absolute value (original) (raw)

Absolute value has a different meaning in the case of complex numbers: for a complex number z∈ℂ, the absolute value |z| of z is defined to be x2+y2, where z=x+y⁢i and x,y∈ℝ are real.

All absolute value functions satisfy the defining properties of a valuation, including:

• •
  |a|≥0 for all a∈R, with equality if and only if a=0
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  |a⁢b|=|a|⋅|b| for all a,b∈R
• •

However, in general they are not literally valuations, because valuations are required to be real valued. In the case of ℝ and ℂ, the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by d⁢(x,y):=|x-y|.