Complex Numbers

Definition

The imaginary unit $i$ is defined as one of the two numbers (the other being $-i$) satisfying $z^2+1=0$, i.e. $i=\sqrt{-1}$. A complex number $z$ and its complex conjugate $\bar{z}$ are

$$z=x+iy,\bar{z}=x-iy,$$

where $x,y\in \mathbb{R}$. We write $x=\mathrm{Re}z$ and $y=\mathrm{Im}z$.

Intuition

The real line is extended to a two-dimensional plane: the real part moves left/right, the imaginary part moves up/down. Multiplication by $i$ rotates a point 90° counterclockwise, and the polar form makes this geometric action explicit — multiplying two complex numbers adds their angles and multiplies their magnitudes.

Formal Description

Extracting real and imaginary parts:

$$\mathrm{Re}z=\frac{1}{2}(z+\bar{z}),\mathrm{Im}z=\frac{1}{2i}(z-\bar{z}).$$

Modulus (absolute value):

$$|z|=(z\bar{z}{)}^{1/2}=\sqrt{x^2+y^2}.$$

Arithmetic: Addition, subtraction, and multiplication follow from $i^2=-1$. Division uses $\frac{z}{w}=\frac{z\bar{w}}{w\bar{w}}$.

Polar form: Representing $z$ in the complex plane with $x=r\cos \theta$, $y=r\sin \theta$ gives

$$z=x+iy=r(\cos \theta +i\sin \theta )=r{e}^{i\theta },$$

where $r=|z|$ and $\theta =\arg z$. The angle $\theta$ is not unique; the principal value satisfies $-\pi <\theta \le \pi$.

Polar multiplication: if $z_1=r_1{e}^{i{\theta }_1}$ and $z_2=r_2{e}^{i{\theta }_2}$, then

$$z_1{z}_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}.$$

Multiplying by $e^{i{\theta }_2}$ (with $r_2=1$) rotates $z_1$ counterclockwise by ${\theta }_2$.

Euler’s formula (derived from the Taylor series of $e^x$, $\cos x$, $\sin x$):

$$e^{i\theta }=\cos \theta +i\sin \theta ,e^{-i\theta }=\cos \theta -i\sin \theta .$$

Cosine and sine from exponentials:

$$\cos x=\frac{e^{ix}+e^{-ix}}{2},\sin x=\frac{e^{ix}-e^{-ix}}{2i}.$$

Euler’s identity (setting $\theta =\pi$):

$$e^{i\pi }+1=0.$$

Applications

Complex numbers are used to represent 2D rotations and scaling in a single multiplication, to analyse AC circuits via phasors, and to solve polynomial equations over $\mathbb{C}$. Fourier analysis relies on $e^{i\omega t}$ as a basis of oscillatory functions.

Trade-offs

$\mathbb{C}$ is not an ordered field — there is no consistent way to compare complex numbers as greater or less than one another. The argument $\theta$ is multi-valued; choosing a branch cut (e.g. the principal value) introduces a discontinuity.

Links

• Trigonometric Functions
• Taylor Series — Euler’s formula is derived from the Taylor expansions of $e^x$, $\cos x$, and $\sin x$