Multiplying is turning

The standard introduction to complex numbers is an apology, and it is the same apology mathematics made for incommensurable lengths two thousand years earlier. Some equations have no solutions, so a solution is invented, called $i$, and declared to satisfy $i^2 = -1$. Everything that follows is presented as bookkeeping with this fictional object, and the word imaginary — Descartes’s, and intended dismissively — reinforces the impression that a sleight of hand has occurred.

The picture makes it clear that nothing was invented. The complex numbers are not an extension bolted onto the reals to patch a deficiency; they are what the plane looks like when it is given a multiplication, and the multiplication is the obvious one. Compare the point added to the plane to make stereographic projection perfect — also called an addition, also better understood as a recognition. The complex numbers were sitting in the plane the whole time, and $i$ is not a fiction. It is a quarter turn.

Two numbers, two jobs #

Every complex number can be described by an arrow from the origin: how long it is, and which way it points. Call those the modulus and the argument.

Multiplying two complex numbers combines them by the simplest possible rule:

• multiply the lengths;
• add the angles.

That is all. Every property of complex arithmetic follows from it, and none of them has to be memorised separately.

Consider what this says about $i$. It has length 1 and points straight up — a quarter turn from the positive real axis. Multiplying by it therefore leaves lengths alone and adds a quarter turn.

So what is $i^2$? Two quarter turns. A half turn. Starting at 1 and turning half a turn lands on $-1$.

$$i^2 = -1$$

is not a stipulation. It is the observation that turning ninety degrees twice leaves a thing facing backwards, which is not a fact anybody needs convincing of. The mystery in $i^2 = -1$ was entirely a consequence of insisting that numbers live on a line, where there is no room to turn.

Whether the spiral grows or shrinks depends only on whether the length exceeds 1. Powers of a number on the unit circle neither grow nor shrink — they just walk around it forever, which is the source of every fact about roots of unity.

A note on the word imaginary, since it does real damage. Descartes introduced it dismissively in 1637, meaning roughly fictitious, and it stuck to a class of numbers that are no less constructible than the negatives — which were themselves treated as absurd for centuries, on the grounds that one cannot have less than nothing of anything. Both objections dissolve the moment the numbers are given somewhere to live: the negatives need a line with two directions, the complex numbers need a plane. Neither was ever fictitious; both were homeless, and a name given during the homelessness outlived the problem.

Roots of unity come out immediately #

“Find the five fifth roots of 1” is a hard-looking algebra problem. It is a trivial geometry problem.

A fifth root of 1 is a number that, multiplied by itself five times, gives 1. Since lengths multiply, its length must satisfy $r^5 = 1$, so $r = 1$: all five roots are on the unit circle. Since angles add, five copies of its angle must total a whole number of full turns. So the angles are the multiples of one fifth of a turn.

The five roots are five equally spaced points around the unit circle. Drawing them takes longer than deriving them.

This generalises without effort: the $n$-th roots of any complex number sit at the vertices of a regular $n$-gon, at radius equal to the real $n$-th root of the modulus. De Moivre’s theorem, $(\cos\theta + i\sin\theta)^n = \cos n\theta + i \sin n\theta$, which looks like an identity to be proved by induction, is the statement that doing a rotation $n$ times rotates by $n$ times as much.

One more fact falls out of the same picture, and it is worth collecting because it does quiet work elsewhere on this site. The $n$-th roots of unity sum to zero, for every $n \ge 2$.

Geometrically that is immediate. They are the vertices of a regular $n$-gon centred at the origin, and the sum of the vectors to a shape’s vertices is $n$ times its centroid, which is the centre, which is $0$. Nothing has to be computed. Algebraically it is equally immediate and by an entirely different route: they are the roots of $z^n - 1 = 0$, the sum of a polynomial’s roots is minus the coefficient of $z^{n-1}$, and $z^n - 1$ has no $z^{n-1}$ term at all. Two arguments, one answer, neither using the other’s machinery — the same-thing-twice situation once more.

