A calculus of the absurd

4.8.3 Algebraically

To write the modulus function as an algebraic expression, we’re interested in functions which are always positive. The one which (probably?) springs to mind is squaring. Using this, we can define the modulus function as 2626 Note that this is just the square root of the magnitude. It is also important to never give into the temptation to say "oh this is just the square root of a square number, it’s just the original number." This is wrong. In the expression \(\sqrt {x^2}\) when we square \(x\), both \(x\) and \(-x\) are mapped to the same value, so stating that the square root of the square is just the original value is not true for negative numbers.

\begin{equation} \label {modulus as sqrt squared} \abs {x} = \sqrt {x^2} \end{equation}

This is really helpful for solving some equations.