A calculus of the absurd
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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 \(0\))?
\(\seteqnumber{0}{4.}{0}\)\begin{equation} \frac {1} {\rbrackets {\frac {1} {x} }} \end{equation}
Here’s a reasonably good way to find the answer - multiply everything by \(1\).
\(\seteqnumber{0}{4.}{1}\)\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.