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History of numbers


Numbers that answer the question "How many?" are 0, 1, 2, 3  and so on. These are

cardinal numbers. When used to indicate position in a sequence they are ordinal numbers.

To the Pythagoreans and Greek mathematician Euclid, the numbers were 2, 3, 4, 5, . . . Euclid did not consider 1 to be a number.

Numbers like , expressible as fractions in which the numerator and denominator are whole numbers, are rational numbers. These make it possible to measure such quantities as two and a quarter gallons and six and a half miles.

In the fifth century BC, one of the ancient Pythagoreans showed that some quantities arising in geometry, including the length of the diagonal of a square, when the unit of measurement is the length of the side of the square, cannot be expressed as rational numbers. If the side of a square were divided into five segments of equal lengths, and if the length of the diagonal of the square were equal to that of exactly seven such short segments (which is in fact a reasonable approximation, but not exact), then those short segments would be what Euclid called a "common measure" of the side and the diagonal. What we today would consider a proof that a number is irrational Euclid called a proof that two lengths arising in geometry have no common measure, or are "incommensurable". Euclid included proofs of incommensurability of lengths arising in geometry in his Elements.

In the Rhind Mathematical Papyrus, a pair of legs walking forward marked addition, and walking away subtraction. They were the first known civilization to use negative numbers.


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