<P> For another axiomatization of R, see Tarski's axiomatization of the reals . </P> <P> The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number, in this case π . For details and other constructions of real numbers, see construction of the real numbers . </P> <P> A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero . Real numbers are used to measure continuous quantities . They may be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823122147...The ellipsis (three dots) indicates that there would still be more digits to come . </P> <P> More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property . The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication . The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound . The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property . </P>

Example of real numbers and not real numbers