Real Numbers System (ℝ)
- a type of number class that can be expressed as an infinite decimal expansion
- are used in measurements of continuously varying quantities such as size and time, in contrast to the natural numbers 1, 2, 3, …, arising from counting
Real Numbers - Axioms
A non-empty set ℝ together with operations +,· and ordering ≤ is called the real numbers if it satisfies:
- (ℝ, +, 0) is an abelian group
- (ℝ\{0}, ·, 1) is an abelian group
- · is distributive over : 𝑥·(𝑦 + 𝑧) = 𝑥·𝑦 + 𝑥·𝑧
- ≤ is a total order, compatible with + and · (i.e. Archimedean property)
- every Cauchy sequence is a convergent sequence
Given a non-empty set ℝ, we define the absolute value operation as follows:
With the absolute value operation we have the following properties:
- |𝑥 · 𝑦| = |𝑥| · |𝑦|
- |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|