charles Hermite proves that e is transcendental,
1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,
1874 - Georg Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his method was not his famous diagonal argument, which he published three years later. (Nor did he formulate set theory at this time.)
1878 - Charles Hermite solves the general quintic equation by means of elliptic and modular functions
1882 - Carl Louis Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge,
1882 - Felix Klein invents the Klein bottle,
1895 - Diederik Korteweg and Gustav de Vries derive the KdV equation to describe the development of long solitary water waves in a canal of rectangular cross section,
1895 - Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis,
1896 - Jacques Hadamard and Charles de La Vall饭Poussin independently prove the prime number theorem,
1899 - Georg Cantor discovers a contradiction in his set theory,
1899 - David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry,
1900 - David Hilbert states his list of 23 problems which show where some further mathematical work is needed,
1901 - ɬie Cartan develops the exterior derivative,
1903 - Carle David Tolme Runge presents a fast Fourier Transform algorithm,
1903 - Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem,
1908 - Ernst Zermelo axiomizes set theory, thus avoiding Cantor's contradictions,
1908 - Josip Plemelj solves the Riemman problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky - Plemelj formulae,
1912 - Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem,
1912 - Josip Plemelj publishes simplified proof for the Fermat's last theorem for exponent n = 5,
1914 - Srinivasa Aaiyangar Ramanujan publishes Modular Equations and Approximations to π,
1919 - Viggo Brun defines Brun's constant B₂ for twin primes,
1928 - John von Neumann begins devising the principles of game theory and proves the minimax theorem,
1930 - Casimir Kuratowski shows that the three cottage problem has no solution,
1931 - Kurt G䥬 proves his incompleteness theorem which shows that every axiomatic system for mathematics is either incomplete or inconsistent,
1931 - Georges De Rham develops theorem in cohomology and characteristic classes,
1933 - Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem,
1933 - Andrey Nikolaevich Kolmogorov publishes his book Basic notions of the calculus of variations (Grundbegriffe der Wahrscheinlichkeitsrechnung) which contains an axiomatization of probability based on measure theory,
1940 - Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory,
1942 - G.C. Danielson and Cornelius Lanczos develop a Fast Fourier Transform algorithm,
1943 - Kenneth Levenberg proposes a method for nonlinear least squares fitting,
1948 - John von Neumann mathematically studies self-reproducing machines,
1949 - John von Neumann computes π to 2,037 decimal places using ENIAC,
1950 - Stanislaw Ulam and John von Neumann present cellular automata dynamical systems,
1953 - Nicholas Metropolis introduces the idea of thermodynamic simulated annealing algorithms,
1955 - Enrico Fermi, John Pasta, and Stanislaw Ulam numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior,
1960 - C. A. R. Hoare invents the quicksort algorithm,
1960 - Irving Reed and Gustave Solomon present the Reed-Solomon error-correcting code,
1961 - Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer,
1962 - Donald Marquardt proposes the Levenberg-Marquardt nonlinear least squares fitting algorithm,
1963 - Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory,
1963 - Martin Kruskal and Norman Zabusky analytically study the Fermi-Pasta-Ulam heat conduction problem in the continuum limit and find that the KdV equation governs this system,
1965 - Martin Kruskal and Norman Zabusky numerically study colliding solitary waves in plasmas and find that they do not disperse after collisions,
1965 - James Cooley and John Tukey present an influential Fast Fourier Transform algorithm,
1966 - E.J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix,
1967 - Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory,
1968 - Michael Atiyah and Isadore Singer prove the Atiyah-Singer index theorem about the index of elliptic operators,
1976 - Kenneth Appel and Wolfgang Haken use a computer to prove the Four-Color Theorem,
1983 - Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's last theorem,
1983 - the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed,
1985 - Louis de Branges de Bourcia proves the Bieberbach conjecture,
1987 - Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places,
1991 - Alain Connes and John W. Lott develop non-commutative geometry,
1994 - Andrew Wiles proves part of the Taniyama-Shimura conjecture and thereby proves Fermat's last theorem,
1998 - Thomas Hales (almost certainly) proves the Kepler conjecture,
1999 - the full Taniyama-Shimura conjecture is proved.
2000 - the Clay Mathematics Institute establishes the seven Millennium Prize Problems of unsolved important classic mathematical questions,
2002 - Manindra Agrawal, Nitin Saxena, and Neeraj Kayal of Indian Institute of Technology (IIT), Kanpur, India, present a unconditional deterministic polynomial time algorithm to determine whether a given number is prime,
2002 - Yasumasa Kanada, Y. Ushiro, Hisayasu Kuroda, Makoto Kudoh and a team of nine more compute π to 1241 billion digits using a Hitachi 64-node supercomputer,

Note

This article is based on a timeline developed by Niel Brandt (1994) who has given permission for its use in Wikipedia. (See Talk:Timeline of mathematics.)

FUNCTIONS , DOMAIN AND RANGE

2007-11-07T19:28:00.000+05:30

Value of a Function
October 6, 2006

Let f:A-->B be a function. Let y0 be an element in B. then y0 is called a value of f provided there is some element, x0 in A, such that y0 = f(x0); that is, y0 is a value of
the function f if it corresponds, with respect to the rule of f, to some x0 in the set A = Dom(f).

Example
Find the value of the following function when x=-2

Find the Domain and Range of this function.

Range and Domain of Functions

let's discuss about range and domain of a function.

Answers

Try to do more problems from your text.

TRIGONOMETRY FOR AIEEE AND CET

2007-11-05T16:00:00.000+05:30

Here is a collection of all trigonometrical results who want to prepare for All india Engineering entracne examination and Karnataka CET. I tried to give all the results. However these results does not include results on Inverse trigonometric function, General solution of trigonometric equation and Complex numbers. Use the following link to download:

http://www.scribd.com/doc/214620/TRIGNOMETRY-FOR-AIEEE-CET-BY-VASUDEVA-KH

JOKES ON MATHS

2007-11-02T22:43:00.000+05:30

IN DIFFERENT VIEWS

Several scientists were all posed the following question: "What is 2 * 2 ?"
The engineer whips out his slide rule (so it's old) and shuffles it back and forth, and finally announces "3.99".
The physicist consults his technical references, sets up the problem on his computer, and announces "it lies between 3.98 and 4.02".
The mathematician cogitates for a while, then announces: "I don't know what the answer is, but I can tell you, an answer exists!".
Philosopher smiles: "But what do you mean by 2 * 2 ?"
Logician replies: "Please define 2 * 2 more precisely."