The consequence is the part that matters. Summing a complex exponential over a full lap gives nothing:

$$\sum_{k=0}^{n-1} e^{2\pi i k m/n} = 0 \quad \text{whenever $m$ is not a multiple of $n$,}$$

because those $n$ terms are the $n$-th roots of unity in some order, each visited once. When $m$ is a multiple of $n$ every term is $1$ and the sum is $n$ instead — at $n = 8$ the sum is $8$ at $m = 0, 8, 16$ and zero at every other $m$, with nothing in between.

That single cancellation is the orthogonality relation the whole of Fourier analysis rests on. It is why one harmonic’s coefficient can be extracted without any of the others contaminating it, and it is the reason the discrete Fourier transform exists. The claim that waves of different frequencies are perpendicular, which arrives in that essay as an integral and in the dot product essay as a projection, is this polygon closing up.

Euler’s formula is a description of walking #

Now the formula that gets called the most beautiful in mathematics, usually without much explanation of what it means:

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

The right side is the point on the unit circle at angle $\theta$ — its horizontal coordinate is $\cos\theta$ and its vertical one is $\sin\theta$, exactly as in a sine wave being a circle seen from the side.

The left side is an exponential. And the property of the exponential, established in the curve that is its own slope, is that its rate of change equals its current value.

Put those together. The formula describes a point whose velocity, at every instant, is its current position multiplied by $i$ — which is to say, its current position rotated a quarter turn. A velocity permanently at right angles to the position vector, and permanently the same magnitude as it, is the definition of moving in a circle at constant speed.

So $e^{i\theta}$ means: start at 1, and walk around the unit circle a distance $\theta$.

Which makes the celebrated identity almost anticlimactic:

$$e^{i\pi} = -1$$

Walk a distance $\pi$ around a circle of radius 1. A full lap is $2\pi$, so $\pi$ is half a lap. Starting at 1 and going half a lap lands at $-1$.

That is the whole content. It is startling only when $e$, $i$ and $\pi$ are held to be three unrelated constants that mysteriously combine. Read geometrically they are three aspects of one situation: $e$ is the base at which growth equals current size, $i$ is the quarter turn that converts growth into circulation, and $\pi$ is how far it is around half a circle. Of course they fit together. They are all descriptions of the same walk.

The three-dimensional version does not exist #

The obvious next question is what happens one dimension up. If giving the plane a multiplication produces something this useful, giving it to space should produce something better.

William Rowan Hamilton spent about fifteen years on it. The requirement was modest: a way to multiply triples $a + bi + cj$ so that lengths multiply, division works, and the usual rules of arithmetic survive. His children are reported to have asked him at breakfast whether he could multiply triples yet, and for years the answer was no.

The answer is no permanently. The failure is not a shortage of ingenuity — it is a theorem.

The clue is in what the plane’s multiplication really was. Complex multiplication by a unit-length number is a rotation, and the unit-length numbers form a circle: a one-dimensional family, which is exactly the number of parameters a rotation of the plane needs. In three dimensions a rotation needs three parameters — an axis, which costs two, and an angle, which costs one — and the unit-length triples form a sphere, which has only two. There are not enough unit vectors in three-dimensional space to name its rotations. The correspondence that made the plane work has no room to happen.

What Hamilton found, on Brougham Bridge in Dublin in 1843, was that four dimensions has exactly enough room, and the price is that multiplication stops commuting: $ij = k$ but $ji = -k$. He carved the relations into the bridge. The quaternions rotate three-dimensional space perfectly well; they just are not three-dimensional themselves, and the rotations they perform are read off a four-dimensional object.

The impossibility was later made precise, and it is sharper than the counting argument suggests. Frobenius proved that the real numbers, the complex numbers and the quaternions are the only finite-dimensional associative division algebras over the reals — the list is three items long and there is nothing else anywhere. Dropping associativity too admits exactly one more, the octonions, in eight dimensions. So the dimensions in which the plane’s trick works at all are $1, 2, 4, 8$, and then it stops forever.

The shape of that argument is worth recognising, because it is the same one that shows there are exactly five regular solids: a budget is counted, the budget runs out, and the list is finished not by exhaustion of effort but by arithmetic. Impossibility results of this kind are the most economical statements in mathematics — one line of counting closes a search that would otherwise never be known to have ended.

