PROBABILITY

PROBABILITY :- Probability theory like many other branches of mathematics, evolved out of practical consideration. It had its origin in the 16th century when an Italian physician and mathematician Jerome Cardan (1501-1576) wrote the first book on the subject "Book on Games of Chance" (Biber de Ludo Aleae). It was published in 1663 after his death. In 1654, a gambler Chevalier de Metre approached the well known French Philosopher and Mathematician Blaise Pascal (1623-1665) for certain dice problem. Pascal became interested in these problems and discussed with famous French Mathematician Pierre de Fermat (1601-1665). Both pascal and Fermat solved the problem independently. Besides, pascal and Fermat, outstanding contributions to probability theory were also made by Christian Huygenes (1629-1665), a Dutchman, J. Bernoulli (1654-1705). De Moivre (1667-1754), a Frenchman Pierre Laplace (1749-1827), the Russian P.L Chebyshev (1821-1897), A. A Markov (1856-1922) and A. N Kolmogorove (1903-1987). Kolmogorove is credited with the axiomatic theory of probability. His book 'Foundations of probability' published in 1933, introduces probability as a set function and is considered a classic...

STATISTICS

STATISTICS :- 'Statistics' is derived from the Latin word 'status' which means a political state. This suggests that statistics is as old as human civilisation. In the year 3050 B.C., perhaps the first census was held in Egypt. In India also, about 2000 year ago, we had an efficient system of collecting administrative statistics, particularly, during the regime of Chandra Gupta Maurya (324-300 B.C.). The system of collecting data related to births and deaths is mentioned in Kautilya's Artishastra(around 300 B.C.) A detailed account of administrative surveys conducted during Akbar's regime is given in Ain-I-Akbari written by Abul Fazl. Captain john Graunt of London (1620-1674) is known as father of vital statistics due to his studies on statistics of births and deaths. Jacob Bernoulli (1654-1705) stated the Law of Large number in his book "Ars Conjectandi', published in 1713. The theoretical development of statistics came during the mid seventeenth century and continued after that with the introduction of theory of games and chance (i.e., probability). Francis Galton (1822-1921), an Englishman, pioneered the use of statistical methods, in the field of Biometry. Karl Pearson (1857-1936) contributed a lot to the development of statistical laboratory in England (1911). Sir Ronald A. Fisher (1890-1962), known as the Father of modern statistics, applied it to various diversified fields such as Genetics, Biometry, Education, Agriculture, etc....

LINEAR INEQUALITIES

LINEAR INEQUALITIES : 1. Two real number or two algebraic expression related by the symbols greater then and less then form an inequality. 2. Equal number may be added to (or subtracted from )both side of an inequality. 3. Both side of an inequality can be multiplied (or divided) by the same positive number, But when both side are multiplied (or divided) by a negative number, then the inequality is reversed. 4. The value of x, which make an inequality a true statement, are called solutions of the inequality. 5. To represent x less then a (or x greatdr then a) on a number line, put a circle on the number a and dark line to the left (or right) of the number a. 6. To represent a number line, put a dark circle on the number a and dark the line to the left (or right) of the number x. 7. If an inequality is having symbol, then the point on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the equality represented by dark line that satisfies and arbitrary point in the part. 8. If an inequality is having greater then or less then symbole. Then the point on the line are not included in the solutions of the inequality and the graph of the inequality lies to the left (below) or right (above) of the graph of the corresponding equality represented by dotted line that satisfies an arbitrary point in that part. 9. The solution region of a system of inequalities is the region which satisfies all the given inequalities in the system simultaneously.

MATHEMATICAL REASONING

MATHEMATICAL REASONING :- The first treatise on logic was written by Aristotle (384 B.C.-322 B.C.). It was a collection of rules for deductive reasoning which would serve as a basis for the study of every branch of knowledge. Later, in the seventeenth century, German mathematician G. W. Leibnitz (1646-1716) conceived the idea of using symbols in logic to mechanise the process of deductive reasoning. His idea was realised in the nineteenth century by the English mathematician George boole (1815-1864) and Augustus De Morgan (1806-1871), who founded the modern subject of symbolic logic....

BINOMIAL THEOREM

The ancient Indian mathematicians knew about the coefficients in the expansions of (x+y)power n . The arrangement of these coefficients was in the form of a diagram called Meru-Prastara, provided by Pingla in his book Chhanda shastra (200B.C.). This triangular arrangement is also found in the work of Chinese mathematician Chu-shi-kie in 1303. The term binongal coefficients was first introduced by the German mathematician, Michael Stipel (1486-1567) in approximately 1544. Bombelli(1572) also gave the coefficients in the expansion of (a+b)power n, for n=1,2..........,7 and oughtred (1631) gave them for n=1,2,......, 10. The arithmetic triangle, popularly known as Pascal's triangle and similar to the Meru-Prastara of Pingla was constructed by the French mathematician Blaise Pascal (1623-1662) in 1665. The present form of the binongal theorem for integral values of n appeared in Trate du triange arithmetic, written by Pascal and published posthumously in 1665....