TOPIC WISE JOKES ON MATHEMATICS

2007-10-29T21:24:00.000+05:30

Q:What is a dilemma?
A: A lemma that proves two results.

Pick-Up Lines to use on Mathematics Chicks

You fascinate me more than the Fundamental Theorem of Calculus.(CALCULUS, INTEGRATION)
Are you a differentiable function? Because I'd like to be tangent to your curves!(DIFFERENTIATION, TANGENT AND NORMAL)
You and I would add up better than a Riemann sum.
My love for you is a monotonic increasing function of time.(CALUCULUS, INCREASING FUNCTION)
Wanna come back to my room and see my copy of Euclid's "Elements"?
I am equivalent to the Empty Set when you are not with me(SET THEORY)

Mathematics Revisited

Life is complex. It has real and imaginary components. (COMPLEX NUMBERS)

2007-10-23T11:19:00.000+05:30

JOKES ON MATHEMATICS-2

2007-10-23T11:09:00.000+05:30

Biologists think they're biochemists.
Biochemists think they're chemists.
Chemists think the're physical chemists.
Physical Chemists think they're physicists.
Physicists think they're God.
God thinks he is a mathematician

.........................................................................................................................

SHOPPING MATH
A man will pay $2 for a $1 item he needs.
A woman will pay $1 for a $2 item that she doesn't need.

.........................................................................................................................

My geometry teacher was sometimes acute, and sometimes obtuse, but always, he was right.

.........................................................................................................................

GENERAL EQUATIONS & STATISTICS
A woman worries about the future until she gets a husband.
A man never worries about the future until he gets a wife.
A successful man is one who makes more money than his wife can spend.
A successful woman is one who can find such a man.
-----------------------------------------------------------------
Mathematics is made up of 50 percent formulas, 50 percent proofs, and 50 percent imagination.

A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying "I differentiate you!"

One day he met a new patient; and true to form he stared at him and said "I differentiate you!", but for once, his victim's expression didn't change.

The new patient calmly looked up and said, "You can differentiate me all you like: I'm e to the x." (CALCULUS)

The functions are sitting in a bar, chatting (how fast they go to zero at infinity etc.). Suddenly, one cries "Beware! Derivation is coming!"
All immediately hide themselves under the tables, only the exponential sits calmly on the chair.

Approximately ten excuses for not doing homework:

I accidentally divided by zero and my paper burst into flames.
I could only get arbitrarily close to my textbook. I couldn't actually reach it.
I have the proof, but there isn't room to write it in this margin.
I have a solar powered calculator and it was cloudy.
I locked the paper in my trunk but a four-dimensional dog got in and ate it.
I couldn't figure out whether I am the square of negative one or I am the square root of negative one.

-----------------------------------------------------------------

JOKES ON MATHS

2007-10-18T22:50:00.000+05:30

The Evolution of Math Teaching

1960s: A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price. What is his profit?
1970s: A farmer sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price, that is, $8. What is his profit?
1970s (new math): A farmer exchanges a set P of potatoes with set M of money. The cardinality of the set M is equal to 10, and each element of M is worth $1. Draw ten big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and give the answer to the question: What is the cardinality of the set of profits?
1980s: A farmer sells a bag of potatoes for $10. His production costs are $8, and his profit is $2. Underline the word "potatoes" and discuss with your classmates.
1990s: A farmer sells a bag of potatoes for $10. His or her production costs are 0.80 of his or her revenue. On your calculator, graph revenue vs. costs. Run the POTATO program to determine the profit. Discuss the result with students in your group. Write a brief essay that analyzes this example in the real world of economics.

Salary Theorem

The less you know, the more you make.

Proof:

WHAT IS MATHEMATICS?

2007-10-09T08:15:00.001+05:30

MATHEMATICS

Mathematics is often defined as the study of quantity, magnitude, and relations of numbers or symbols. It embraces the subjects of arithmetic, geometry, algebra, calculus, probability, statistics, and many other special areas of research.

There are two major divisions of mathematics: pure and applied. Pure mathematics investigates the subject solely for its theoretical interest. Applied mathematics develops tools and techniques for solving specific problems of business and engineering or for highly theoretical applications in the sciences.

Mathematics is pervasive throughout modern life. Baking acake or building a house involves the use of numbers, geometry, measures, and space. The design of precision instruments, the development of new technologies, and advanced computers all use more technical mathematics.

Mathematics is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter.

HISTORY

Mathematics first arose from the practical need to measure time and to count. Thus, the history of mathematics begins with the origins of numbers and recognition of the dimensions and properties of space and time. The earliest evidence of primitive forms of counting occurs in notched bones and scored pieces of wood and stone. Early uses of geometry are revealed in patterns found on ancient cave walls and pottery.

Ancient Periods

As civilizations arose in Asia and the Near East, the field of mathematics evolved. Both sophisticated number systems and basic knowledge of arithmetic, geometry, and algebra began to develop.

HISTORY

Ancient Periods

Egypt and Mesopotamia.

The earliest continuous records of mathematical activity that have survived in written form are from the 2nd millennium BC. The Egyptian pyramids reveal evidence of a fundamental knowledge of surveying and geometry as early as 2900 BC. Written testimony of what the Egyptians knew, however, is known from documents drawn up about 1,000 years later.

Two of the best-known sources for our current knowledge of ancient Egyptian mathematics are the Rhind papyrus and the Moscow papyrus. These present many different kinds of practical mathematical problems, including applications to surveying, salary distributions, calculations of the areas of simple geometric surfaces and volumes such as the truncated pyramid, and simple solutions for first- and second-degree equations.

Egyptian arithmetic, based on counting in groups of ten, was relatively simple. Base-10 systems, the most widespread throughout the world, probably arose for biological reasons. The fingers of both hands facilitated natural counting in groups of ten. Numbers are sometimes called digits from the Latin word for finger. In the Egyptians' base-10 arithmetic, hieroglyphs stood for individual units and groups of tens, hundreds, and thousands. Higher powers of ten made it possible to count numbers into the millions. Unlike our familiar number system, which is both decimal and positional (23 is not the same as 32), the Egyptians' arithmetic was not positional but additive.

