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

• 
  `(1)/(z)`

• 
  `(1)/(x + iy)`

• 
  `(1)/(x + iy) * ( x - iy)/(x - iy)`

• 
  `\frac{x - iy}{ {x}^(2) - {i}^(2)y {}^(2) }`

• 
  `\frac{x - iy}{ {x}^(2) + {y}^(2) }`

• 
  `\frac{x}{ {x}^(2) + {y}^(2) } + i \frac{ - y}{ {x}^(2) + {y}^(2) }`

now, we can compare it with general form, where

• 
  `a = \frac{x}{ {x}^(2) + {y}^(2) }`

• 
  `b = \frac{ - y}{ {x}^(2) + {y}^(2) }`

In general form :

• 
  `\frac{x}{ {x}^(2) + {y}^(2) } + i \frac{ - y}{ {x}^(2) + {y}^(2) }`