Preliminaries
REAL NUMBERS
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 in theory 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) indicate that there would still be more digits to come.
More formally, 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 nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely in the context of the usual set theory, and allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real coefficients has a real root, and that if you add the square root of −1 to the real numbers, obtaining the complex numbers, the resulting field is algebraically closed.