MATHS_ERA

J.J. SYLVESTER

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SYLVESTER

James Joseph Sylvester was the most influential mathematician in America in the 19th century. Sylvester was born on September 3, 1814, in London and showed his mathematical genius early. At the age of 14, he studies under De Morgan and won several prizes for his mathematics, and at the age of 25, he was elected a Fellow of the ROYAL SOCIETY.

After receiving B.A. and M.A. degrees from Trinity College in Dublin in 1841, Sylvester began a professional life that would include academics, law and actuarial careers. In 1876, at the age of 62, he was appointed to a prestigious position at the newly founded Johns Hopkins University. During his seven years at Johns Hopkins, Sylvester pursued research in pure mathematics, the first ever done in America, with tremendous vigor and enthusiasm. He also founded the American Journal of Mathematics, the first journal in America devoted to mathematical research. Sylvester returned to England in 1884 to a professorship at Oxford, a position he hold until his death on March 15, 1897.

Sylvester's major contributions to mathematics were in the theory of equations, matrix theory, determinant theory, and invariant theory (which is founded with Caylay). His writings and lectures- flowery and eloquent, pervaded with poetic flights, emotional expressions, bizarre utterances, and paradoxes-reflected the personality of this sensitive, excitable and enthusiastic man. We quote three of his students. E.W. Davis commented on Sylvester's teaching methods.

Sylvester's methods! He had none. "Three lectures will be delivered on a New Universal Algebra," he would say; then, "The course must be extended to twelve." It did last all the rest of that year. The following year the course was to be Substitutions- Theorie, by Netto. We all got the text. He lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. "I must give one lecture a week on those," he said. He could not confine himself to the hour, nor to Jo one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again. Statements like the following were not infrequent in his lectures: 'I haven't proved this, but I am as sure as I can be of anything that it must be so. From this it will follow, etc." At the next lecture it turned out that what he was so sure of was false. Never mind, he kept on forever guessing and trying, and presently a wonderful discovery followed, then an-other and another. Afterward he would go back and work it all over again, and surprise us with all sorts of side lights. He then made another leap in the dark, more treasures were discovered, and so on forever.

Sylvester's enthusiasm for teaching and his influence on his students are captured in the following passage written by Sylvester's first student at Johns Hopkins, G.B.Halsted.

A short, broad man of tremendous vitality.... Sylvester's captions head was ever lost on the highest cloud-lands of pure mathematics. Often in the dead of night he would get his favorite pupil, that he might communicate the very last product of his creative thought. Everything he saw suggested to him something new in higher algebra. This transmutation of everything into new mathematics was a revelation to those who knew him intimately. They began to do it themselves.

Another characteristic of Sylvester, which is very unusual among mathematicians, was his apparent inability to remember mathematics! W.P. Durfee had the following to say:

Sylvester had one remarkable peculiarity. He seldom remembered theorems, propositions, etc., but had always to deduce them when he wished to use them. In this he was the very antithesis of Caylay, who was thoroughly conversant with everything that had been done in every branch of mathematics.

I remember once submitting to Sylvester some investigations that I had been engaged on, and he immediately denied my first statement, saying that such a proposition had never been heard of, let alone proved. To his astonishment, I showed him a paper of his own in which he had proved that proposition; in fact, I belive the object of his paper had been the very proof which was so strange to him.

LOWEST COMMON MULTIPLE

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L.C.M. stands for Lowest Common.

LCM

So let us first discuss about what are multiples of the given number.

Multiple: The number obtained by multiplying the given number is called the multiple of given number.

For example: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc. therefore 2, 4, 6, 8, etc. are the multiples of 2.

Lowest Common Multiple: The L.C.M. of two or more given numbers is the lowest (smallest) number that is a multiple of each of the given numbers.

Thus, it is the smallest number which is exactly divisible by each of the given numbers.

For example: L.C.M. of 4 and 6 is 12 as 12 is the least multiple common multiple of 4 as well as 6.

Methods of finding L.C.M.: The following three methods are most commonly used to find the L.C.M.

1) Common Multiple Method:

Step 1: Find a few multiples of each given number.

Step 2: From the multiples obtained in Step 1, select the common ones.

Step 3: The lowest number (multiple) obtained in Step 2 is the required lowest common multiple of the given numbers.

For example: Find L.C.M. of 4, 5 and 10.

Step 1: Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ......

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45 ,50, 55, .....

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, .......

Step 2: Common multiples of 4, 5 and 10 = 20, 40, 60, 80, .....

Step 3: smallest common multiple obtained in step 2 is 20.

So, L.C.M. of 4, 5 and 10 is 20.

2) Prime Factor Method:

Step 1: Express each of the given number as a product of its prime factors and then in index form.

Step 2: Product of all the prime factors obtained in step 1 with highest power of each is the L.C.M. of given numbers.

For example: Find L.C.M. of 18 , 24 and 36.

Step 1: 18 = 2 × 3 × 3 = 2¹ × 3²,

24 = 2 × 2 × 2 × 3 = 2³ × 3¹,

36 = 2 × 2 × 3 × 3 = 2² × 3²

Step 2: Since, the prime factors 2 and 3, obtained above, with highest power are 2³ and 3².

Therefore, L.C.M. = 2³ × 3² = 2 × 2 × 2 × 3 × 3 = 72.

3) Common Division Method:

Step 1: Write all the given numbers in a horizontal line, separating them by commas.

Step 2: Divide by a suitable number, that exactly divides at least two of the given numbers. And, write down the quotients and the undivided numbers obtained, below the first line.

Step 3: Repeat the process until we get a line of numbers that are prime to one-another.

Step 4: The product of all the divisors and the numbers obtained in the last line will be the required L.C.M.

For example: Find L.C.M. of 16, 20 and 24.

16, 20, 24

2 16, 20, 24

8, 10, 12

2 16, 20, 24

2 8, 10, 12

2 4, 5, 6

2, 5, 3

Therefore, L.C.M = 2 × 2 × 2 × 2 × 3 × 5 = 240.

HIGHEST COMMON FACTOR

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As we know H.C.F. stands for highest common factor.

HCF

Let us first discuss about factors.

Factors: When two or more natural numbers are multiplied together, the result is referred as their product, and each of the numbers multiplied is called a factor of this product.

For example: product of 3 and 7 is 21, therefore 3 and 7 are factors of 21, product of 2, 3 and 5 is 30, therefore each of 2, 3 and 5 is a factor of 30.

In other words

Any natural number that divides a given number completely is called a factor of the given number.

For example: 5 divides 20 completely → 5 is a factor of 20,

7 divides 42 completely → 7 is a factor of 42.

As each of 1, 2, 3, 4, 6, 8, 12 and 24 divides 24 completely, so factors of 24 are 1, 2, 3, 4, 6, 12 and 24.

Factors of F24 = 1, 2, 3, 4, 6, 12, 24.

1 (one) is a factor of every number.
Every number is a factor of itself.
Zero (0) can not be a factor of any number.

Prime factors: Any prime number that divides completely a given natural number is called a prime factor of the given number.

For example: factors of 24 are 1, 2, 3, 4, 6, 12 and 24. Out of these 2 and 3 are the prime numbers and so the factors 2 and 3 are called the prime factors of 24.

HIGHEST COMMON FACTOR: The H.C.F. of two or more given numbers is the greatest number that divides each of the given numbers.

For example: The greatest number that can divide both 18 and 24 completely is 6; therefore, H.C.F. of 18 and 24 = 6.

METHODS OF FINDING H.C.F.

For finding the H.C.F. of two or more given numbers, any of the following three methods can be used:

1) Common Factor Method:

Step 1: Firstly find all the possible factors of each given number.

Step 2: From the factors obtained in Step 1, select the common factors.

Step 3: Out of the common factors, obtained in Step 2, take the highest factor, which is the Highest Common Factor (H.C.F.) of the given numbers.

For example:

Step 1: Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 and

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

Step 2: Factors that are common to 36 and 48 = 1, 2, 3, 4, 6, 12

Step 3: From the result of Step 2, the highest common factor = 12

2) Prime factor method:

Step 1: Split each given number into its prime factors.

