number

The natural numbers are 0, 1, 2, 3… The integers are…–3, –2, –1, 0, 1, 2, 3…The rationals, as their name implies, measure ratios: any number that can be written as a/b, where a and b are integers, and b ? 0 (in other words, values of x that give solutions of an equation b x – a = 0 where a and b are integers). The real numbers contain all the rational numbers, but also numbers such as √2 or π that are not rational. The reals can be thought of as the points of a line, with the integers equally spaced along the line. Every real number can be expressed as an infinite decimal. There are more reals than rationals ( Cantor's theorem ), and a number of rigorous ways of defining the set of reals (see also diagonal argument, Dedekind cut ). Transfinite numbers measure the size of infinite sets. The first transfinite number is aleph-null, written א₀, which measures the set of natural numbers. See also continuum hypothesis . The cardinal numbers measure the size of sets: the cardinality of a set is the number of its elements. The ordinal numbers measure the length of a well-ordering (see ordering relation ). The difference is not apparent in finite cases, where an ordering is bigger simply if it has more members, but in transfinite cases the notions come apart. Thus the natural numbers can be ordered in the standard way: 1, 2, 3, 4…. The length of this ordering is ω. But they can also be ordered in ways that themselves tail off to infinite successions 1, 3, 5…; 2, 4, 6…. Even although this ordering contains just the same elements, there is no order-preserving one-to-one correspondence between members of the two orderings, and it is of greater length, 2ω.