Unlike the Egyptians, the Babylonians of ancient Mesopotamia developed flexible techniques for dealing with fractions. They also succeeded in developing a more sophisticated base-10 arithmetic that was positional, and they kept mathematical records on clay tablets. The most remarkable feature of Babylonian arithmetic was its use of a sexagesimal (base 60) place-valued system in addition to a decimal system. Thus the Babylonians counted in groups of sixty as well as ten. Babylonian mathematics is still used to tell time—an hour consists of 60 minutes, and each minute is divided into 60 seconds—and circles are measured in divisions of 360 degrees.

The Babylonians apparently adopted their base-60 number system for economic reasons. Their principal units of weight and money were the mina, consisting of 60 shekels, and the talent, consisting of 60 mina. This sexagesimal arithmetic was used in commerce and astronomy. Surviving tablets also show the Babylonians' facility in computing compound interest, squares, and square roots.

Because their base-60 system was especially flexible for computation and handling fractions, the Babylonians wereparticularly strong in algebra and number theory. Tablets survive giving solutions to first-, second-, and some third-degree equations. Despite rudimentary knowledge of geometry, the Babylonians knew many cases of the Pythagorean theorem for right triangles. They also knew accurate area formulas for triangles and trapezoids. Since they used a crude approximation of three for the value of pi, they achieved only rough estimates for the areas of circles.

Greece and Rome.

The Greeks were the first to develop a truly mathematical spirit. They were interested not only in the applications of mathematics but in its philosophical significance, which was especially appreciated by Plato (see Plato).

The Greeks developed the idea of using mathematical formulas to prove the validity of a proposition. Some Greeks, like Aristotle, engaged in the theoretical study of logic, the analysis of correct reasoning (see Aristotle). No previous mathematics had dealt with abstract entities or the idea of a mathematical proof.

Pythagoras provided one of the first proofs in mathematicsand discovered incommensurable magnitudes, or irrational numbers (see Pythagoras). The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks: since the lengths of diagonals of squares could not be expressed by rational numbers of the form a/b, the Greek number system was inadequate for describing them. Due to the incompleteness of their number system, the Greeks developed geometry at the expense of algebra. The only systematic contribution to algebra was made much later inantiquity by Diophantus. Called the father of algebra, he devised symbols to represent operations, unknown quantities, and frequently occurring constants.

Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However, the mathematics of Euclid, Apollonius of Perga, and Archimedes—the three greatest mathematicians of antiquity—remains as valid today as it was more than 2,000 years ago (see Apollonius of Perga; Archimedes; Euclid). Euclid's ‘Elements of Geometry' used logic and deductive reasoning to set up axioms, postulates, and a collection oftheorems related to plane and solid geometry, as well as atheory of proportions used to resolve the difficulty of irrational numbers. Despite its flaws, the ‘Elements' remains a historic example of how to establish universally agreed-upon knowledge by following a rigorous course of deductive logic. Apollonius, best known for his work on conic sections, coined the terms parabola, hyperbola, and ellipse. Another great figure was Ptolemy, who contributed to the development of trigonometry and mathematical astronomy.

Roman mathematicians, in contrast to the Greeks, are renowned for being very practical. The Roman mind did not favor the abstract side of mathematics, which had so delighted the Greeks. The Romans cared instead for the usefulness of mathematics in measuring and counting. As the fortunes of the Roman Empire declined, a rising interest in mathematics developed elsewhere, in India and among Arab scholars.

HISTORY

The Middle Ages

Indian mathematicians were especially skilled in arithmetic, methods of calculation, algebra, and trigonometry. Aryabhata calculated pi to a very accurate value of 3.1416, and Brahmagupta and Bhaskara II advanced the study of indeterminate equations. Because Indian mathematicians were not concerned with such theoretical problems as irrational numbers, they were ableto make great strides in algebra. Their decimal place-valued number system, including zero, was especially suited for easy calculation. Indian mathematicians, however, lacked interest in a sense of proof. Most of their results were presented simply as useful techniques for given situations, especially in astronomical or astrological computations.

One of the greatest scientific minds of Islam was al-Khwarizmi, who introduced the name (al-jabr) that became known as algebra. Consequently, the numbers familiar to most people are still referred to as Arabic numerals. Arab mathematicians also translated and commented on Ptolemy's astronomy before it was broughtto the attention of Europeans. Islamic scholars not only translated the works of Euclid, Archimedes, Apollonius, and Ptolemy into Arabic but advanced beyond what the Greek mathematicians had done to provide new results of their own.

By the end of the 8th century the influence of Islam had extended as far west as Spain. It was there, primarily, that Arabic, Jewish, and Western scholars eventually translated Greek and Islamic manuscripts into Latin. By the 13th century, original mathematical work by European authors had begun to appear.

Renaissance Period

Most of the early mathematical activity of the Renaissance was centered in Italy, where the mathematician Luca Pacioli wrote a standard text on arithmetic, algebra, and geometry that served to introduce the subject to students for generations. The solution of the cubic equation instigated great rivalries and priority claims between Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano. Among the advances inalgebra made during the 16th century, the use of letters of the alphabet to denote constants, variables, and unknowns in equations is notable. This symbolic algebra later proved to be the key to advances in geometry, algebra, and the infinitesimal calculus.

HISTORY

17th Century

Mathematics received considerable stimulus in the 17th century from astronomical problems. The astronomer Johannes Kepler, for example, who discovered the elliptical shape of the planetary orbits, was especially interested in the problem of determining areas bounded by curved figures (see Kepler). Kepler and other mathematicians used infinitesimal methods of one sort oranother to find a general solution for the problem of areas. In connection with such questions, the French mathematician Pierre de Fermat investigated properties ofmaxima and minima. He also discovered a method of determining tangents to curves, a problem closely related to the almost simultaneous development of the differentialand integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz later in the century (see Fermat; Leibniz; Newton).

Of equal importance to the invention of the calculus was the independent discovery of analytic geometry by Fermat and René Descartes (see Descartes). Of the two, Descartes used a better notation and devised superior techniques. Above all, he showed how the solution of simultaneous equations was facilitated through the application of analytic geometry. Many geometric problems could be translated directly into equivalent algebraic terms for solution.

Developed in the 17th century, projective geometry involves, in part, the analysis of conic sections in terms of their projections. Its value was not fully appreciated until the 19th century. The study of probability as related to games of chance had also begun.

The greatest achievement of the century was the discoveryof methods that applied mathematics to the study of motion. An example is Galileo's analysis of the parabolic path of projectiles, published in 1638. At the same time, the Dutch mathematician Christiaan Huygens was publishing works on the analysis of conic sections and special curves. He also presented theorems related to the paths of quickest descent of falling objects (see Huygens).

The unsurpassed master of the application of mathematics to problems of physics was Isaac Newton, who used analytic geometry, infinite series, and calculus to make numerous mathematical discoveries. Newton also developed his method of fluxions and fluents—the differential and integral calculus. He showed that the two methods—derivatives and integrals—were inversely related to one another (see Newton). Newton and Leibniz were studying similar problems of physics and mathematics at the same time. Having made his own discovery of the calculus in 1674, Leibniz published a rather obscure version of his methods in 1684, some years before Newton published a full version of his own methods. The sequence of mathematical developments that flows out of the discovery of the calculus is called analysis.

Although the new calculus was an immediate success, its methods were sharply criticized because infinitesimals were sometimes treated as if they were finite and, at other times, as if they were zero. Doubts about the foundations of the calculus were unresolved until the 19th century.

18th Century

The discovery of analytic geometry and invention of the calculus made possible the application of mathematics toa wide range of problems in the 18th century. The Bernoullis, a Swiss family of mathematicians, were pioneers in the application of the calculus to physics. However, they were not the only ones to advance the calculus in the 18th century. Mathematicians in France andEngland also tried to extend the range of the work of Newton and Leibniz.

The greatest development of mathematics in the 18th century took place on the Continent, where monarchs suchas Louis XIV, Frederick the Great, and the Empress Catherine the Great of Russia provided generous support for science, including mathematics. The most prolific 18th-century mathematician was Leonhard Euler of Switzerland (see Euler). He published hundreds of research papers, and his major books dealt with both the differential and integral infinitesimal calculus as well as with algebra, geometry, mechanics, and the calculus of variations.

Joseph-Louis Lagrange contributed to mechanics, foundations of the calculus, the calculus of variations, probability theory, and the theories of numbers and equations (see Lagrange). While analysis was being developed by some French mathematicians, others were turning to geometry and probability theory. The French astronomer Pierre-Simon Laplace succeeded in applying probability theory and analysis to the Newtonian theory of celestial mechanics. He was thereby able to establish the dynamic stability of the solar system (see Laplace).

19th Century

The 19th century witnessed tremendous change in mathematicswith increased specialization and new theories of algebra and number theory. The entire scope of mathematics was enriched by the discovery of controversial areas of study such as non-Euclidean geometries and transfinite set theory. Non-Euclidean geometries, in showing that consistent geometries could be developed for which Euclid's parallel postulate did not hold, raised significant questions pertaining to the foundation of mathematics.

In Germany, Carl Friedrich Gauss discovered the law of quadratic reciprocity, proved the fundamental theorem of algebra, and developed the theory of complex numbers. The Norwegian mathematician Niels Henrik Abel also made great strides during the 19th century, particularly with his theory of integrals of algebraic functions and a theorem that led to the Abelian functions, later advanced by Karl Gustav Jacobi. (See also Abel; Gauss.)

The German mathematician Karl Weierstrass brought new levels of rigor to analysis by reducing its elements to arithmetic principles and by using power series as a foundation for the theory of complex functions. August Möbius, also from Germany, worked in the area of analytic geometry and was a pioneer in topology. He discovered the Möbius strip, a topological space obtained by twisting one end of a rectangular strip and pasting it to the other.

Mathematicians in England slowly began to take an interest in advances made on the Continent during the previous century. The Analytic Society was formed in 1812 to promote the new notation and ideas of the calculus commonly used by the French.A form of noncommutative algebra called quaternions was discovered by William Rowan Hamilton, and other mathematical forms were applied to the theory of electromagnetism (see Hamilton, William Rowan).

In the United States indigenous groups of mathematicians were beginning to form, particularly in the areas of linear associative algebra and logic. In France mathematicians made significant contributions to work in geometry and analysis, especially analysis of elliptic functions. Other advances were made in complex analysis, modular functions, number theory, and invariant theory. Augustin-Louis Cauchy advanced nearly every branch of mathematics, but especially real and complex analysis. Henri Poincaré made significant contributions to mathematical physics, automorphic functions, differential equations, topology, probability theory, and the foundations of mathematics (see Poincaré). Italian mathematics in the 19th century tended to stress geometry and analysis.

Two related areas of mathematics established in the 19th century proved to be of major significance in the 20th century: set theory and mathematical logic. These were closely related to questions concerning the foundations of mathematics and the continuum of real numbers as investigated by Richard Dedekind and Georg Cantor (see Cantor). It was Cantor who created set theory and the theory of transfinite numbers.

Modern Times

Twentieth-century mathematics is highly specialized and abstract. The advance of set theory and discoveries involving infinite sets, transfinite numbers, and purely logical paradoxes caused much concern as to the foundations of mathematics. In addition to purely theoretical developments, devices such as high-speed computers influenced both the content and the teaching of mathematics. Among the areas of mathematical research that were developed in the 20th century are abstract algebra, non-Euclidean geometry, abstract analysis, mathematical logic, and the foundations of mathematics.

Modern abstract algebra includes the study of groups, rings, algebras, lattices, and a host of other subjects developed from a formal, abstract point of view. This approach formed the cornerstone of the work of a group of mathematicians called Bourbaki. Bourbaki uses abstract algebra in an axiomatic framework to develop virtually all branches of higher mathematics, including set theory, algebra, and general topology.

The significance of non-Euclidean geometry was realized early in the 20th century when the geometry was applied in mathematical physics. It has come to play an essential role in the theory of relativity and has also raised controversial philosophical questions about the nature of mathematics and itsfoundations.

