Square Roots of a Complex Number (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

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 $z^2=a+b\mathrm{i}$ then $z=c+d\mathrm{i}$

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

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

• Then expand and simplify:

  • $a+b\mathrm{i}=c^2+2cd\mathrm{i}+d^2{\mathrm{i}}^2$

  • $a+b\mathrm{i}=c^2+2cd\mathrm{i}-d^2$

• 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 ($c+d\mathrm{i}$)

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

  • i.e.  $c+d\mathrm{i}$  and  $-c-d\mathrm{i}$