1.13 Complex numbers: polar and exponential formsIB Maths: Applications and Interpretation HL: Revision notes

A complex number $z=a+bi$ is a point on the Argand diagram. Its modulus is $r=|z|=\sqrt{a^2+b^2}$ and its argument $\theta=\arg z$ is the angle from the positive real axis, measured anticlockwise. The principal argument satisfies $-\pi<\theta\leq\pi$. $z=r(\cos\theta+i\sin\theta)=r\,\mathrm{cis}\,\theta.$ To convert from Cartesian to polar form, find $r$, then find $\theta$ from $\tan^{-1}\left|\frac{b}{a}\right|$ and place it in the correct quadrant. Example: $z=-1+\sqrt3\,i$ has $r=2$ and lies in the second quadrant, so $\theta=\pi-\frac{\pi}{3}=\frac{2\pi}{3}$ and $z=2\,\mathrm{cis}\frac{2\pi}{3}$. To go back, use $a=r\cos\theta$ and $b=r\sin\theta$.

Euler's formula says $e^{i\theta}=\cos\theta+i\sin\theta$, so a number in polar form can be written in exponential form $z=re^{i\theta}.$ Examples: $e^{i\pi}=-1$, $2e^{\frac{\pi}{2}i}=2i$ and $-1+\sqrt3\,i=2e^{\frac{2\pi}{3}i}$. The angle $\theta$ must be in radians. Your GDC will convert between $a+bi$, $r\,\mathrm{cis}\,\theta$ and $re^{i\theta}$: check that it is in radian mode, and for a negative real part check the quadrant of the argument it returns. A number is unchanged if you add $2\pi$ to $\theta$, so $e^{i\theta}=e^{i(\theta+2\pi)}$.

In polar or exponential form the algebra is simple. For $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$: $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)},\qquad \frac{z_1}{z_2}=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)},\qquad z^n=r^ne^{in\theta}\ (n\in\mathbb{Z}).$ So moduli multiply (or divide) and arguments add (or subtract). Example: $\left(3e^{\frac{\pi}{4}i}\right)\left(2e^{\frac{\pi}{12}i}\right)=6e^{\frac{\pi}{3}i}$. Powers: $1+i=\sqrt2e^{\frac{\pi}{4}i}$, so $(1+i)^{10}=32e^{\frac{5\pi}{2}i}=32i$. Finding roots of complex numbers is not required.

On an Argand diagram, $z=a+bi$ is the point $(a,b)$ or the vector $\begin{pmatrix}a\\ b\end{pmatrix}$.

• Addition and subtraction are vector addition and subtraction: $z_1+z_2$ is the fourth vertex of the parallelogram formed by $z_1$ and $z_2$, and $z_1-z_2$ is the vector from $z_2$ to $z_1$.
• Multiplication by $w=re^{i\theta}$ is a stretch of scale factor $r$ (centre the origin) combined with an anticlockwise rotation of $\theta$ about the origin. Multiplying by $i=e^{\frac{\pi}{2}i}$ is a rotation of $\frac{\pi}{2}$ with no stretch. Example: multiplying $z$ by $2e^{\frac{\pi}{12}i}$ doubles its distance from the origin and rotates it $\frac{\pi}{12}$ anticlockwise.

A voltage $V=a\cos(\omega t+\phi)$ is the real part of $ae^{i\phi}e^{i\omega t}$. So two sources with the same frequency but different phase shifts add by adding their complex amplitudes: $a_1\cos(\omega t+\phi_1)+a_2\cos(\omega t+\phi_2)=\text{Re}\left[\left(a_1e^{i\phi_1}+a_2e^{i\phi_2}\right)e^{i\omega t}\right]=A\cos(\omega t+B),$ where $Ae^{iB}=a_1e^{i\phi_1}+a_2e^{i\phi_2}$. Example: $10\cos(40t)+20\cos(40t+0.5)$. The amplitude is $10+20e^{0.5i}=27.55\ldots+9.59\ldots i$, with modulus $29.2$ and argument $0.335$, so the total is $29.2\cos(40t+0.335)$. The phase shift is $B$ and the amplitude $A$ is the maximum voltage.