By Hardy G.H., Wright E.M.

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4. THE RATIONAL NUMBERS 1. Historical. Division, as the inverse of multiplication, cannot be done without restriction in the domain of integers. Fractions, which make division always possible, were already considered in early times. They were never surrounded by such mystery as were the negative numbers, which were thought of as being in some never-never land below "nothing," or the irrational and imaginary numbers, which we still have to discuss. The first systematic treatment of rationals is found in Book VII of EUCLID'S Elements, which deals with the ratios of natural numbers.

In this political turmoil, HIPPASUS is presumed to have played an important role (see IAMBLICHUS [14], p. 77, 6f; also FRITZ [10], HELLER [11]). The treatment of ratios of line-segments had come out of traditionally employed practices in measurement. A segment a of a line had traditionally been measured by laying along the line unit measures e, one after the other, along the line, as many times as were necessary: a ------ = e + ... + e = m . e. mtimes Two segments ao and al are said to be commensurable if they can both be measured, in this sense, with the same unit of measurement e, so that ao m .

C as well. The natural numbers other than zero are thus the integers> 0, the socalled positive integers. A number a is said to be negative whenever -a is positive. Remarks. Every commutative ring R expressible as a disjoint union R = -p U {OJ U P where P is additively and multiplicatively closed, can be totally ordered by the relation a::; b if b - a E P U {OJ. Historically, it was also DEDEKIND who introduced the idea of defining integers by pairs from N x N. In a letter from the 82-year-old mathematician written in 1913 to a former student, DEDEKIND ([10], p.