A calculus of the absurd

Chapter 4 Concrete algebra

This is not a chapter about “modern” algebra, but algebra (usually about real numbers, or some subset thereof).

4.1 Fractions

4.1.1 Reciprocals of fractions

Fractions can be surprisingly confusing. For example, what is the value of the expression directly below (assuming \(x \ne 0\), as we can’t divide by \(0\))?

\begin{equation} \frac {1} {\rbrackets {\frac {1} {x} }} \end{equation}

Here’s a reasonably good way to find the answer - multiply everything by \(1\).

\begin{align} \frac {1} {\rbrackets {\frac {1} {x} }} & = \frac {1} {\rbrackets {\frac {1} {x} }} \times \frac {x}{x} \\ & = \frac {x} {\rbrackets {\frac {x} {x} }} \\ & = x \end{align}

We can then apply this principle to more complex fractions.