Another area of mathematics, abstract analysis, has produced theories of the derivatives and integrals in abstract and infinite-dimensional spaces. There are many areas of special interest in the field of abstract analysis, including functional analysis, harmonic analysis, families of functions, integral equations, divergent and asymptotic series, summability, and the study of functions of a complex variable. In recent years, analysis has advanced with the introduction of nonstandard analysis. By developing infinitesimals this theory provides an alternative to the traditional approach of using limits in the calculus.

The most notable development in the area of logic began in the 20th century with the work of two English logicians and philosophers, Bertrand Russell and Alfred North Whitehead. Theobject of their three-volume publication, ‘Principia Mathematica' (1910–13), was to show that mathematics can be deduced from a very small number of logical principles. In the 1930s questions about the logical consistency and completeness of axiomatic systems helped to spark interest in mathematical logic and concern for the foundations of mathematics. Since the 1940s mathematical logic has become increasingly specialized.

The foundations of mathematics have many “schools.” At the beginning of the 20th century, David Hilbert was determined to preserve the powerful methods of transfinite set theory and the use of the infinite in mathematics, despite apparent paradoxes and numerous objections (see Hilbert, David). He believed it was possible to find finite means of establishing the truth of mathematical propositions, even when the infinite was involved. To this end Hilbert devoted considerable effort to developing a metamathematical theory of proofs. His program was virtually abandoned in the 1930s when Kurt Gödel demonstrated that for any general axiomatic system there are always theorems that cannot be proved or disproved (see Gödel, Kurt).

Hilbert's followers, known as formalists, view mathematics in terms of abstract structures. The axioms are developed as arbitrary rules. When applied to the unspecified elements of the theory, they can be used to establish the validity of theorems. Mathematical “truth” is thus reduced to the question of logical self-consistency. Those opposed to the formalist view, called intuitionists, believe that the basic truths of mathematics presentthemselves as fundamental intuitions of thought. The oldest philosophy of mathematics is usually ascribed to Plato. Platonism asserts the existence of eternal truths, independent ofthe human mind. In this philosophy the truths of mathematics arise from an abstract, ideal reality.

SUBDIVISIONS OF MATHEMATICS

Throughout history mathematics has become increasingly complex and diversified. At the same time, however, it has become increasingly general and abstract. Among the major subdivisions of modern mathematics are the following:

Arithmetic

Arithmetic comes from the word arithmos, meaning “number” in Greek. It is the study of the nature and properties of numbers. It includes study of the algorithms of calculation with numbers, namely the basic operations of addition, subtraction, multiplication, and division, as well as the taking of powers and roots. Arithmetic is often applied in the calculation of fractions, ratios, percentages, and proportions.

Algebra

Algebra has often been described as “arithmetic with letters.” Unlike arithmetic, which deals with specific numbers, algebra introduces variables that greatly extend the generality and scope of arithmetic. The algebra taught in high schools involves techniques for solving relatively simple equations.

Modern algebra, or abstract algebra, is a more general branch of mathematics that analyzes algebraic axioms and operations with arbitrary sets of symbols. Special areas of abstract algebra include the study of groups, rings, fields, the algebra of matrices, and a large variety of nonassociative and noncommutative algebras. Special algebras of sets and vectors and Boolean algebras arise in the study of logic (see Boole). Algebra is used in the calculation of compound interest, in the solution of distance-rate-time problems, or in any situation in the sciences where the determination of unknown quantities from a body of known data is required.

Geometry

The word geometry is derived from the Greek meaning “earth measurement.” Although geometry originated for practical purposes in ancient Egypt and Babylonia, the Greeks investigated it in a more systematic and general way.

In the 19th century, Euclidean geometry's status as the primary geometry was challenged by the discovery of non-Euclidean geometries. These inspired a new approach to the subject by presenting theorems in terms of axioms applied to properties assigned to undefined elements called points and lines. This led to many new geometries, including elliptical, hyperbolic, and parabolic geometries. Modern abstract geometry deals with very general questions of space, shape, size, and other properties of figures. Projective geometry, for example, is an abstract geometry concerned with the geometric properties that remain invariant under the projection of figures onto other figures, as in the case of mathematical perspective.

A very useful approach to geometry is found in topology, the studyof the properties of a geometric figure that remain the same when a figure is subjected to continuous transformation without loss of identity of any of its parts. Differential geometry is the studyof geometry in terms of infinitesimals.

Analytic Geometry and Trigonometry

Analytic geometry combines the generality of algebra with the precision of geometry. It is sometimes called Cartesian geometry, after Descartes, who was the first to exploit the methods of algebra in geometry. Analytic geometry addresses geometric problems from an algebraic point of view by associating any curve with variables by means of a coordinate system. For example, in a two-dimensional coordinate system, any point on a curve can be associated with a pair of points (a,b). General properties of such curves can then be studied in terms of their algebraic properties.

Trigonometry is the study of triangles, angles, and their relations.It also involves the study of trigonometric functions. There are six trigonometric ratios associated with an angle: sine, cosine, tangent, cotangent, secant, and cosecant. These are especially useful in determining unknown angles or the sides of triangles based upon known trigonometric ratios. In antiquity, trigonometrywas used with considerable success by surveyors and astronomers.

Calculus

The calculus discovered in the 17th century by Newton and Leibniz used infinitesimal quantities to determine tangents to curves and to facilitate calculation of lengths and areas of curved figures. These operations were found to be inversely related. Newton called them “fluxions” and “fluents,” corresponding to what are now termed derivatives and integrals. Leibniz called them “differences” and “sums.”

In the 19th century, in response to questions about its rigorous foundations, the calculus was developed in terms of a theory of limits. Analysis—differential and integral calculus—was subsequently approached even more rigorously by those who sought to establish its results by strictly arithmetic means. This required an exact definition of the continuity of the real numbers. Others extended the power of analysis with very general theories of measure.

Analysis gives primary emphasis to functions, convergence of sequences, series, continuity, differentiability, and questions about the completeness of the real numbers. Introductory courses in calculus generally include study of logarithms, exponential functions, trigonometric functions, and transcendental functions.

