Square Root of Complex Number

To find

$\sqrt{a+bi}$

, assume the answer has the form

$x + yi$

and square both sides:

$(x+yi)^2 = x^2 - y^2 + 2xyi = a + bi$

Matching real and imaginary parts gives

$x^2 - y^2 = a$

and

$2xy = b$

. Combined with the modulus relation

$x^2 + y^2 = \sqrt{a^2+b^2}$

, you can solve for

$x^2$

and

$y^2$

directly, then pick the signs of

$x$

and

$y$

so that

$2xy$

matches the sign of

$b$

.

Example:

$\sqrt{3+4i}$

. Here

$a=3, b=4$

, modulus

$=5$

. Then

$x^2 = \tfrac{5+3}{2} = 4$

and

$y^2 = \tfrac{5-3}{2} = 1$

, so

$x=2, y=1$

(same sign, since

$b>0$

). Check:

$(2+i)^2 = 4+4i-1 = 3+4i$

. ✓