Step 2: Select the common prime factors.

Step 3: Multiply the prime factors obtained in Step 2.

The product so obtained is the H.C.F. of the given numbers.

For example: Find H.C.F. of 15 and 25.

Step 1: Prime factors of 15 are 3 and 5, since 5 × 3 = 15, prime factors of 25 are 5 and 5, since 5 × 5 = 25

Step 2: Since the common prime factor is 5 only,

Step 3: Therefore H.C.F. of 15 and 25 is 5.

Any two numbers that do not have a common prime factor are called co-prime numbers. e.g 2 and 3, 5 and 7.

The H.C.F. of two co-prime numbers is always 1.

3) Division method:

Step 1: Divide the greater number by the smaller number.

Step 2: By the remainder of division in Step 1, divide the smaller number.

Step 3: By the remainder in Step 2, divide the remainder obtained in Step 1.

Step 4: Continue in the same way till no remainder is left. The last divisor is the required H.C.F.

For example: Find the H.C.F. of 36 and 60.

Step 1: 60 = 36 × 1 + 24

Step 2: 36 = 24 × 1 + 12

Step 3: 12 = 12 × 2 + 0

Since the last divisor is 12, therefore H.C.F. is 12.

ÉVARISTE GALOIS

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Galois

Évariste Galois was born on October 25, 1811, near Paris. He took his first mathematics course when he was 15 and quickly mastered the work of Legendre and Lagrange. At the age of 18, Galois wrote his important research on the theory of equations and submitted it to the French Academy of Sciences for publication. The paper was given to Cauchy for refereeing. Cauchy impressed by the paper, agreed to present it to the academy, but never did. At the age of 19, Galois Prize in Mathematics, given by the French Academy of Sciences. The paper was given to Fourier, who died shortly thereafter. Galois's paper was never seen again.

"Galois at seventeen was making discovering of epochal significance in the theory of equations, discoveries whose consequences are not yet exhausted after more than a century."

E.T. Bell, Men of Mathematics

Galois stamp

Galois twice failed his entrance examination to 1'École Polytechnique. He did not know some basic mathematics, and he did mathematics almost entirely in his head, to the annoyance of the examiner. Legend has it that Galois became so enraged at the stupidity of the examiner that he threw an eraser at him.

Galois spent most of the last year and half of his life in prison for revolutionary political offenses. While in prison, he attempted suicide, and prophesied his death in a duel. On may 30, 1832, Galois was shot in a duel and died the next day at the age of 20. The life and death of Galois have long been a source of fascination and speculation for mathematics historians. One article [1] argues that three of the most widely read accounts of Galois's life are highly fictitious.

Among the many concepts introduced by Galois are normal subgroups, isomorphism, simple groups, finite fields, and Galois theory. His work provided a method for disposing of several famous constructability problems, such as trisecting an arbitrary angle and doubling a cube. Galois's entire collected work full only 60 pages.

VARIOUS TYPES OF NUMBERS

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NUMBER SYSTEM

Natural numbers: Counting numbers are called natural number.

Thus, 1, 2, 3, 4, ...... are all natural number.

Whole numbers: All counting numbers, together with 0, form the set of whole numbers.

Thus, 0, 1, 2, 3, 4, ..... are all whole numbers.

Integers: All counting numbers, zero and negatives of counting numbers, form the set of integers.

Thus, ....., -3, -2, -1, 0, 1, 2, 3,...... are all integers.

Set of positive integers = { 1, 2, 3, 4, 5, .......... }

Set of negative integers = { -1, -2, -3, -4, ....... }

Set of non-negative integers = { 0, 1, 2, 3, 4, .......}

Set of non-positive integers = { 0, -1, -2, -2, -4, .......}

Even numbers: All multiples of two are called even numbers.

Since, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...... is a multiple of 2.

Therefore each of 2, 4, 6, 8, 10, 16, 18, is an even number.

Every even number is divisible by 2 or 2 is a factor of every even number.

For example: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc. therefore 2, 4, 6, 8, etc. are the multiples of 2.

Multiples: The number obtained by multiplying the given number is called the multiple of given number.

