structural:summarization, structural:reformatting, linguistic:paraphrasing
Dawn of arithmetic

The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 BC) 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.

The Pythagorean tradition spoke also of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc., are seen now as more natural than triangular numbers, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th centuries).

We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in each. The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later solved by Āryabhaṭa 's Kuṭṭaka – see below.) The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao 's 1247 Mathematical Treatise in Nine Sections which was translated into English in early 19th century by British missionary Alexander Wylie.

There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere.