Let \(x,y\in \mathbb R\). Based on the ordering relation for real numbers, we define a function \(|~|:\mathbb R\times \mathbb R\mapsto \mathbb R\) by
\[|x-y| := \begin{cases} x-y & \text{ if } x\ge y \\ y-x & \text{ if } x < y \end{cases}\]
and call it the distance of \(x\) and \(y\).
The distance of any real number \(x\) from \(0\)
\[|x| := |x-0|= \begin{cases} x & \text{ if } x\ge 0 \\ -x & \text{ if } x < 0 \end{cases}\]
is called the absolute value of \(x\).
Corollaries: 1
Corollaries: 1
Definitions: 2 3 4 5 6 7 8
Examples: 9
Lemmas: 10
Parts: 11
Proofs: 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Propositions: 26 27 28 29 30 31 32 33
Theorems: 34
