Question

The multiplicative inverse of (1 + i) is ____________.

Answer 1

For z = 1 + i
Let us suppose multiplicative inverse of 1 + i is a + ib
then (1 + i) (a + ib) = 1
i.e a + ib + ai + i²b = 1
i.e a + ib + ia – b = 1
i.e a – b + i(a + b) = 1 + i0
On comparing, real and imaginary part, we get
a – b = 1 and a + b = 0
i.e a – b = 1 and a = – b
i.e a + a = 1
$$\begin{gathered} \mathrm{i}.\mathrm{e}a=\frac{1}{2} \\ b=-\frac{1}{2} \end{gathered}$$
i.e multiplicative inverse of 1 + i is $\frac{1}{2}-\frac{i}{2}$