Odd numbers: Numbers which are not divisible by 2 are called odd numbers.

Since, none of 1, 3, 5, 7, 9, 11, 13, .... is divisible by 2, so each of 1, 3, 5, 7, 9, 11, 13, ...... is an odd number.

Every number is either even or odd.
A number cannot be both even as well as odd.
All prime numbers except 2 are odd.

Prime number: A natural number that is divisible by not one (1) and itself is called a prime number.

For example: 2 is divisible only by 1 and itself, therefore 2 is prime number; 5 is divisible by only 1 and itself, therefore 5 is prime number.

To be more clear, note:

If a natural number has only two factors, it is a prime number.
1 (one) is not a prime number as it has only one factor.
3 is a prime number as it has only two factors, which are 1 and 7.
10 is not a prime number as it has more than two factors i.e.1, 2, 5, 10.

All prime numbers less than 100 are 2, 3, 5, 7, 11, 13 ,17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Composite numbers: Every natural number that has more than two factors is called a composite number.

Since, factors of 10 are 1, 2, 5 and 10.

Therefore, 10 is a composite number.

Similarly, each of 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, etc. is a composite number.

Every even number greater than 2 is a composite number.

Perfect number: A number, the sum of whose factors (except the number itself), is equal to the number, is called a perfect number.

For example: The factors of 6 are 1, 2, 3 and 6. And, 1+ 2+ 3 = 6, so 6 is perfect number.

Co-prime numbers: Any two numbers that do not have a common prime factor are called co-prime numbers.

For example: 2 and 3, 5 and 7 are the examples of co-prime numbers.

Twin prime: The two prime numbers whose difference is 2 are called twin prime numbers.

Important facts:

The smallest prime number is 2.
The only even prime number is 2.
The first odd prime number is 3.
1 is unique number which is neither prime nor composite.
The least composite number is 4.
The least odd composite number is 9.
All natural numbers are whole numbers.
All whole numbers are not natural numbers as 0 is a whole number but not a natural number.

What are irrationals?

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Irrational numbers

A number that can not be written in the form p/q, where p and q are integers and q is non zero. So we can say a number which is not rational is irrational. Thus, √2, √5, π, etc. are such examples of irrational numbers.

In the view of decimal expansion

As we know the decimal expansion of √2 =. 1.4142135623730950488016887242096..., π = 3.14159265358979323846264338327950...., e = 2.71828183, etc., which are irrational numbers.

In above examples we have seen that decimal expansion is non terminating as well as non repeating. Now, let us summarize our results in the following form:

The decimal expansion of a irrational number is neither terminating nor non terminating recurring (non repeating). Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.

The collection of irrational numbers is denoted by Q' (Compliment of Q).

The Greek genius Archimedes(287 BC - 212 BC) was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 AD - 550 AD), the great Indian mathematician and astronomer, found the value of π correct up to four decimal places. Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places.

Properties of irrational numbers:

No fraction is irrational number.
No natural number is irrational number.
The integer 0 is not irrational number.
All integers are not irrational numbers.
Here we can't say what is next irrational number, i.e. we can't say successor or predecessor of a irrational number as there are infinitely many irrational numbers, even uncountable.
Irrationals have a denseness property, i.e. the closure of a collection of irrational numbers is closed.
Between any two given irrational number, there are infinitely many rationals as well as irrationals.
Square-root of every every positive integer which is not perfect square is irrational number.
There is nothing common between rationals and irrationals.

PETER LUDWIG MEJDELL SYLOW

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sylow

Ludwig Sylow was born on December 12, 1832, in Christiania (now Oslo), Norway. While a student at Christiania University, Sylow won a gold medal for competitive problem solving. In 1855, he became a high school teacher, and despite the long hours required by this teaching duties, Sylow found time to study the papers of Abel. During the school year 1862-1863, Sylow received a temporary appointment at Christiana University and gave lectures on Galois's theory and permutation groups. Among his students that year was the great mathematician Sophus Lie, after who's Lie algebras and Lie groups are named. From 1873 to 1881, Sylow, with some help from Lie, prepared a new edition of Abel's work. In 1902, Sylow and Ellington Hoslt published Abel's correspondence.

