The Cabinet of Curios

Complex Numbers

You know numbers as points on a line. Complex numbers are points on a plane, and multiplying by one turns and stretches the whole plane around zero. Start on the number line, meet i, the square root of −1, and then watch arithmetic turn into rotation.

Points you can drag can also be clicked or reached with Tab, then nudged with the arrow keys. Hold Shift for bigger steps.

Station 1

The number line

Every ordinary number has a spot on one line. Multiplying by a positive number stretches or shrinks a number along the line. Multiplying by a negative number also flips it to the other side of 0, which is the same as swinging it a half turn around 0. Drag m and watch 1 hop to 1 × m and then hop again to m × m. Every square you make leaves a mark, and you can zoom out for bigger ones.

m
m²

Show up to

Now try to drag m so that m² lands on the ring at −1. You can't. A stretch twice is still a stretch, and a flip twice faces forward again, so no number on the line times itself makes −1. A hop that stopped halfway through its half turn would point straight up, off the line. Station 2 makes room for it.

Station 2

Meet i

Stand a second number line upright through 0 and the line becomes a plane. Now m can leave the line. How should multiplying work out here? Here's a guess: keep doing what it did on the line, where multiplying by m turns by m's angle and stretches by m's distance from 0. On the line those angles were only ever 0° for a positive number and 180° for a negative one, which is why all they could do was stretch and flip. Off the line, any angle is possible. Drag m and try again to make m² land on −1.

	number	length	angle
m
m²
−1	−1	1	180°

Show up to

Two quarter turns

1 × i = ia quarter turn, up

i × i = −1another quarter turn, over

To reach −1 the two hops have to add up to a half turn without stretching. Two equal turns that make a half turn are quarter turns, and a quarter turn from 1 points one step straight up. That spot is i. It isn't on the number line, which is why no ordinary number could do it, but it's a number all the same: i × i = −1.

Every point on the plane is some a + bi: go a along the number line, then b up, as the dashed lines under m show. The number line is still here, running across the middle. Its numbers are the ones with no step up.

But that turning rule was only a guess. All we actually know about i is that i × i = −1. Station 3 checks the guess using nothing else.

Station 3

Check the guess

Station 2 found i by guessing that multiplying turns things. All we really know about i is that i × i = −1. Is multiplying by i a quarter turn for every point, or only for 1 and i? Pick numbers below to multiply by i, or drag z anywhere you like. The working uses ordinary algebra plus that one fact, and each answer gets plotted next to the number it came from.

	number	length	angle
z
iz

Station 4

A quarter turn

Station 3's answers all landed a quarter turn ahead. Here's why that happens for every point. Multiplying it out, i(a + bi) = ai + bi² = −b + ai, so the two parts swap places and one flips sign. Drag z and compare the dashed triangle under z with the one under iz: it's the same triangle, stood on its side. Press × i to make the turn.

z
iz

So multiplying by i sends the point (x, y) to (−y, x). The two legs of the dashed triangle swap places and the one that lands across the axis flips sign. That's a quarter turn.

Station 5

Turn and stretch

Now multiply by any number w, not just i. Drag z and w. The product zw is z turned by w's angle and stretched by w's length, exactly the rule Station 2 guessed. Lengths multiply and angles add. The two shaded triangles always have the same shape, which is why it works for every w. The faint grid shows the whole plane going along.

	number	length	angle
z
w
zw

Keep w on the circle, so it only turns Show the turned grid

Repeat

Station 6

Rotating coordinates

There are two ways to rotate. You can leave the axes alone and turn the point, or leave the point alone and turn the axes and then ask where the point is now. Both are one complex multiplication, and both are the rotation matrix in disguise. Drag P anywhere.

Angle θ

Station 7

Many small turns

Multiplying by 1 + iε pushes a point sideways by ε of its length, because iz is at right angles to z. For a small ε that's nearly a pure turn. Split the angle θ into n of those nudges, starting from 1. The more nudges, the closer you land to the circle, at cos θ + i sin θ. That point is e^iθ, and Exploring e shows where the e comes from.

Angle θ

Steps n

Things to notice

i is a quarter turn. There's nothing imaginary about it. Multiplying by i turns every point a quarter turn counterclockwise around zero, so i × i = −1 just says that two quarter turns leave you facing backwards. The square root of −1 is half of turning around.
−1 has two square roots. Turning a quarter turn clockwise twice also gets you to −1, so (−i)² = −1 as well. That matches the number line, where 4 has two square roots, 2 and −2. The two roots of any number sit on opposite sides of 0.
"Real" and "imaginary" are just names. Both kinds of number are equally real. The vertical line isn't made up any more than negative numbers are. People once distrusted negatives too, and for the same reason: you can't count a pile of −3 apples, but you can walk 3 steps backwards.
Lengths multiply and angles add. That's all complex multiplication is. A number with length 1 only turns, a positive real number only stretches, and everything else does a bit of both.
Two triangles with the same shape. In Station 5 the triangle from 0 to 1 to w has the same shape as the triangle from 0 to z to zw. Multiplying by w moves 1 to w, so it has to move z to the matching corner.
Turns stack by multiplying. Set w to the 30° turn and repeat 12 times. The points land on a clock face and the 12th lands back on z, because (cos 30° + i sin 30°)¹² = 1. Those 12 clock positions are the twelve 12th roots of 1.
Turning the axes turns the point backwards. Rotate your axes by θ and every point seems to swing by −θ. That's why rotating coordinates multiplies by cos θ − i sin θ, and why the matrix that rotates the axes has its two sines swapped compared to the matrix that rotates the point.
The matrix is the multiplication written out. Multiply (x + yi)(c + si) and the real part is cx − sy and the imaginary part is sx + cy. Those are the two rows of the rotation matrix. The four numbers in the matrix are just the two numbers c and s, each written twice.
Programs rotate this way. A 2D heading can be stored as the pair (cos θ, sin θ), which is a complex number of length 1. Turning is one multiplication, combining two turns is another, and there's no angle to wrap past 360°. The Box2D physics engine stores each body's rotation as a cosine and sine pair, and its source notes that this is like using a complex number. To turn a body by a small angle ε it multiplies by 1 + iε, the nudge from Station 7, then rescales the result back to length 1 to take out the small stretch.
Why radians. In Station 7 each nudge pushes the point sideways by θ/n of its length, which turns it by almost exactly θ/n radians. A radian is the angle whose arc is as long as the radius, so sideways distance and angle match up. That's why Euler's formula, e^iθ = cos θ + i sin θ, only works with θ in radians. Station 7 is that formula: e^x is what (1 + x/n)ⁿ settles on as n grows.

A little history

Useful before they made sense. In 1572 Rafael Bombelli's L'Algebra gave rules for calculating with square roots of negative numbers. The formula for solving cubic equations sometimes passed through them on its way to a perfectly ordinary answer, so he worked with them without saying what they were.
"Imaginary" was a put-down. René Descartes called such roots imaginary in La Géométrie in 1637. He meant that you could imagine them, but no actual quantity matched them. The name stuck.
A surveyor drew the picture. Caspar Wessel, a surveyor working on the map of Denmark, presented numbers as arrows on a plane to the Royal Danish Academy in 1797, with multiplication as adding angles and multiplying lengths. His paper was published in Danish in 1799 and went almost unread for a century, until it was rediscovered in 1895 and translated into French in 1897. Jean-Robert Argand found the same picture on his own and published it anonymously in 1806, which is why it's often called the Argand diagram. Carl Friedrich Gauss had been using it privately, and in 1831 he backed it in print and gave the numbers their name: complex.
Euler's formula appeared in Leonhard Euler's Introductio in analysin infinitorum in 1748, decades before anyone drew the plane. He reached it with algebra, not pictures: take a tiny angle, use it an infinitely large number of times, and see what the powers settle on. Station 7 is a picture of the same idea.
There's no three-part version. William Rowan Hamilton spent years looking for numbers with three parts that would do for 3D rotation what complex numbers do in 2D. He later wrote to his son that every morning at breakfast the boys asked, "Well, Papa, can you multiply triplets?" and he'd shake his head. On 16 October 1843, walking beside the Royal Canal in Dublin, he saw that it took four parts, not three, and scratched i² = j² = k² = ijk = −1 into Broom Bridge. His quaternions now turn spacecraft, phones, and game cameras. Ken Shoemake brought them to computer animation in 1985.

Sources

Rafael Bombelli, L'Algebra (Bologna, 1572).
René Descartes, La Géométrie (1637), an appendix to the Discourse on Method.
Caspar Wessel, "Om Directionens analytiske Betegning," Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter (1799); French translation by H. G. Zeuthen, Essai sur la représentation analytique de la direction (Copenhagen, 1897); English translation by Flemming Damhus, On the Analytical Representation of Direction, ed. Bodil Branner and Jesper Lützen (Royal Danish Academy, 1999).
Jean-Robert Argand, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques (Paris, 1806), published anonymously.
Leonhard Euler, Introductio in analysin infinitorum, volume 1, chapter 8 (Lausanne, 1748).
Carl Friedrich Gauss, "Theoria residuorum biquadraticorum, commentatio secunda" (presented 1831, printed 1832), and his notice of it in the Göttingische gelehrte Anzeigen (23 April 1831).
William Rowan Hamilton, letter to his son Archibald (1865), recalling the walk to Broom Bridge.
Ken Shoemake, "Animating Rotation with Quaternion Curves," SIGGRAPH '85.
Box2D, math_functions.h, for b2Rot and b2IntegrateRotation.
Paul J. Nahin, An Imaginary Tale: The Story of √−1 (Princeton University Press, 1998).