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

Square roots of a complex number using exponential form

Previously we looked at how to find the square roots of a complex number in Cartesian form (a+bi). We can also find square roots using polar ($r\left(\cos \theta +\mathrm{i}\mathrm{ }\sin \theta \right)$) and exponential form ($r{e}^{\mathrm{i}\theta }$).

How do I find a square root of a complex number in exponential form?

• Let $w=r{e}^{\mathrm{i}\theta }$ be a square root of $z=n{e}^{\mathrm{i\alpha }}$

  • $w\times w=z$

  • ${\left(r{e}^{\mathrm{i}\theta }\right)}^2=n{e}^{\mathrm{i\alpha }}$  

• Applying rules of indices: 

  • $r^2{e}^{2i\theta }=n{e}^{\mathrm{i}\alpha }$

• Comparing the coefficients of e (moduli) and powers of e (arguments) we can state:

  • $n=r^2$

    • $r=\sqrt{n}$

  • $\alpha =2\theta$

    • $\theta =\frac{\alpha }{2}$

• A square root of $z=n{e}^{\mathrm{i\alpha }}$ is $w=\sqrt{n}{e}^{\frac{\mathrm{\alpha }}{2}\mathrm{i}}$

  • Square root the modulus

  • Halve the argument

How do I find the second square root?

• The second square root is the first one multiplied by -1

  • $w=\sqrt{n}{e}^{\frac{\mathrm{\alpha }}{2}\mathrm{i}}$ and $-w=-\sqrt{n}{e}^{\frac{\mathrm{\alpha }}{2}\mathrm{i}}$

• We can write the second one in polar or exponential form too

• Adding 2π to the argument of a complex number still gives the same complex number

  • So we could also say that $n{e}^{\mathrm{i}(\alpha +2\pi )}=r^2{e}^{2\mathrm{i}\theta }$

  • Therefore $\alpha +2\pi =2\theta$ is another possibility

    • $\theta =\frac{\alpha }{2}+\pi$

• So the two square roots of ($n{e}^{\mathrm{i}\alpha }$) are:

  • $z_1=\sqrt{n} e^{\frac{\alpha }{2}\mathrm{i}}$

  • $z_2=\sqrt{n} e^{\left(\frac{\alpha }{2}+\mathrm{\pi }\right)\mathrm{i}}$

• You should notice that the two square roots are π radians apart from each other

  • This is always true when finding square roots

• And if you were to write them in cartesian form they would be negatives of one another

  • E.g. a+bi and -a-bi

  • This is also always true when finding square roots

• This approach can be extended to find higher order roots (e.g. cube roots) by knowing that the nth roots will be $\frac{2\mathrm{\pi }}{\mathrm{n}}$ radians apart from each other, however this is beyond the specification of this course