Sylow's great discovery, Sylow's Theorem, came in 1872. Upon learning of Sylow's result, C. Jorden called it "one of the essential points in the theory of permutations." The result took on greater importance when the theory of abstract groups flowered on the late 19th century and early 20th century.

In 1869, Sylow's was offered a professorship at Christiania University, but turned it down. Upon Sylow's retirement at the age 6t from high school teaching, Lie mounted a successful campaign to establish a chair for Sylow's at Christiania University. Sylow held this position until his death on September 7, 1918.

WHAT ARE RATIONALS & PROPERTIES ?

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rational numbers

Rational numbers: A number that can be written in the form p/q, where p and q are integers and q is non zero. Thus, 2/3, 5/9, 31/57, -73/1134 are such examples of rational numbers.

In the view of decimal expansion

As we know that any fraction can be converted into decimal number upon dividing the numerator by its denominator. For example: 1/2 = 0.5, 1/3 = 0.33333....., 1/4 = 0.25, 14/11= 1.272727272...., etc.

In above examples we have seen either the decimal expansion is terminating or repeating but what about the decimal expansion which are neither terminating nor repeating. About the later case we'll discuss later on. Now, let us summarise our results in the following form:

The decimal expansion of a rational number is either terminating or non terminating recurring (repeating). Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.

The collection of rational numbers is denoted by Q.

Properties of rational numbers:

All fractions are rational numbers.
All natural numbers are rational numbers but all rational numbers are not natural numbers.
The integer 0 is a rational number.
All integers are rational numbers.
Here we can't say what is next rational number, i.e. we can't say successor or predecessor of a rational number as there are infinitely many rational numbers, even countably infinite.
Rationals have a denseness property, i.e. the closure of a collection of rational numbers is closed.
Between any two given rational numbers, we can always find another rational number and hence infinitely many rationals between any two given rationals.

ALL ABOUT FRACTIONS

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Fractions represent parts of a whole.

fractions

For example, if an apple is divided into three equal parts, each part is called one third and is denoted by 1/3. We call 1/3 a fraction. Similalry, 1/4 , 2/5 , 3/7 , and 5/4 are also fractions.

In any fraction a/b, the dividend a is called the numerator and dividend b is called the denominator. For example, in the fraction 5/9, the numerator is 5 and the denominator is 9.

Classification of Fractions

Common, simple or vulgar fraction

The numerator and the denominator are integers in a common fraction, also known as a simple fraction or vulgar fraction.

Example 3/5 , 7/9 , 11/7 , 3/2 are some simple fractions.

Complex fraction

A fraction in which the numerator or the denominator or the both of them also contain fractions, is called a complex fractions.

For example: (4/5)/(7/9), (3/5)/(8/7), etc.

Proper fraction

A vulgar fraction in which the numerator is less than the denominator.

For example: 3/5, 4/9, etc.

Improper fraction

A vulgar fraction in which the numerator is greater than the denominator.

For example: 4/3, 8/5, etc.

Mixed fraction

An integer together with a proper fraction, is called a mixed fraction.

From example: 67/5, etc.

A mixed fraction can easily be converted into an improper fraction as follows.

Mixed fraction = [ (integral part) × (denominator of the proper fraction) + (numerator of the proper fraction)] / denominator.

Mixed fraction = quotient in (numerator ÷ denominator) + remainder/ denominator

Equivalent or equal fractions

If the numerator and denominator of a fraction are multiplied or divided by the same non-zero number, then we get an equivalent or equal fraction.

For example: 2/5 = 6/15, 7/9 = 42/54, etc.

This brings us to conclusion that the value of a fraction remains same of both its numerator and denominator are multiplied or divided by the same number.

• Two fractions a/b and c/d are equivalent fractions if ad = BC.

Simplest form of fraction: A fraction is said to be in the simplest form of if its numerator and denominator have no factor in common except 1.

For example: 2/3, 7/9, 11/51, etc.

Like and unlike fractions: Two or more fractions are called like fractions of they have the same denominator. Fractions that do not have the same denominator are called unlike fractions.

