Editing Multiplication (section)

=== Product of two real numbers ===

There are several equivalent ways to define formally the real numbers; see [[Construction of the real numbers]]. The definition of multiplication is a part of all these definitions. 

A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by [[rational number]]s. A standard way for expressing this is that every real number is the [[least upper bound]] of a set of rational numbers. In particular, every positive real number is the least upper bound of the [[truncation]]s of its infinite [[decimal representation]]; for example, <math>\pi</math> is the least upper bound of <math>\{3,\; 3.1,\; 3.14,\; 3.141,\ldots\}.</math>

A fundamental property of real numbers is that rational approximations are compatible with [[arithmetic operation]]s, and, in particular, with multiplication. This means that, if {{mvar|a}} and {{mvar|b}} are positive real numbers such that <math>a=\sup_{x\in A} x</math> and <math>b=\sup_{y\in B} y,</math> then <math>a\cdot b=\sup_{x\in A, y\in B}x\cdot y.</math> In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the [[sequence]]s of their decimal representations.

As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in {{slink|#Product of two integers}}. The construction of the real numbers through [[Cauchy sequence]]s is often preferred in order to avoid consideration of the four possible sign configurations.