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”)

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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.
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JARASANDHA NUMBERS

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

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

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

• 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."
•