[Linear Algebra] Why is it useful to write z⁻¹ = z̅ / |z|² ?

In developing general properties of complex numbers, it is often helpful to proceed without having to decompose the complex numbers into their real and imaginary components. For instance, when proving that |v + w| ≤ |v| + |w|, we can get at this result by just working with v, w, |v|, |w|, and the complex conjugates of v and w. We don’t need to use the forms v = a + ib, w = c + id. In this approach, z⁻¹ = z̅ / |z|² is one of the results which has a role to play.