Complex Numbers for Fourier

Why This Matters

The Fast Fourier Transform, the Fourier Neural Operator, and the spectral theory of operators are all written in the language of complex exponentials. If you don't know what $e^{i\theta}$ means, the page that tells you the FFT decomposes a sequence into "frequency components" reads as ritual.

This page exists to ground that ritual. It is not complex analysis. There is no Cauchy-Riemann, no contour integration, no residue theorem. The goal is narrower: enough vocabulary and a few identities so that $e^{i\omega t}$, $|z|$, and $\arg(z)$ all parse without friction.

The Field

Definition

Complex Numbers

The complex numbers $\mathbb{C}$ are the set of formal expressions $z = a + b i$ with $a, b \in \mathbb{R}$, equipped with addition and multiplication that extend the real-number rules subject to the single relation $i^2 = -1$. Concretely, for $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$: $z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2) i,$ $z_1 \cdot z_2 = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) i.$ Here $a$ is the real part $\mathrm{Re}(z)$ and $b$ is the imaginary part $\mathrm{Im}(z)$.

The symbol $i$ is the imaginary unit. It is defined by the property $i^2 = -1$. Nothing else. It is not a "number you can't see"; it is a formal symbol that, once you adjoin it to $\mathbb{R}$ with this single rule, gives you a field. That field happens to be algebraically closed: every nonconstant polynomial with complex coefficients has a root in $\mathbb{C}$. This is the fundamental theorem of algebra and is why $\mathbb{C}$, not $\mathbb{R}$, is the natural setting for spectra of matrices.

Definition

Modulus and Argument

For $z = a + b i$:

• The conjugate is $\bar{z} = a - b i$.
• The modulus (absolute value) is $|z| = \sqrt{a^2 + b^2} = \sqrt{z \bar{z}}$.
• The argument $\arg(z)$ is the angle $\theta \in (-\pi, \pi]$ such that $z = |z|(\cos\theta + i \sin\theta)$, defined for $z \neq 0$.

Identities: $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$, $|z_1 z_2| = |z_1||z_2|$, and $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \pmod{2\pi}$.

Geometrically, $\mathbb{C}$ is $\mathbb{R}^2$ with extra structure. The modulus is the Euclidean norm. The argument is the angle from the positive real axis. Multiplication by a complex number $w$ rotates by $\arg(w)$ and scales by $|w|$. This rotation-and-scaling picture is the source of all the geometric intuition you will ever need.

Polar Form and Euler's Formula

The polar form $z = r(\cos\theta + i \sin\theta)$, with $r = |z|$ and $\theta = \arg(z)$, makes multiplication trivial: $z_1 z_2 = r_1 r_2 \big(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\big)$. The compact form is Euler's formula.

Theorem

Euler's Formula

Statement

For every $\theta \in \mathbb{R}$, $e^{i\theta} = \cos\theta + i \sin\theta.$ In particular $e^{i\pi} = -1$, and $|e^{i\theta}| = 1$ for all $\theta$.

Intuition

Define $f(\theta) = \cos\theta + i \sin\theta$. Differentiate: $f'(\theta) = -\sin\theta + i \cos\theta = i \cdot f(\theta)$, with $f(0) = 1$. The unique solution to $y' = i y$, $y(0) = 1$ is $y(\theta) = e^{i\theta}$, where the complex exponential is defined by its power series. So $f(\theta) = e^{i\theta}$.

Proof Sketch

The complex exponential is defined by the absolutely convergent series $e^{z} = \sum_{n=0}^{\infty} z^n / n!$ for $z \in \mathbb{C}$. Substituting $z = i\theta$ and grouping by parity, the even powers contribute the Taylor series of $\cos\theta$ and the odd powers contribute $i \sin\theta$. This proves $e^{i\theta} = \cos\theta + i\sin\theta$ termwise.

Why It Matters

Every sinusoid is the real part of a complex exponential: $\cos(\omega t) = \mathrm{Re}(e^{i\omega t})$. Linear time-invariant systems acting on sinusoids become diagonal in the basis of complex exponentials, which is the entire reason Fourier methods exist. The eigenfunction property $\frac{d}{dt} e^{i\omega t} = i\omega \cdot e^{i\omega t}$ is also why differentiation becomes multiplication in the Fourier domain, the trick the Fourier Neural Operator exploits.

The polar form is then $z = r e^{i\theta}$. Multiplication is $r_1 r_2 e^{i(\theta_1 + \theta_2)}$, division flips the sign of the angle, and powers are $z^n = r^n e^{i n \theta}$ (de Moivre).

Roots of Unity

The equation $z^N = 1$ has exactly $N$ complex solutions, called the $N$th roots of unity: $\omega_N^k = e^{2\pi i k / N}, \quad k = 0, 1, \ldots, N-1.$ They lie on the unit circle, evenly spaced. The DFT matrix is built entirely out of these roots: the $(j, k)$ entry of the $N \times N$ DFT is $\omega_N^{j k}$, scaled by $1/\sqrt{N}$ in the unitary convention. Every fact about the FFT, and the $O(N \log N)$ divide-and-conquer that makes it tractable, comes from the multiplicative structure $\omega_N^{j k} = \omega_N^{j(k \bmod N)}$.

Worked Example: Multiplying Two Complex Numbers

Let $z_1 = 1 + i$ and $z_2 = \sqrt{3} + i$. Then $|z_1| = \sqrt{2}$, $\arg(z_1) = \pi/4$, and $|z_2| = 2$, $\arg(z_2) = \pi/6$. By the polar identities $|z_1 z_2| = 2\sqrt{2}$ and $\arg(z_1 z_2) = 5\pi/12$. Direct multiplication gives $z_1 z_2 = (\sqrt{3} - 1) + (\sqrt{3} + 1) i$. Check: $|z_1 z_2|^2 = (\sqrt{3} - 1)^2 + (\sqrt{3} + 1)^2 = 4 + 4 = 8$, so $|z_1 z_2| = 2\sqrt{2}$. The two computations agree, as they must.

Common Confusions

Watch Out

i is not 'sqrt(-1)'

The notation $\sqrt{-1}$ is misleading because the square-root function on the negative reals is not single-valued: both $i$ and $-i$ square to $-1$. The correct definition is the rule $i^2 = -1$, treating $i$ as a formal symbol. Writing $\sqrt{a}\sqrt{b} = \sqrt{ab}$ for negative reals leads to contradictions like $1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1} = -1$. The identity $\sqrt{a}\sqrt{b} = \sqrt{ab}$ holds only when $a, b \geq 0$.

Watch Out

The argument is multivalued

For $z \neq 0$, the angles $\theta$ and $\theta + 2\pi k$ for any integer $k$ all describe the same point. The convention $\arg(z) \in (-\pi, \pi]$ is the principal value. This matters when you take logarithms ($\log z = \log|z| + i \arg(z)$) or fractional powers, where branch choices change the answer.

Exercises

ExerciseCore

Problem

Verify Euler's formula at four checkpoints by direct computation: $\theta = 0$, $\pi/2$, $\pi$, $3\pi/2$. State each value of $e^{i\theta}$ in standard form $a + b i$.

Hint

Reveal Solution

ExerciseCore

Problem

Show that the $N$ roots of unity sum to zero: $\sum_{k=0}^{N-1} e^{2\pi i k / N} = 0$ for $N \geq 2$.

Hint

Reveal Solution

References

Related Topics

• Fast Fourier Transform
• Fourier Neural Operator
• Spectral Theory of Operators
• Wave-Based Neural Networks
• Vectors, Matrices, and Linear Maps