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Mathematics

Complex numbers

when numbers step off the line onto the plane

The equation x² + 1 = 0 has no solution among ordinary numbers. But allow a number i with i² = −1, and the number line unfolds into a plane. Multiplication becomes rotation, Euler's formula becomes a spiral in 3D, and functions become colorful landscapes with pits where their zeros are.

z = a + bi eiφ = cos φ + i sin φ eiπ + 1 = 0
hue = arg f(z), rings = |f(z)|,  f(z) = (z³ − 1) / (z² + 0.4)

A number line that rotates

“Impossible” numbers

For centuries the square root of a negative number was considered nonsense. What let it into mathematics was… cubic equations.

  • 1545Girolamo Cardano, “Ars Magna”: publishes formulas for cubic equations. In the problem “divide 10 into two parts whose product is 40” he gets 5 + √−15 and 5 − √−15, and calls such calculations as subtle as they are useless.
  • 1572Rafael Bombelli, “L’Algebra”: gives rules for calculating with √−1 and shows that the road to the ordinary answer x = 4 runs through “impossible” numbers (worked out below).
  • 1637René Descartes calls such roots imaginaires — “imaginary.” The name sounded dismissive, yet it stuck.
  • 1748Leonhard Euler publishes the formula eix = cos x + i sin x.
  • 1777Euler starts writing √−1 as the letter i (the paper appeared in print only after his death).
  • 1799Caspar Wessel (Denmark) represents complex numbers as arrows on a plane — his work goes unnoticed for a long time.
  • 1806Jean-Robert Argand independently does the same; the plane is sometimes called the Argand diagram today.
  • 1831Carl Friedrich Gauss popularizes the picture of the plane and the name “complex numbers.” Hence the Polish name for it, the “Gaussian plane.”

Bombelli's trick: x³ = 15x + 4

  1. Cardano's formula for the equation x³ = px + q:
    x = ∛(q/2 + √((q/2)² − (p/3)³))+ ∛(q/2 − √((q/2)² − (p/3)³))
  2. For p = 15, q = 4 the number under the square root is (2)² − (5)³ = 4 − 125 = −121. The square root of a negative number! And yet x = 4 works: 4³ = 64 = 15 · 4 + 4.
  3. Bombelli carries on as if nothing had happened: √−121 = 11√−1 = 11i, so
    x = ∛(2 + 11i) + ∛(2 − 11i)
  4. He guesses the cubes:
    (2 + i)³ = 8 + 12i + 6i² + i³= 8 + 12i − 6 − i = 2 + 11i
    and likewise (2 − i)³ = 2 − 11i.
  5. So x = (2 + i) + (2 − i) = 4. The imaginary parts cancel — the “impossible” numbers were just a footbridge leading to the real answer.

How it works

A complex number is a pair of real numbers, written as z = a + bi: the real part Re z = a and the imaginary part Im z = b. On the plane it is the point (a, b) or an arrow from zero. The only new rule: i² = −1.

Modulus, argument, conjugate

|z| = √(a² + b²)arg z = φ,  tan φ = b/az̄ = a − bi,  zz̄ = |z|²

Three forms of the same number

z = a + bi= |z|(cos φ + i sin φ)= |z|eiφalgebraic · trigonometric · exponential

Multiplication is rotation and scaling

|zw| = |z|·|w|arg(zw) = arg z + arg wmoduli multiply, angles add (up to a full turn)

De Moivre’s formula and roots

(cos φ + i sin φ)n = cos nφ + i sin nφ
zk = n√|c| · ei(φ + 2kπ)/nk = 0, 1, …, n − 1

Euler's formula

eiφ = cos φ + i sin φeiπ + 1 = 0

Complex functions

A function f(z) maps a plane to a plane — we would need four dimensions to draw its graph. So we cheat: hue tells us the argument of the result, and height (or the brightness of the rings) the modulus. Zeros are pits, poles are towers. Around a zero of order k the colors go around the wheel k times.

Every polynomial of degree n ≥ 1 has n complex rootsthe fundamental theorem of algebra (counting multiplicity)

Simplifications: heights in the landscape are compressed (pole towers are gently clipped), the coloring is computed at a finite resolution, and the RLC circuit is an ideal model.

See also

Trigonometry Mandelbrot set Fourier Circuits Power plant Derivative and integral The number π

Challenges

  1. Multiply by i four times and return to the start“Where did i come from?”: × i four times in a row.not completed
  2. Find the sixth roots of unity“Plane” → Roots: drag c to 1 and set n = 6.not completed
  3. (1 + iπ/n)n closer to −1 than 0.01“Euler's formula in 3D”: choose the number of steps n.not completed
  4. Find all the zeros of z³ − 1 in the landscape“Functions”: click the three pits to plant flags.not completed
  5. One deep pitChoose “polynomial from roots” (at least 3) and slide all the roots into one spot.not completed
  6. Make an RLC circuit resonate“Where they're useful”: tune the frequency until the current and voltage are in phase.not completed

Did you know…?

  • The word “imaginary” started out as a jibe: in 1637 Descartes contrasted “imaginary” roots (imaginaires) with “real” ones. Today, without these “imaginary” numbers, you couldn't design a power grid or a smartphone.
  • Euler's identity eiπ + 1 = 0 links five of the most important constants: 0, 1, π, e, and i. In a poll of readers of the journal “The Mathematical Intelligencer” (1988) it came first as the most beautiful formula.
  • The Mandelbrot set lives on the complex plane: it is made up of the numbers c for which the sequence 0, c, c² + c, … does not run off to infinity.

Keys

Q W E R T
switch modes (← → on the tabs)
I N 0
history: × i, × (−1), restart
1–4
plane and uses: sub-tabs; Euler: views; functions: |f|, log, Re, Im (5 — Riemann sphere)
Space
Euler: play / pause; uses: pause; on the plane: next point
← ↑ → ↓
on the canvas: 3D rotation or moving a point
+ −
zoom in 3D views
N Enter
functions (on the plane): next root; flag at the probe point
Esc
stops all animations