quality:corruption
Construction of the real numbers

The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (R {\displaystyle \mathbb {R} }), together with two binary operations denoted + and ⋅, and a total order denoted ≤. The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique complete ordered field, in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers Q {\displaystyle \mathbb {Q} }) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least upper bound property (see below).