| real number One of the infinitely divisible range of values between positive and negative , used to represent continuous physical quantities such as distance, time and temperature. Between any two real numbers there are infinitely many more real numbers. The ("counting numbers") are real numbers with no fractional part and real numbers ("measuring numbers") are with no imaginary part. Real numbers can be divided into and . Real numbers are usually represented (approximately) by computers as numbers. Strictly, real numbers are the of the of under the "~", where a ~ b if and only if a-b is with limit 0. The real numbers are the minimal containing the rational field. A sequence, r, of rationals (i.e. a function, r, from the to the rationals) is said to be Cauchy precisely if, for any tolerance delta there is a size, N, beyond which: for any n, m exceeding N, | r[n] - r[m] | < delta A Cauchy sequence, r, has limit x precisely if, for any tolerance delta there is a size, N, beyond which: for any n exceeding N, | r[n] - x | < delta (i.e. r would remain Cauchy if any of its elements, no matter how late, were replaced by x). It is possible to perform addition on the reals, because the equivalence class of a sum of two sequences can be shown to be the equivalence class of the sum of any two sequences equivalent to the given originals: ie, a~b and c~d implies a+c~b+d; likewise a.c~b.d so we can perform multiplication. Indeed, there is a natural of the rationals in the reals (via, for any rational, the sequence which takes no other value than that rational) which suffices, when extended via continuity, to import most of the algebraic properties of the rationals to the reals. (1997-03-12) |
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