For example: 2/9, 5/9, 7/9 are like fractions where as 2/5, 3/7,6/11 are unlike fractions.

JOSEPH LOUIS LAGRANGE

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Lagrange

JOSEPH LOUIS LAGRANGE was born on Italy of Franchise ancestry on January 25, 1736. He became captivated by mathematics at an early age when he read on essay on Halley on Newton's calculus. At the age of 19, he became professor of mathematics at the Royal Artillery School in Turin. Lagrange made significant contributions to many branches of mathematics and physics, among them the theory of numbers, the theory of equations, ordinary and partial differential equations, the calculus of variations, analytic geometry, fluid dynamics, and celestial mechanics. His methods for solving third and fourth degree polynomial equations by radicals laid the groundwork for the group-theoretic approach to solving polynomials taken by Galois. Lagrange was a very careful writer with a clear and elegant style.

Lagrange stamp

"Lagrange is the Lofty Pyramid of the Mathematical Sciences."

NAPOLEON BONAPARTE

At the age of 40, Lagrange was appointed Head of Berlin Academy, succeeding Euler

In offering this appointment, Frederick The Great proclaimed that the "greatest king in Europe" ought to have the "greatest mathematician in Europe" at his court. In 1787, Lagrange was invited to Paris by Louis XVI and became a good friend of the king and his wife, Marie Antoinette. In 1793, Lagrange headed a commission, which included Laplace and Lavoisier, to devise a new system of weights and measures. Out of this came the metric system. Late in his life he made a count by Napoleon. Lagrange died on April 10, 1813.

WHAT ARE INTEGERS AND PROPERTIES ?

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Integers

As we are now we'll known with natural numbers & for each natural number a (say), there is a number -a such that a+(-a) = 0. This, 1+(-1)= 0, 2+(-2)= 0, ......

The numbers ....,-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ... are called integers. 1, 2, 3, 4, ... are called position integers, while -1, -2, -3, -4, -5, ... are called negative integers. The integer 0(zero) is neither positive nore negative.

The set of integers is denoted by Z or I. It is clear that all the natural numbers and all the whole numbers are integers but all the integers are neither natural nor whole numbers.

Some properties of integers:

Every integer is greater than every Negative integers
Zero is greater than every negative integer and is less than every positive integer.
Every negative integer is smaller than every positive integer.
Given two positive integers, the one with the bigger absolute value is greater.
Given two negative integers, the one with smaller absolute value is greater.

Absolute value: The absolute value of an integer is the whole number obtained by disregarding its sign. The absolute value of an integer a is denoted by |a|

Thus |-51| = (absolute value of -51) = 51,

|0|= (absolute value of 0) = 0,a|.

|-2| = (absolute value of -2) = 2,

|43| = (absolute value of 43) =43.

WHAT ARE WHOLE NUMBERS ?

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Whole numbers

The natural numbers along with 0 (zero) form a collection of whole numbers.

Whole numbers = 0, 1, 2, 3, 4, 5, 6, .....

A whole number is either zero or a natural number.

Properties of whole numbers:

The first and smallest whole number is 0 (zero).
The last and the greatest whole number can not be obtained, like natural numbers, whole number are also infinite.
The difference between two consecutive whole numbers is 1 (one).
By adding 1 to any whole number, its next whole number is obtained.
All natural numbers are whole numbers, but all whole numbers are not natural numbers.

AUGUSTIN LOUIS CAUCHY

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cauchy

AUGUSTIN LOUIS CAUCHY was born on August 21, 1789, in Paris, the oldest of six children. By the time he was 11, both Laplace and Lagrange had recognised Cauchy's extraordinary talent for mathematics. In school he won prizes for Greek, Latin and the humanities. At the age of 21, he was given a comemission in Napoleon's army as a civil engineer. For the next few years, Cauchy attend to his engineering duties while carrying out brilliant mathematical research on the side.

You saw that little young man? Well! He will supplant all of us in so far as we are mathematicians.

