Question

let z be any non zero complex number. Express 1/z in the form a+ib.​

Answer 1

let z be a complex number and z = x + iy

now, let's solve for 1/z

• $\dfrac{1}{z}$

• $\dfrac{1}{x + iy}$

• $\dfrac{1}{x + iy} \times \dfrac{ x - iy}{x - iy}$

• $\dfrac{x - iy}{ {x}^{2} - {i}^{2}y {}^{2} }$

• $\dfrac{x - iy}{ {x}^{2} + {y}^{2} }$

• $\dfrac{x}{ {x}^{2} + {y}^{2} } + i \dfrac{ - y}{ {x}^{2} + {y}^{2} }$

now, we can compare it with general form, where

• $a = \dfrac{x}{ {x}^{2} + {y}^{2} }$

• $b = \dfrac{ - y}{ {x}^{2} + {y}^{2} }$

In general form :

• $\dfrac{x}{ {x}^{2} + {y}^{2} } + i \dfrac{ - y}{ {x}^{2} + {y}^{2} }$