INTERSTING NUMBER 37

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

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

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

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

0

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

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

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

CLICK ON THE FOLLOWING LINK FOR
REGIONAL MATHEMATICS OLYMPIAD QUESTION PAPER 2008
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What is the best way to study mathematics

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

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
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