Complex Analysis: Theory and Applications · Problem 1.2.12

Complex Numbers — Howell & Matthews

Mathematics — Complex Analysis · multiplicative-inverse-verification

Problem

Verify that, if

z = (x, y)

, with

x

and

y

not both 0, then

z^{-1} = \dfrac{(1,0)}{z}

\bigl(

i.e.,

z^{-1} = \dfrac{1}{z}\bigr)

. Hint: Let

z = (x, y)

and use the (ordered pair) definition for division to compute

z^{-1} = \dfrac{(1,0)}{(x,y)}

. Then, with the result you obtained, use the (ordered pair) definition for multiplication to confirm that

zz^{-1} = (1, 0) = 1