Modulus-Argument (Polar) Form (DP IB Applications & Interpretation (AI): HL): Revision Note

Modulus-argument (polar) form

What is modulus-argument (polar) form?

• The modulus-argument (polar) form of a complex number $z$ is

  • $z=r\left(\cos \theta +\mathrm{isin}\theta \right)$

    • sometimes written $z=r\mathrm{cis}\theta$

  • where

    • $r=\left|z\right|$

    • $\theta =\arg z$

• e.g. $z=1+\sqrt{3}\mathrm{i}$ has a modulus of 2 and an argument of $\frac{\pi }{3}$

  • so $z=2\left(\cos \frac{\pi }{3}+\mathrm{isin}\frac{\pi }{3}\right)$

• You can also convert back to Cartesian form

• e.g. $z=4\left(\cos \frac{\pi }{4}+\mathrm{isin}\frac{\pi }{4}\right)$ is $4\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)=2\sqrt{2}+2\sqrt{2}\mathrm{i}$

How do I write a complex conjugate in modulus-argument (polar) form?

• The complex conjugate of $z=r\left(\cos \theta +\mathrm{isin}\theta \right)$ is

  • $z^{*}=r\left(\cos \left(-\theta \right)+\mathrm{isin}\left(-\theta \right)\right)$

    • also written as $r\mathrm{cis}\left(-\theta \right)$

• The modulus is the same

  • but the argument changes sign

• This works because, in general,

  • $\sin \left(-\theta \right)=-\sin \theta$ and $\cos \left(-\theta \right)=\cos \theta$

  • so $r\left(\cos \left(-\theta \right)+\mathrm{isin}\left(-\theta \right)\right)=r\left(\cos \theta -\mathrm{isin}\theta \right)=x-\mathrm{i}y$

How do I multiply complex numbers in modulus-argument (polar) form?

• To multiply two complex numbers in modulus-argument (polar) form

  • multiply their moduli

    • $\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|$

  • and add their arguments

    • $\arg \left(z_1{z}_2\right)=\arg z_1+\arg z_2$

• So if $z_1=r_1\mathrm{cis}{\theta }_1$ and $z_2=r_2\mathrm{cis}{\theta }_2$

  • then $z_1{z}_2=r_1{r}_2\mathrm{cis}\left({\theta }_1+{\theta }_2\right)$

• These rules work for

  • multiplying more than two complex numbers, e.g. $z_1{z}_2{z}_3$

  • powers of complex numbers, e.g. $z^2$, $z^3$, ...

    • using $z^2=z\times z$ etc

How do I divide complex numbers in modulus-argument (polar) form?

• To divide two complex numbers in modulus-argument (polar) form

  • divide their moduli

    • $\left|\frac{z_1}{z_2}\right| =\frac{\left|z_1\right|}{\left|z_2\right|}$

  • and subtract their arguments

    • $\arg \left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2$

• So if $z_1=r_1\mathrm{cis}{\theta }_1$ and $z_2=r_2\mathrm{cis}{\theta }_2$

  • then $\frac{z_1}{z_2}=\frac{r_1}{r_2}\mathrm{cis}\left({\theta }_1-{\theta }_2\right)$

What if the new argument is out of range?

• Sometimes the new argument does not lie in the range $-\pi <\theta \le \pi$

  • so adjust it by either adding or subtracting $2\pi$

  • E.g. If ${\theta }_1=\frac{2\pi }{3}$ and ${\theta }_2=\frac{\pi }{2}$  then  ${\theta }_1+{\theta }_2=\frac{7\pi }{6}$ 

  • This is currently not in the range $-\pi <\theta \le \pi$

  • Subtracting $2\pi$ from $\frac{7\pi }{6}$ gives $-\frac{5\pi }{6}$ which

    • is in range

    • and represents the same angle