# A calculus of the absurd

### Chapter 4 Concrete algebra

This is not a chapter about “modern” algebra, but algebra (usually about the )

#### 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 $$0$$)?

$$\frac {1} {\rbrackets {\frac {1} {x} }}$$

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.