The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality when applied to a matrix, it denotes its determinant. In programming languages and computational software packages, the absolute value of x is generally represented by abs( x), or a similar expression. Other names for absolute value include numerical value and magnitude. The notation | x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English. In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for the complex absolute value, and it was borrowed into English in 1866 as the Latin equivalent modulus. 2.2.1 Proof of the complex triangle inequality.
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