INTRODUCTIONS TO THREE DIMENSIONAL GEOMETRY

Rene' Descartes (1596-1650), the father of analytical geometry, essentially dealt with plane geometry only in 1637. The same is true of his co-inventor Pierre Fermat (1601-1665) and La Hire (1640-1718). Although suggestions for the three dimensional coordinate geometry can be found in their work but no develop it J. Bernoulli (1667-1748) in a letter of 1715 to Leibnitz introduced the three coor-dinate planes which we use today. It was Antoinne Parent (1666-1716), who gave a systematic development of analytical solid geometry for the first time in a paper presented to the French Academy in 1700. L. Euler (1707-1783) took up systematically the three dimensional coordinate geometry, in Chapter 5 of the appendix to the second volume of his "INTRODUCTION TO GEOMETRY " in 1748. It was not until the middle of the nineteenth century that geometry was extended to more than three dimensions, the well-known application of which is in the Space-Time Continuum of Einstein's Theory of Relativity....

SEQUENCE AND SERIES

Evidence is found that Babylonians, some 4000 year ago, knew of arithmetic and geometric sequences. According to Boethius(510), arithmetic and geometric sequences were known to early Greek writer. Among the Indian mathematician, Aryabhatta (476 was the first to give the formula for the sum of squares and cubes of natural number in his famous work Aryabhatiyam, written around 499. He also gave the formula for finding the sum to n terms of an arithmetic sequence starting with Pth term. Noted Indian mathematicians Brahmgupta (598), Mahavira (850) and bhaskara (1114-1185) also considered the sum of squares and cubes. Another specific type of sequence having important applications in mathematics, called Fibonacci sequence, was discovered by Italian mathematician Leonardo Fibonacci (1170-1250). Seventeenth century witnessed the classification of series into specific forms. In 1671 James Gregory used the term infinite series in connection with infinitic sequence. It was only through the rigorous development of algebraic and set theoretic tools that the concepts related to sequence and series could formulated suitably....

PERMUTATIONS AND COMBINATIONS.

The concepts of permutations and combinations can be traced back to the advent of jainism in India and perhaps even earlier. The credit, howevere, goes to the jains who treated its subject matter as a self-contained topic in mathematics, under the name vikalpa. Among the jains, mahavira, (around 850) is perhaps the world's first mathematician credited with providing the general formulae for permutations and combinations. In the 6th century B.C., Sushruta, in his medicinal work, Sushruta Samhita, asserts that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc. Pingala, a Sanskrit scholar around third century B.C., gives the method of determining the number of combinations of a given number of letters, taken one at a time, two at a time, etc. In his work Chhanda Sutra. Bhaskaracharya(born 1114) treated the subject matter of permutations and combinations under the name Anka Pasha in his famous work Lilavati. In mahavira, Bhaskaracharya gives several important theorems andrestl concerning the subject...

COMPLEX NUMBER AND QUADRCTIC EQUATION

The fact that square root of a negative number does not exist in the real number system was recognised by the Greeks. But the credit goes to the Indian mathematician Mahavira(850) who first stated this difficulty clearly. "He mentions in his work 'Ganitasara Sangraha' as in the nature of things a negative (quantity) is not a square (quantity)',it has therefore, no square root". Bhaskara, another Indian mathematician, also writes in his work Bijaganita, written in 1150. 'There is no square root of a negative quantity, for it is not a square. "Cardan(1545) considered the problem of solving x+y= 10, xy= 40. He obtained x=5+root-15 as the solution of it, which was discarded by saying that these number are 'useless'. Albert Girard (about 1625) accepted square root of negative number and said that this will enable us to get as many roots as the degree of the polynomial equation. Euler was the first to introduce the symbol ¡ For root-1 and W.R. Hamilton(about1830) regarded the complex number a+¡b as an ordered pair of real numbers (a,b)thus giving it a purely mathematical definition and avoiding use of the so called 'imaginary number'......

PRINCIPLE OF MATHEMATICAL INDUCTION

Unlike other concepts and methods proof by mathematical induction is not the invention of a particular individual at a fixed moment. It said that the principle of mathematical induction was known by the pythagoreans. The french mathematician Blaise pascal is credited with the origin of the principle of mathematical induction. The name induction was used by the English mathematician John Wallis. Later the principle was employed to provide a proof of the binomial theorem. De Morgan contributed many accomplishments in the field of mathematics on many different subject. He was the first person to define and name "mathematical induction " and developed De morgan's rule to determine the convergence of a mathematical series. G. Peano undertook the task of deducing the properties of natural numbers from a set of explicitly stated assumptions, now known as Peano's axioms. The principle of mathematical induction is a restatement of one of the peano's axioms.....