Finding square roots of complex numbers

Example: find w so that w^2 = -16 + 30i

Method 1: calculator

Convert to r ∠ θ form: w2 ≈ 34 cis 2.06

Mentally, square-root the modulus and halve the argument

w ≈ √34 cis 1.03

Convert back to a+bi form: w ≈ 3+5i (round your answer back to a neat value, to correct the calculator error introduced when halving a decimal approximation of the argument)

Check that you’ve rounded back to an exactly correct answer:

(3+5i)2 = -16 + 30i

Write your answer: w = ± (3+5i)

(See below for a tweak to this method)

Method 2: Equate modulus and real part

Let square root = w = a + bi

Then |w^2| = |w|^2 = a^2 + b^2 = 34

\mathrm{Re}(w^2) = a^2 - b^2 = -16

Solve those simultaneous equations for a2 and b2

a2=9 and b2=25

Only problem now is to work out the valid signs for a and b. \mathrm{Im}(w^2) = 2ab . In this case a and b must have the same sign (both negative or both positive) because \mathrm{Im}(w^2) > 0 . (If \mathrm{Im}(w^2) < 0 then a and b would have to have opposite signs, one positive, one negative).

So w = ± (3+5i)

Method 3: Equate real and imaginary parts (the method in the old Edexcel textbook)

Let square root = w = a + bi

Then \mathrm{Re}(w^2) = a^2 - b^2 = -16

\mathrm{Im}(w^2) = 2ab = 30

Substitute b = \frac{15}{a} from the second equation into the first

a^2 - \frac{225}{a^2} = -16

Multiply up to get a quadratic equation in a^2

a^4 +16a^2 - 225 = 0

Solve: a2 = 9, so a = ± 3, and b = ± 5

So w = ± (3+5i)

Method 1 (calculator) tweaked

You can minimise the approximation problem in the calculator method by using the calculator rather than your head to halve the argument. As in these images (poor quality, sorry):

This will usually give you the exact answer in the exact form you want it. The likely exception is if you’re finding something like √(−1 + 2√2 i), where the answer is (1 + √2 i).

The calculator will give you (1 + 1.414213562 i), and it won’t convert that to (1 + √2 i) even if you press the S-D key.

Workaround: if you get an answer for real or imaginary part of the square root which isn’t a whole number, square your answer to see if it’s an exact square root of a whole number (in A level, it pretty much always will be).

It’s worth memorising the facts that

√2 ≈ 1.414

√3 ≈ 1.732

√5 ≈ 2.236

Often come in useful, like it comes in useful to be able to remember your way home from school, or to a friend’s house, without having to look at Google Maps.