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mathematics One of the infinitely divisible range of values
between positive and negative infinity
, used to represent
continuous physical quantities such as distance, time and
Between any two real numbers there are infinitely many more
real numbers. The integers
("counting numbers") are real
numbers with no fractional part and real numbers ("measuring
Real numbers are usually represented (approximately) by
containing the rational field.
A sequence, r, of rationals (i.e. a function, r, from the
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
| 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 embedding
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.