J.J. SYLVESTER

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

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.

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.

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.

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.

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.

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.