. Square root of complex number formula

Best Answer

square root of complex number a + ib is

√(a + ib) = x + iy,

(x + iy)² = a + ib.

Derivation of Square Root of Complex Number Formula

Let us assume that the square root of complex number a + ib to be x + iy,

i.e. √(a + ib) = x + iy.

squaring both the sides, we have

[√(a + ib)]² = (x + iy)²

⇒ a + ib = x² + (iy)² + i²xy

⇒ a + ib = x² - y² + i2xy [Because i² = -1]

Comparing real and imaginary parts of the above equation, we have

a = x² - y², b = 2xy

We know that (x² + y²)² = (x² - y²)² + 4x²y²

⇒ (x² + y²)² = a² + b²

⇒ x² + y² = √(a² + b²)

Now, we have x² + y² = √(a² + b²) and a = x² - y². Solving for these two values, we have

x = ± √{[√(a² + b²) + a]/2} and y = ± √{[√(a² + b²) - a]/2}

Hence, the square root of complex number a + ib (b ≠ 0) is given by

√(a + ib) = ± (√{[√(a² + b²) + a]/2} + (ib/|b|) √{[√(a² + b²) - a]/2})

Thus, the formula to determine the square root of complex number is: