Number theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, 'Mathematics is the queen of the sciences—and number theory is the queen of mathematics.' Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).'In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad.'The theory of the division of the circle...which is treated in sec. 7 does not belongby itself to arithmetic, but its principles can only be drawn from higher arithmetic. the question 'how was the tablet calculated?' does not have to have the same answer as the question 'what problems does the tablet set?' The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems (Robson 2001, p. 202). Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to 'work for a living', and would not have belonged to a 'leisured middle class') could have been motivated by his own 'idle curiosity' in the absence of a 'market for new mathematics'.(Robson 2001, pp. 199–200) Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. Answer: 23. Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. Answer: Male.This is the last problem in Sunzi's otherwise matter-of-fact treatise. Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, 'Mathematics is the queen of the sciences—and number theory is the queen of mathematics.' Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation). The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by 'number theory'. (The word 'arithmetic' is used by the general public to mean 'elementary calculations'; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic. The world's oldest document about Mathematics is the Berlin Papyrus 6619 from the Middle Kingdom, second half of the 12th (c. 1990–1800 BC) or 13th dynasty (c. 1800BC–1649BC), and had a problem similar to the Pythagorean theorem before Pythagoras lived and much before Euclid (300BC). Another early historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 BCE) contains a list of 'Pythagorean triples', that is, integers ( a , b , c ) {displaystyle (a,b,c)} such that a 2 + b 2 = c 2 {displaystyle a^{2}+b^{2}=c^{2}} .The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: 'The takiltum of the diagonal which has been subtracted such that the width...' The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by c / a {displaystyle c/a} , presumably for actual use as a 'table', for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of 'algebra') was exceptionally well developed. Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is very probably Pythagorean; it is very simple material ('odd times even is even', 'if an odd number measures an even number, then it also measures half of it'), but it is all that is needed to prove that 2 {displaystyle {sqrt {2}}} is irrational. Pythagorean mystics gave great importance to the odd and the even.The discovery that 2 {displaystyle {sqrt {2}}} is irrational is credited to the early Pythagoreans (pre-Theodorus). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect. This forced a distinction between numbers (integers and the rationals—the subjects of arithmetic), on the one hand, and lengths and proportions (which we would identify with real numbers, whether rational or not), on the other hand.