That is a rare kind of result, and it is worth pausing on what it does to the original question. Why are the complex numbers two-dimensional? has the feel of a question with a historical answer — because that is where $\sqrt{-1}$ happened to lead. It has an answer of a completely different character: two is one of only four sizes at which such a thing can exist, and the reason is not about square roots at all.

What the turning costs #

Every extension of a number system is a trade, and this one is usually reported with the gains listed and the price left off.

The gain is enormous and easy to state. Over the reals, some polynomials have roots and some do not, and which is which requires inspection. Over the complex numbers every non-constant polynomial has a root, every one of degree $n$ has exactly $n$ counted properly, and the exceptions are gone. A whole category of question — does a solution exist — stops being a question.

The price is order. The real numbers are ordered, and the order is compatible with the arithmetic: adding the same thing to both sides of an inequality preserves it, and multiplying by a positive number does too. No such ordering exists on the complex numbers, and the proof takes one line. In any ordered field a square is never negative. But $i^2 = -1$, so $-1$ would have to be both a square and negative. There is no clever ordering waiting to be found; the requirement is self-contradictory.

Losing the order is not a technicality, because a great deal of real analysis is built on it. Bisection — halving an interval and asking which half contains the root — needs a notion of between. Monotone sequences converge because they are increasing and bounded, and neither word means anything here. Optimisation asks for a largest value, and there is no largest. Every one of those tools has to be replaced by something keyed on the modulus instead, which is a real number and does carry an order. Even chopping an interval into ever finer pieces — the move that defines the integral — presupposes that the pieces lie in a sequence, and in the plane they do not.

So the picture is subtly misleading in a way worth naming. It draws the complex numbers as the plane and the reals as a line through it, which makes the extension look like pure gain — the same objects, with more room. It is not. Something was handed over at the door, and what was handed over is the reason a line is easier to reason about than a plane.

What the picture cannot show #

The complex plane draws numbers as points, which works perfectly for addition and multiplication and stops working the moment functions are involved.

A real function can be graphed: input along one axis, output along the other. A complex function needs two dimensions for its input and two for its output, so its graph lives in four dimensions and cannot be drawn. Everything interesting about complex analysis — that differentiable functions are automatically infinitely differentiable, that a function is determined by its values on any small disc, that contour integrals depend only on what is enclosed — concerns objects these pictures cannot represent at all.

The plane is also silent about its own scarcity. Nothing in a drawing of two dimensions can indicate that four and eight are the only sizes above it where the same construction survives, or that three — the dimension the reader is sitting in — is not one of them. An impossibility is invisible by construction: a figure can show a thing existing, and has no vocabulary for a thing that cannot.

The ladder from here #

Rungs above: roots of unity in full, and the regular polygons they draw. De Moivre and the multiple-angle formulas, derived rather than memorised. The fundamental theorem of algebra, and why the plane is where polynomials finally behave. Möbius transformations, on the Riemann sphere. Complex numbers as $2\times2$ matrices, made precise. The quaternions in their own right, and the rotation of space they perform without being a part of it. Analytic continuation. And the Mandelbrot set, which is nothing but the question of whether iterating $z^2 + c$ keeps the point bounded.

The other reason this matters #

There is a practical payoff that is easy to miss behind the elegance.

Oscillations are hard to compute with in trigonometric form. Adding two sine waves of the same frequency but different phase and amplitude requires an identity nobody remembers. Represent them as complex numbers instead and it is a single addition of arrows, done by adding coordinates.

This is why electrical engineers describe alternating current with complex impedances — alternating current being a circle seen from the side — why quantum mechanics is written over the complex numbers, and why signal processing lives there. In each case the underlying quantity is real and observable; the complex representation is a change of coordinates in which rotation is multiplication, and therefore in which the algebra becomes easy.

The connection back to matrices as pictures of the grid is exact, and worth noticing. The map “multiply by the complex number $a + bi$” is a linear transformation of the plane, and as a matrix it is

$$\begin{pmatrix} a & -b \\ b & a \end{pmatrix},$$

whose determinant is $a^2 + b^2$ — the square of the modulus, which is the factor by which areas scale. The complex numbers are exactly the $2 \times 2$ matrices of that shape: the rotation-and-scaling ones. Nothing imaginary about any of it.