# Finding square roots of complex numbers

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

Method 1: calculator

Find the modulus of the number to be square-rooted, by mental arithmetic or the calculate: |w| = 34

Use the calculator to find the argument of w

calculate $\mathrm{Ans}\div 2$

calculate $\sqrt{\mathrm{mod}} \, \angle \, \mathrm{Ans}$

convert back to cartesian form: in this case you get 3+5i

and check (just to make sure no rounding error on the calculator)

and write the negative of that answer (in this case -3-5i) as the other square root

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)

See below for calculator images.

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) – images

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.