Complex Numbers Visualization | Add, Multiply, Euler, Roots

The idea: a complex number is a point

A complex number z = a + bi has a real part a and an imaginary part b, where i is a number with i² = −1. Draw it as the point (a, b) on the complex plane: real part across, imaginary part up. Or draw it as an arrow from 0 to that point. The arrow has a length, the modulus r = |z| = √(a² + b²), and an angle from the positive real axis, the argument θ.

z = 3 + 4i is the point (3, 4): length r = 5, angle θ = 53.13°.

Addition is vector addition

To add, add the real parts and the imaginary parts: (3 + 4i) + (1 + 2i) = 4 + 6i. On the plane that is adding two arrows head to tail: slide a copy of w so its tail sits on the tip of z, and its tip lands on z + w, the far corner of the parallelogram on z and w. Subtraction z − w is the arrow from w to z (Demo: add (3 + 4i) + (1 + 2i)).

Multiplication rotates and scales

Multiply out like any two brackets and use i² = −1:

(3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i.

The same product has a picture. The moduli multiply and the angles add: |3 + 4i| = 5, |1 + 2i| = √5 ≈ 2.24, and 5 × 2.24 = 11.18 = |−5 + 10i|. The angles are 53.13° and 63.43°, and 53.13° + 63.43° = 116.57°, the angle of −5 + 10i. So multiplying by w rotates by arg w and scales by |w| (Demo: multiply (3 + 4i)(1 + 2i)).

z = 3 + 4i at 53.13 degrees, w = 1 + 2i at 63.43 degrees, and z times w = minus 5 + 10i at 116.57 degrees. A dashed arc shows z rotated by 63.43 degrees, then stretched by 2.24 to reach the product — z · w: turn z by w's angle, then stretch by w's length. 53.13° + 63.43° = 116.57°, 5 × 2.24 = 11.18.

Operation	Rectangular a + bi	Polar r·e^iθ
z + w	add parts: (a + c) + (b + d)i	no simple rule
z · w	(ac − bd) + (ad + bc)i	r₁r₂·e^{i(θ₁ + θ₂)}
zⁿ	expand n brackets	rⁿ·e^inθ

Multiplying by i is a quarter turn

i has length 1 and angle 90°, so multiplying by i keeps the length and turns by 90°: (a + bi)·i = −b + ai. Start at 1 and keep multiplying: 1 → i → −1 → −i → 1. Two quarter turns are a half turn, and half a turn from 1 points at −1. That is the picture behind i² = −1: no real number squares to −1, but a rotation done twice can reverse a direction (Demo: powers of i).

Polar form and Euler's formula

A number with modulus r and angle θ is r(cos θ + i sin θ). Euler's formula says this is the exponential:

e^iθ = cos θ + i sin θ

As θ grows, e^iθ runs round the unit circle: cos θ along the real axis, sin θ up. So every complex number can be written in polar form z = r·e^iθ, and the multiplication rule is just the exponent rule e^iα·e^iβ = e^{i(α + β)}. Angles are in radians: half a turn is θ = π, which gives e^iπ = −1, or e^iπ + 1 = 0. A full turn gives e^2πi = 1 (Demo: Euler's formula).

Roots of unity: a regular polygon

The equation zⁿ = 1 has exactly n complex solutions, the nth roots of unity. With ω = e^2πi/n they are ω⁰ = 1, ω¹, ω², …, ωⁿ⁻¹: all on the unit circle, 360°/n apart. Each multiplication by ω turns one step, and n steps make a full turn, so (ω^k)ⁿ = e^2πik = 1. Joined in order they form a regular n-gon, and by symmetry they add up to 0 (Demo: the 8th roots of unity).

The eight 8th roots of unity on the unit circle, 45 degrees apart: 1, 0.71 + 0.71i, i, minus 0.71 + 0.71i, minus 1, minus 0.71 minus 0.71i, minus i, 0.71 minus 0.71i, joined into a regular octagon — The 8th roots of unity: powers of ω = e^(i·45°), a regular octagon.

Where it is used

The Fast Fourier Transform multiplies by roots of unity, the twiddle factors ω^k = e^−2πik/n. The FFT works because ω² for n points is ω for n/2 points: halving the circle halves the problem.
Multiplying by e^iθ is the 2D rotation by θ. Written as a matrix it is [[cos θ, −sin θ], [sin θ, cos θ]], the matrix in Rotate and Scale in 2D; multiplying by r·e^iθ rotates and scales at once.
AC circuits, signal processing and quantum mechanics write waves as e^iωt, so a phase shift is a multiplication.

See also Vectors and the Dot Product (the same head-to-tail addition) and Taylor Series (where the series for e^x, cos and sin explain Euler's formula).

See also quadratic equations, where a negative discriminant gives a conjugate pair of complex roots such as −1 ± 2i. The point e^(iθ) = cos θ + i sin θ is the point P on the unit circle; see The Unit Circle: sin, cos, tan for how its coordinates unroll into the sine and cosine graphs.

Square a complex number and add c, again and again: whether that orbit stays bounded decides The Mandelbrot Set, and for a fixed c it draws Julia sets.

What the page leaves out

Division (divide the moduli, subtract the angles), the conjugate a − bi and |z|² = z·conj(z), and why Euler's formula holds (compare the power series of e^x, cos and sin). Roots of numbers other than 1, the many values of log z, and complex functions such as z² as maps of the plane are not drawn. The page uses degrees for the numbers; mathematics normally uses radians (180° = π).