Complex Analysis

Complex analysis extends the methods of analysis from real to complex variables. Complex numbers first arose to permit general solutions to algebraic equations. They take the form a + bi, where a and b are real numbers. The variable a is called the real part of the number; b, the imaginary part of the number; and i represents the complex, or “imaginary,” number signified by the square root of –1. Because complex numbers have two independent components, a and b, they are especially useful in applications whenever two variables must be treated simultaneously. For example, complex analysis has proven particularly valuable in applications to fluid dynamics, where bothpressure and velocity vary from point to point. Complex numbers were made more acceptable to many in the 19th century when they were given a geometric interpretation.

Number Theory

It has been said that any unsolved mathematical problem that is over a century old and is still considered interesting belongs to number theory. This branch of mathematics involves the study of the properties of numbers and the structure of different number systems. It is concerned with integers, or whole numbers. Many problems in number theory deal with prime numbers. These are integers larger than 1 that have only themselves and 1 as factors.

Questions about highest common factors, least common multiples, decompositions into primes, and the representation of natural numbers in certain forms as well as their divisibility are all the province of number theory. Computers have recently been applied to the solution of certain number-theory problems.

Probability Theory and Statistics

The branch of mathematics concerned with the analysis of random phenomena is called probability theory. The entire set of possible outcomes of a random event is called the sample space. Each outcome in this space is assigned a probability, a number indicating the likelihood that the particular event will arise in a single instance. An example of a random experiment isthe tossing of a coin. The sample space consists of the two outcomes, heads or tails, and the probability assigned to each isone half.

Statistics applies probability theory to real cases and involves theanalysis of empirical data. The word statistics reflects the original application of mathematical methods to data collected for purposes of the state. Such studies led to general techniquesfor analyzing data and computing various values, drawing correlations, using methods of sampling, counting, estimating, and ranking data according to certain criteria.

Set Theory

Created in the 19th century by the German mathematician Georg Cantor, set theory was originally meant to provide techniques for the mathematical analysis of the infinite. Set theory deals with theproperties of well-defined collections of objects. Sets may be finite or infinite. A finite set has a definite number of members; such a set might consist of all the integers from 1 to 1,000. An infinite set has an endless number of members. For example, allof the positive integers compose an infinite set.

Cantor developed a theory of infinite numbers and transfinite arithmetic to go along with them. His ‘Continuum Hypothesis' conjectures that the set of all real numbers is the second smallest infinite set. The smallest infinite set is composed of the integers or any set equivalent to it.

Early in the 20th century certain contradictions of set theory concerning infinite sets, transfinite numbers, and purely logical paradoxes brought about attempts to axiomatize set theory in hopes of eliminating such difficulties. When Kurt Gödel showed that, for any axiomatic system, propositions could be devised thatwere neither true nor false, it seemed that the traditional certainty of mathematics had been suddenly lost.

In the 1960s Paul Cohen succeeded in showing the independence of the ‘Continuum Hypothesis', namely that it could be neither proved nor disproved within a given axiomatization of set theory. This meant that it was possible to contemplate non-Cantorian set theories in which the ‘Continuum Hypothesis' might be negated, much as non-Euclidean geometries treat geometry without assuming the necessary validity of Euclid's parallel postulate.

Logic

Logic is the study of the way in which valid conclusions may be drawn from given premises. It was first treated systematically by Aristotle and later developed in terms of an algebra of logic. Symbolic logic arose from traditional logic by using symbols to stand for propositions and relations between them. Modern logicians use algebraic and formal methods to study the relations between logical propositions. This has led to model theory and model logic.

EUCLID

It has been said that, next to the Bible, the ‘Elements' of Euclid is the most-translated, -published, and -studied book in the Western world. Of the author himself almost nothing is known. It is recorded that he founded and taughtat a school of mathematics in Alexandria, Egypt, during thereign of Ptolemy I Soter, who ruled from 323 to about 283 BC. It is assumed from his books that he was not a first-class mathematician, but he was a first-rate teacher of geometry and arithmetic. The ‘Elements' remained unchallenged for more than 2,000 years. Not until the mid-19th century was a non-Euclidean geometry devised. (See also Geometry).

To compile his ‘Elements' Euclid relied on the work of several predecessors, so the book is of uneven quality. Once it was published, the ‘Elements' superseded all previous mathematical treatises and became the standard text. During the Middle Ages three Arabic translations were made of the book, and it was through these that it became known in Europe. The English traveler and philosopher Adelard of Bath went to Spain disguised as a Muslim student and obtained an Arabic copy, from which he made a Latin translation in 1120. The first Latin translation of the Greek without an Arabic intermediary was made by Bartolomeo Zamberti and published in Venice in 1505. There have been several more recent translations into other languages. Among Euclid's other works on geometry were ‘Data' and ‘On Divisions'. He also wrote ‘Optics' and ‘The Elements of Music'. Some of his writings have been lost, while other works are wrongly credited to him.

ARYABHATA 1

(476–550?), Indian astronomer and mathematician. Aryabhata I was the earliest Hindu mathematician whose work and history are available to modern scholars, and he was one of the first to use algebra. In 499 he finished the Aryabhatiya, summarizing mathematics as known in his time. Most of this work deals with astronomy and spherical trigonometry; the remainder consists of 33 rules of arithmetic, algebra, and plane trigonometry. He also created a table of sines and worked with indeterminate equations.

SREENIVASA RAMANUJAN

(1887–1920). The Indian mathematician Srinivasa Ramanujan made profound contributions to the theory of numbers. He was the first Indian to be elected to the Royal Society of London, and when he died he was widely recognized by mathematicians as a phenomenal genius.

Ramanujan was born on Dec. 22, 1887, in Erode, India. When he was 15 years old, he began tutoring himself froman old mathematics manual and soon began developing his own theorems and ideas.

In 1911 he published the first of his papers in the Journal of the Indian Mathematical Society. His genius gained recognition, leading to a special scholarship from the University of Madras and a grant from Trinity College of Cambridge University. Ramanujan traveled to England in 1914, where the British mathematician Godfrey H. Hardy tutored him privately and collaborated with him in some research. His papers were published in English and European journals.

