Square roots of a complex number

Square roots of a complex number

Let the square root of a + ib be x + iy

That is = √[a+ib] = x + iy where x, y ∈ R

a + ib =  ( x + iy )²  = x²  - y²  + i2xy

Equating real and imaginary parts, we get 

x² - y ² = a and 2xy = b

(x²  + y² )² =  (x² - y² ) ²  + 4x² y²  = a²  + b²

x²  + y² =  √[a²+b²], since x² + y² is positive 

Solving x² + y² = a  and  x²+ y² =√[a²+b²], we get

Since 2xy = b  it is clear that both x and y will have the same sign when b is positive, and x and y have different signs when b is negative.

Formula for finding square root of a complex number

Note

Example 2.17

Find the square root of 6 - 8i.

Solution

We compute |6 - 8i| = √[6² + (-8)²] = 10

and applying the formula for square root, we get