Absolute value

It is defined by cases: \(|x|=x\) if \(x\ge0\) and \(|x|=-x\) if \(x<0\). The basic properties follow: \(|x|\ge0\), \(|x|=0\) only if \(x=0\), \(|xy|=|x||y|\) and \(-|x|\le x\le|x|\). A frequent toolbox: \(|x|<c\) is equivalent to \(-c<x<c\), and \(|x-a|<\varepsilon\) is equivalent to \(a-\varepsilon<x<a+\varepsilon\): the points at distance less than \(\varepsilon\) from \(a\).

The central property is the triangle inequality, \(|x+y|\le|x|+|y|\). It is proved by adding \(-|x|\le x\le|x|\) and \(-|y|\le y\le|y|\). A useful consequence is the reverse version, \(\big||x|-|y|\big|\le|x-y|\), which gives lower bounds.

For example, if \(|x|=3\) and \(|y|=4\), then \(|x+y|\) can be anywhere between \(1\) and \(7\); the maximum occurs when both have the same sign. The typical mistake is writing \(|x+y|=|x|+|y|\) or \(|x-y|=|x|-|y|\); both fail in general.

Another constant use: if \(|x|<\varepsilon\) for every \(\varepsilon>0\), then \(x=0\). This argument appears when proving that a limit is unique. The absolute value gives the distance \(d(x,y)=|x-y|\) that will be used to define convergent sequences.