Ramanujan's knowledge of mathematics (most of which he had worked out for himself) was startling. Although almost completely ignorant of what had been developed, his mastery of certain areas of mathematics was unequaled by any living mathematician. He had only the vaguest idea, however, of what constitutes a mathematicalproof. Some of his theorems on the theory of prime numbers, though brilliant, were completely wrong. In 1917 Ramanujan contracted tuberculosis. He returned to India in 1919 and died on April 26, 1920, in Kumbakonam, India.

BHASKARA II

(1114–85?), Indian mathematician. Bhaskara II was born in 1114 in Biddur, India. He wrote the first work with a full and systematic use of the decimal number system . Bhaskara was the lineal successor of Brahmagupta (598–665?) as head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He anticipated modernsign conventions and found solutions for quadratic equations and a good value for pi. (See also Mathematics.)

PYTHOGORUS

(580? BC–500? BC). The man who played a crucial role in formulating principles that influenced Plato and Aristotle was the Greek philosopher and mathematician Pythagoras. He founded the Pythagorean brotherhood, a group of his followers whose beliefs and ideas were rediscovered during the Renaissance and contributed to the development of mathematics and Western rational philosophy.

Pythagoras was born in about 580 BC on the island of Samos, in the Aegean Sea. It is said he spent his early years traveling widely in search of wisdom. He settled in Crotona, a Greek colony in southern Italy, about 530 BC. A brotherhood of disciples soon gathered around him, inspired by his teachings. The group was strongly religious and devoted to reformation of political, moral, and social life. The order was influential in the region, but eventually its involvement in politics resulted in suppression of the brotherhood. Pythagoras was forced toretire and leave the area. He went to Metapontum, a Greek city in southern Italy. He died there in about 500 BC.

Because none of the writings of Pythagoras have survived, it is difficult to distinguish his teachings from those of his disciples. Among the basic tenets of the Pythagoreans arethe beliefs that reality, at its deepest level, is mathematicalin nature; that philosophy can be used for spiritual purification; that the soul can rise to union with the divine; and that certain symbols have a mystical significance. Pythagoras is generally credited with the theory of the functional significance of numbers in the objective world and in music. His followers are credited with the development of the Pythagorean theorem in geometry andthe application of number relationships to music theory, acoustics, and astronomy.

RENE DESCARTES

(1596–1650). Both modern philosophy and modern mathematics began with the work of René Descartes. His analytic method of thinking focused attentionon the problem of how we know, which has occupied philosophers ever since. His invention of coordinate geometry prepared the way for advances in mathematics. Descartes offered one of the first modern theories to account for the origin of the solar system of the Earth.

René Descartes was born on March 31, 1596, at La Haye in the Touraine region of France. At the renowned Jesuit school of La Flèche, René was taught philosophy, the humanities, science, and mathematics. After getting a law degree at the University of Poitiers in 1616, he served as a volunteer in Dutch and Bavarian armies to broaden his experience. He resumed his study of mathematics and science when his duties permitted. Dissatisfied with the haphazard methods of science then in use, he began to doubt all but mathematical knowledge.

In 1619 Descartes arrived at the conclusion that the universe has a mathematically logical structure and that a single method of reasoning could apply to all natural sciences, providing a unified body of knowledge. He believed he had discovered such a method by breaking a problem down into parts, accepting as true only clear, distinct ideas that could not be doubted, and systematically deducing one conclusion from another.

Descartes soon gave up army life. Living on private means, he spent several years traveling and applying his analytical system to mathematics and science. Finding, however, that the sciences rested on disputed philosophical ideas, he determined to discover a first principle, which could not be doubted, on which to build knowledge. Retiring to seclusion in Holland in 1629, he methodically doubted all accepted traditions and evidenceabout the universe and mankind. He could not doubt the statement “I think, therefore I am,” and thus his first principle was established.

Descartes's major writings on methodology and philosophy were his ‘Discourse on Method' (published in 1637) and ‘Meditations' (1641). His application of algebra to geometry appeared in his ‘Geometry' (1637). He also published works on his studies in natural science.

Descartes's work brought him both fame and controversy. In 1649 he was invited to teach philosophy to the queen of Sweden. Unused to the climate, he became ill and died in Stockholm on Feb. 11, 1650.

LEONOHARD EULER

(1707–83). The Swiss mathematician and physicist Leonhard Euler not only made important contributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in observa- tional astronomy. A founder of pure mathematics, he also demonstrated useful applications of mathematics in technology and public affairs.

Euler was born in Basel, Switzerland, on April 15, 1707. In 1727 he became, upon the invitation of Catherine I of Russia, an associate of the Academy of Sciences at St. Petersburg. In 1730 he became professor of physics. The author of innumerable papers, Euler overtaxed himself and in 1735 lost the sight of one eye. Invited by Frederick the Great, Euler in 1741 became a member of the Academy of Sciences at Berlin, where for 25 years he poured forth a steady stream of publications. When Frederick became less cordial, Euler in 1766 returned to Russia. Soon after his arrival in St. Petersburg, a cataract formed in the other eye, and Euler spent the last years of his life in total blindness. Despite this and other misfortunes, his productivity continued undiminished, sustained by an uncommon memory and a remarkable ability to compute mentally.

The mathematician J.L. Lagrange, rather than Euler, is often regarded as the greatest mathematician of the 18th century. But Euler never has been excelled either in productivity or in the skillful and imaginative use of computational devices for solving problems.

Fermat, Pierre de

(1601–65). One of the leading mathematicians of the 17th century was the Frenchman Pierre de Fermat. His work was all the more remarkable because mathematics was only his hobby. His profession was law. Independently of his great contemporary, René Descartes, he discovered the fundamental principles of analytic geometry. He is alsoregarded as the inventor of differential calculus, and in association with Blaise Pascal, he was a cofounder of the theory of probability.

Pierre de Fermat is reported to have been born on Aug. 17, 1601, in Beaumont-de-Lomagne, France. He studied law and received his degree from the University of Orléans in 1631. He served in the legislature at Toulouse and was named to the criminal court in 1638. He henceforth made his living as a jurist. Fermat seldom gave proofs of his many mathematical theories, but his work was taken up and elaborated by other great mathematicians such as Christiaan Huygens, Gottfried Leibniz, Isaac Newton, Leonhard Euler, and Carl Friedrich Gauss. Much of his work was published by his son in 1679 under the title ‘Opera mathematica' (Mathematical Works).

