Corollary: Properties of the Absolute Value

(related to Definition: Absolute Value of Real Numbers (Modulus))

The absolute value has the following properties1:

  1. \(|-x|=|x|\) for all \(x\in\mathbb R\).
  2. \(|xy|=|x||y|\) for all \(x,y\in\mathbb R\).
  3. \(|\frac xy|=\frac{|x|}{|y|}\) for all \(x,y\in\mathbb R\), \(y\neq 0\).

Proofs: 1

Proofs: 1 2 3 4 5

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  1. Please note that another, very important property of the absolute value is the triangle inequality \(|x+y|\le |x|+|y|\) for all \(x,y\in\mathbb R\), which is proven here