Question

1/(x+yi) ,make it like a+bi please.
Step by step.

Answer 1

x/(x^2+y^2)-yi/(x^2+y^2) is that right ?

Answer 2

\[\frac{1}{x+iy} = \frac{1}{x+iy} \times \frac{x-iy}{x-iy}\]
now, multiplying the denominators and using the rule:
\[ (m+n) times (m-n) = m^2 - n^2 \]
we get:
\[\frac{1}{x+iy} = \frac{x-iy}{x^2-(iy)^2}\]
\[\frac{1}{x+iy} = \frac{x-iy}{x^2-i^2y^2}\]
\[\frac{1}{x+iy} = \frac{x-iy}{x^2-(-1)y^2}\]
\[\frac{1}{x+iy} = \frac{x-iy}{x^2+y^2}\]
\[\frac{1}{x+iy} = \frac{x}{x^2+y^2} + i\frac{-y}{x^2+y^2}\]