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 x0x\neq 0, as we can’t divide by 0)?

1(1x)\frac{1}{\left(\frac{1}{x}\right)} (4.1)

Here’s a reasonably good way to find the answer - multiply everything by 11.

1(1x)\displaystyle\frac{1}{\left(\frac{1}{x}\right)} =1(1x)×xx\displaystyle=\frac{1}{\left(\frac{1}{x}\right)}\times\frac{x}{x} (4.2)
=x(xx)\displaystyle=\frac{x}{\left(\frac{x}{x}\right)} (4.3)
=x\displaystyle=x (4.4)

We can then apply this principle to more complex fractions.