SIR ISAAC NEWTON

(1642–1727). The chief figure of the scientific revolution of the 17th century was Sir Isaac Newton. He was a physicist and mathematician who laid the foundations of calculus, extended the understanding of color and light, studied the mechanics of planetary motion, and discovered the law of gravitation. His work established the commonly held scientific view of the world until Albert Einstein underminedit in the early 20th century (see Einstein).

Isaac Newton was born on Dec. 25, 1642, in Woolsthorpe, England. His father died before Isaac was born and left thefamily with little money. Isaac's mother soon remarried and had three more children. She expected Isaac to manage her considerable property after she was widoweda second time. Since he paid little attention to the family farm, Isaac was sent back to grammar school at Grantham to prepare for the university. When he arrived at Trinity College, University of Cambridge in 1661, he learned of the scientific revolution that had been going on in Europe through the work of Galileo, Nicolaus Copernicus, Johannes Kepler, and René Descartes. Abandoning the university's Aristotelian teachings for studies in natural philosophy, Newton became intrigued with atomists, who proposed that everything in nature is ultimately made of indivisible particles of matter—atoms. At the same time Newton created the basis for his mathematical discoveries. In 1665 Newton returned to Woolsthorpe and continued his study of light, gravity, and mathematics that led him to three of the greatest discoveries in the history of science.

Newton's experiments with light showed that white light passed through a prism broke up into a wide color band, called a spectrum. Passed through another prism, the color band became white light again. Next he passed a single color through a prism. It remained unchanged. From this he concluded that white light is a mixture of pure colors. He also formulated the corpuscular theory of light, which states that light is made up of tiny particles, or corpuscles, traveling in straight lines at great speeds. (See also Color; Light; Optics.)

The general law of gravitation arose from Newton's question: what keeps the moon in its regular path around the Earth? He decided that only the attraction of the Earth and the moon for each other could account for it (see Gravitation; Moon).

In mathematics, Newton used the concepts of time and infinity to calculate the slopes of curves and the areas under curves. His fluxional method—later known as calculus—was developed in 1669 but was not published until 1704 (see Calculus).

Newton continued his scientific research when he was appointed professor of mathematics at Cambridge in 1669. Three years later he invented the reflecting telescope (see Telescope). In 1687 he published his major work, ‘Principia' (Philosophiae Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy), setting forth the theory of gravitation. He also served a term in Parliament.

In 1696 Newton was appointed warden of the mint. At that time a complete recoinage and standardization of coins was taking place. When the project was completed in 1699, he was made master of the mint. He was elected president of the Royal Society in 1703 and was knighted in 1705. Newton died in London on March 20, 1727, and was the first scientist to be honored with burial in Westminster Abbey.

Asimov, Isaac. Realm of Numbers (Fawcett, 1981).

Boyer, C.B. A History of Mathematics (Princeton, 1985).

Courant, R. and Robbins, H. What Is Mathematics? (Oxford, 1978).

Dauben, J.W., ed. The History of Mathematics from Antiquity to the Present (Garland, 1985).

Hershey, R.L. How to Think with Numbers (Kaufmann, 1982).

Hoffman, Paul. Archimedes‘ Revenge. The Joys and Perils of Mathematics (Norton, 1988).

James, E. and Barkin, C. What Do You Mean by “Average”? Means, Medians, and Modes (Lothrop, 1978).

Kline, Morris. Mathematical Thought from Ancient to Modern Times (Oxford, 1972).

Struik, D.J. A Concise History of Mathematics, 3rd rev. ed. (Dover, 1967).

SPIRIT OF MATHEMATICS

NATIONAL MATHEMATICAL YEAR 2O12

PM declares 2012 as 'National Mathematical Year'

AN INTERESTING FACT IN FEBRAURY 2010

A WEDDING PROPOSAL BY Mr. ALGEBRA TO Mr. CALCULUS

SALARY THEOREM

MATHEMATICS-LANGUAGE OF THE UNIVERSE

INTERSTING NUMBER 37

YOUR AGE BY CHOCOLATE MATHS

MATH FUN

FIND OUT MISTAKE

APPLICATION OF DIFFERENTIATION

Uses of Differentiation

Increasing and Decreasing Functions

Examples

Stationary Points

Example

Maximum, Minimum or Point of Inflection?

Example

Solving Practical Problems

Example

REGIONAL MATHEMATICS OLYMPIAD QUESTION PAPER 2008

A GOOD BLOG FOR IIT/AIEEE ASPIRANTS

What is the best way to study mathematics

A review for Mathematics site

JARASANDHA NUMBERS

HOW TO FIND THE LAST DIGIT OF ANY NUMBER

STORY OF ZERO

CALCULUS JOKES

What is interesting about the number 2008? and mathematical horoscope for 2008

What is interesting about the number 2008?

HOW TO STUDY MATHEMATICS

A MATHS STUDENT LOVE LETTER

SOME QUOTATIONS ON MATHS(COLLECTIONS)

Infinity

Life is …

Mathematician and Politician-an anology

WHY NOBLE PRIZE IS NOT GIVEN FOR MATHEMATICS?

· No Nobel for Maths because...

MATHEMATICAL LINKS FOR OLYMPIAD

MATH JOKES

MATHEMATICAL OLYMPIAD

MATH JOKES

RANKING OF ENGINEERING COLLEGES

KAPREKAR CONSTANT

Timeline of mathematics

Timeline of mathematics

Note

FUNCTIONS , DOMAIN AND RANGE

Value of a Function October 6, 2006

TRIGONOMETRY FOR AIEEE AND CET

JOKES ON MATHS

TOPIC WISE JOKES ON MATHEMATICS

JOKES ON MATHEMATICS-2

JOKES ON MATHS

WHAT IS MATHEMATICS?

HISTORY

Ancient Periods

Ancient Periods

HISTORY

The Middle Ages

Renaissance Period

HISTORY

Modern Times

Arithmetic

Geometry

Calculus

Number Theory

Set Theory

EUCLID

SREENIVASA RAMANUJAN

PYTHOGORUS

RENE DESCARTES

Value of a Function
October 6, 2006