SPOKEN BY LAGRANGE TO LAPLACE

ABOUT THE 11-YEAR-OLD CAUCHY

Cauchy stamp

In 1815, at the age of 26, Cauchy was made Professor of Mathematics at the École polytechnique and was recognised as the leading mathematics in France. Cauchy and his contemporary Gauss were the last men to know the whole of mathematics as known at their time, and both made important contributions to nearly branch, both pure and applied, as well as to physics and astronomy.

Cauchy introduced a new level of rigor into mathematical analysis. We owe our contemporary notions of limit and continuity to him. He gave the first proof of the Fundamental Theorem of Calculus. Cauchy was the founder of complex function theory and a pioneer in the theory of permutation groups and determinants. His total written output of mathematics fills 24 large volumes and is second only to that of Euler. He wrote over 500 research papers after the age of 50. Cauchy died at the age of 67 on May 23, 1857.

What are natural numbers ?

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Natural numbers

Natural numbers are those numbers by which we can count things in nature like 4 pens, 5 girls, 8 trees, etc.

These are the numbers used for counting purpose.

So, Natural numbers = 1, 2, 3, 4, 5, 6, 7, .....

Properties of natural numbers:

The first and smallest natural number is 1(one).
The last and greatest natural number can not be obtained, infact there are infinite numbers, like counting of stars in nature.
The difference between any two consecutive natural numbers is obtained.
The fractions like: 15/43, -34/51, etc. are not natural numbers.
The decimal numbers like: 4.4, 2.36, 8.94,etc. are not natural numbers.
0 (zero) is not a natural numbers.
No natural number is negative i.e. none of -2, -53, -712, etc. is a natural number.
If n is any natural number, then 2n is even natural number and (2n-1) is an odd natural number.

NUMBERS IN INDIAN AND INTERNATIONAL SYSTEM

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INTRODUCING UNIT, NUMBER, NUMERAL AND NUMERATION

NUMBERS IN INDIAN AND INTERNATIONAL SYSTEM

In mathematics unit means a single thing.

For example, a pen, a book, a girl, etc.

A unit is the first and lowest natural number as a standard of measurement.

The number written before the name of a unit indicates how many times that unit is taken.

For example, 6kg means; the number 6 is written before the unit kg.

Unit kg (kilogram) is taken 6 times.

NUMERAL AND NUMERATION

A numeral is a symbol representing a given number and numeration represents that number in words.

Number Numeral Numeration
2 2 two
36 36 thirty-six
0 0 zero

HINDU-ARABIC (INDIAN) SYSTEM OF NUMERATION

The Indian system of numeration is in fact the decimal system that is in use all over the world. This system was developed by the ancient Hindu-Mathematicians in India and was carried to West by the Arabs. For this reason, it is called the Hindu-Arabic system of numeration.

In India system, which is also known as denary system, the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used to write a numeral (number). Each of these ten symbols is called a digit.

In Indian system, also known as Hindu Arabic System and in International sysytem of numeration, to read and write large quantities with ease, the group's are made with certain periods as shown below.

Indian system	Figures	International system	Figures
Unit	1	Ones	1
Tens	10	Tens	10
1 Hundred	100	1 Hundred	100
1 Thousand	1000	1 Thousand	1000
10 Thousands	10000	10 Thousands	10000
1 Lakh	100000	100 Thousands	100000
10 Lakhs	1000000	1 Million	1000000
1 Crore	10000000	10 Millions	10000000
10 Crores	100000000	100 Millions	100000000
1 Arab	1000000000	1 Billion	1000000000
10 Arabs	10000000000	10 Billions	10000000000
1 Kharab	100000000000	100 Billions	100000000000
10 Kharabs	1000000000000	1 Trillion	1000000000000
1 Neel	10000000000000	10 Trillions	10000000000000
10 Neels	100000000000000	100 Trillions	100000000000000
1 Padam	1000000000000000	1 Quadrillion	1000000000000000
10 Padam	10000000000000000	10 Quadrillions	10000000000000000
1 Shankh	100000000000000000	100 Quadrillions	100000000000000000
10 Shankh	1000000000000000000	1 Quintillion	1000000000000000000
Maha-Shankh	10000000000000000000	10 Quintillions	10000000000000000000

Number	Numeral	Numeration
2	2	two
36	36	thirty-six
0	0	zero