We now define \(\frac{z}{w}\), provided \(w \neq 0\).
Let&nbsp;\(z = x + iy\) and \(w = u + iv\). Then
\[\begin{aligned} \frac{z}{w} &= \frac{x + iy}{u + iv} \\ &= \frac{x + iy}{u + iv} \cdot \frac{u - iv}{u - iv} \\ &= \frac{(x + iy)(u - iv)}{(u + iv)(u - iv)} \\ &= \frac{(xu + yv) + i(yu - xv)}{u^2 - (iv)^2} \\ &= \frac{(xu + yv) + i(yu - xv)}{u^2 + v^2} \end{aligned}\]
Notice that this working is very similar to the action of rationalizing a surd:
\[\frac{1}{3 - \sqrt{2}} = \frac{1}{3 - \sqrt{2}} \cdot \frac{3 + \sqrt{2}}{3 + \sqrt{2}} = \frac{3 + \sqrt{2}}{3^2 - (\sqrt{2})^2} = \frac{3 + \sqrt{2}}{9 - 2} = \frac{1}{7}(3 + \sqrt{2})\]
The only difference is that operationally \(i^2 = -1\) and it turns the difference of squares in the denominator term into the sum of squares! This is remarkable!

We now define \(\frac{z}{w}\), provided \(w \neq 0\).

Let \(z = x + iy\) and \(w = u + iv\). Then

\[\begin{aligned} \frac{z}{w} &= \frac{x + iy}{u + iv} \\ &= \frac{x + iy}{u + iv} \cdot \frac{u - iv}{u - iv} \\ &= \frac{(x + iy)(u - iv)}{(u + iv)(u - iv)} \\ &= \frac{(xu + yv) + i(yu - xv)}{u^2 - (iv)^2} \\ &= \frac{(xu + yv) + i(yu - xv)}{u^2 + v^2} \end{aligned}\]

Notice that this working is very similar to the action of rationalizing a surd:

\[\frac{1}{3 - \sqrt{2}} = \frac{1}{3 - \sqrt{2}} \cdot \frac{3 + \sqrt{2}}{3 + \sqrt{2}} = \frac{3 + \sqrt{2}}{3^2 - (\sqrt{2})^2} = \frac{3 + \sqrt{2}}{9 - 2} = \frac{1}{7}(3 + \sqrt{2})\]

The only difference is that operationally \(i^2 = -1\) and it turns the difference of squares in the denominator term into the sum of squares! This is remarkable!

Complex division

Complex division