Complex Numbers & the Complex Plane
Complex Numbers & the Complex Plane
The complex numbers $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$ extend $\mathbb{R}$ to an algebraically closed field: every polynomial has a root. The complex plane (Argand diagram) represents $z=a+bi$ as the point $(a,b)$, with polar form $z=re^{i\theta}=r(\cos\theta+i\sin\theta)$ where $r=|z|$ and $\theta=\arg z$. Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ unifies exponential and trigonometric functions.
Complex Numbers & Polar Form
$z=x+iy\in\mathbb{C}$ has real part $\text{Re}(z)=x$, imaginary part $\text{Im}(z)=y$, modulus $|z|=\sqrt{x^2+y^2}$, argument $\arg z=\arctan(y/x)$ (adjusted for quadrant), and conjugate $\bar{z}=x-iy$. Key identities: $z\bar{z}=|z|^2$; $|z_1 z_2|=|z_1||z_2|$; $\arg(z_1 z_2)=\arg z_1+\arg z_2$ (mod $2\pi$). Polar form: $z=re^{i\theta}$; multiplication rotates and scales: $z_1 z_2=r_1 r_2 e^{i(\theta_1+\theta_2)}$.
De Moivre's Theorem
For any $n\in\mathbb{Z}$ and $z=re^{i\theta}$: $(re^{i\theta})^n=r^n e^{in\theta}$, i.e., $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$. The $n$-th roots of unity are $\omega_k=e^{2\pi ik/n}$ for $k=0,1,\ldots,n-1$, equally spaced on the unit circle. The $n$-th roots of $w=Re^{i\phi}$ are $R^{1/n}e^{i(\phi+2\pi k)/n}$ for $k=0,\ldots,n-1$.
Example 1
Express $(1+i)^{10}$ in Cartesian form.
Solution: $1+i=\sqrt{2}e^{i\pi/4}$. So $(1+i)^{10}=(\sqrt{2})^{10}e^{i10\pi/4}=32e^{i5\pi/2}=32e^{i\pi/2}=32i$.
Example 2
Find all cube roots of $-8$.
Solution: $-8=8e^{i\pi}$. Cube roots: $8^{1/3}e^{i(\pi+2\pi k)/3}=2e^{i\pi/3},2e^{i\pi},2e^{i5\pi/3}$. These are $2(\frac{1}{2}+\frac{\sqrt{3}}{2}i)=1+\sqrt{3}i$; $-2$; $1-\sqrt{3}i$.
Practice
- Prove $|z_1+z_2|\leq|z_1|+|z_2|$ (triangle inequality) using $|z|^2=z\bar{z}$.
- Show $\cos 3\theta=4\cos^3\theta-3\cos\theta$ using De Moivre's theorem.
- Find all solutions to $z^4+1=0$ in Cartesian form.
- Prove the identity $e^{i\pi}+1=0$ (Euler's identity) from the series definition of $e^z$.
Show Answer Key
1. $|z_1+z_2|^2=(z_1+z_2)\overline{(z_1+z_2)}=|z_1|^2+z_1\bar{z}_2+\bar{z}_1 z_2+|z_2|^2=|z_1|^2+2\text{Re}(z_1\bar{z}_2)+|z_2|^2\le|z_1|^2+2|z_1||z_2|+|z_2|^2=(|z_1|+|z_2|)^2$.
2. $(\cos\theta+i\sin\theta)^3=\cos3\theta+i\sin3\theta$. Expand the LHS using the binomial theorem: $\cos^3\theta+3i\cos^2\theta\sin\theta-3\cos\theta\sin^2\theta-i\sin^3\theta$. Real part: $\cos3\theta=\cos^3\theta-3\cos\theta\sin^2\theta=4\cos^3\theta-3\cos\theta$.
3. $z^4=-1=e^{i(\pi+2k\pi)}$, so $z=e^{i(\pi+2k\pi)/4}$ for $k=0,1,2,3$. Solutions: $\frac{\sqrt{2}}{2}(1+i),\;\frac{\sqrt{2}}{2}(-1+i),\;\frac{\sqrt{2}}{2}(-1-i),\;\frac{\sqrt{2}}{2}(1-i)$.
4. $e^{i\pi}=\sum_{n=0}^\infty\frac{(i\pi)^n}{n!}=\cos\pi+i\sin\pi=-1+0i=-1$, so $e^{i\pi}+1=0$.