Square Roots of a Complex Number (Cambridge International A-Level Mathematics)

Square roots of a complex number

How do I find the square root of a complex number?

• The square roots of a complex number will themselves be complex:

  • i.e. if $z^{2}=a+bi$ then $z=c+di$

• We can then square ($c+di$) and equate it to the original complex number ($a+bi$), as they both describe $z^{2}$:

  • $a+b \mathrm{i} =(c+d \mathrm{i} )²$

• Then expand and simplify:

  • $a+bi=c^{2}+2cdi+d^{2}i^{2}$
  • $a+bi=c^{2}+2cdi-d^{2}$

• As both sides are equal we are able to equate real and imaginary parts:

  • Equating the real components: $a=c^{2}-d^{2}$ (1)
  • Equating the imaginary components: $b=2cd$ (2)

• These equations can then be solved simultaneously to find the real and imaginary components of the square root

  • In general, we can rearrange (2) to make $\frac{b}{2d}=c$ and then substitute into (1)
  • This will lead to a quartic equation in terms of d; which can be solved by making a substitution to turn it into a quadratic (see 1.1.5 Further Solving Quadratic Equations (Hidden Quadratics))

• The values of $d$ can then be used to find the corresponding values of $c$, so we now have both components of both square roots ($c+di$)

• Note that one root will be the negative of the other root

  • i.e. $c+di$ and $-c-di$

Worked Example

Exam Hint

• Most calculators used at A-Level can handle complex numbers.
• Once you have found the square roots algebraically; use your calculator to square them and make sure you get the number you were originally trying to square-root!