Twisty Numbers for a Screwy Universe

If new kinds of numbers were like new consumer products, mathematicians would have every right to fire the marketing company that came up with the names “complex numbers” and “imaginary numbers”. I mean, what kind of sales pitch goes with that branding? “Psst: wanna buy a number? It’s really hard to understand, and best of all, it doesn’t even exist!”?

We mathematicians have nobody but ourselves to blame, since it was one of our own (René Descartes) who saddled numbers like sqrt(−1) with the term “imaginary” and another mathematician (Carl-Friedrich Gauss) who dubbed numbers like 2+sqrt(−1) “complex”. Now it’s several centuries too late for us to ask everybody to use different words. But since those centuries have given us a clearer understanding of what these new sorts of numbers are good for, I can’t help wishing that, instead of calling them “complex numbers”, we’d called them — well, I’ll come to that in a bit.

Mind you, I totally get why sqrt(−1) got called imaginary. “sqrt(−1)” signifies a number x with the property that x² = −1, but no respectable number behaves that way. A law-abiding number is positive, negative, or zero. If x is positive, x² will be positive too. If x is negative, x² will still be positive, since a negative number times a negative number is a positive number (see my essay “Going Negative, part 1” and other Mathematical Enchantments essays about negative numbers if you’re wondering why the product of two negative numbers is positive). And if x is zero, x² will be zero. In none of the three allowed cases is x² negative, so you can’t have x² equal to −1. Sorry; it’s an impossible equation. And you might think that that would end the matter …

… except that five hundred years ago algebraists learned, to their astonishment, that expressions involving the square roots of negative numbers can be useful intermediate stages of certain calculations that have sensible final answers. So mathematicians grudgingly invited square roots of negative numbers into the house of mathematics but only through the back door, and chose nomenclature that would let those impossible square roots know in no uncertain terms that they were second-class citizens who could mix with other algebraic expressions but who needed to be out the door when their work was done, before respectable company arrived. (See Endnote #1.)

MATH AND MYSTICISM

Isaac Asimov, in his essay “The imaginary that isn’t” (from his book Adding A Dimension), describes an encounter he had with a sociology professor while he was an undergraduate in the 1930s. The professor sorted humankind into two groups, “realists” and “mystics”, and asserted that mathematicians belong to the latter camp because “they believe in numbers that have no reality.” When young Asimov asked him to explain, the professor cited the example of the square root of minus one, saying “It has no existence. Mathematicians call it imaginary. But they believe it has some kind of existence in a mystical way.” Asimov protested that imaginary numbers are just as real as any other kind of number, and the professor challenged the upstart to hand him the square root of minus one pieces of chalk, and … but I’ll break off the story there for now, because I like to imagine it going a different way: I like to imagine young Asimov asking, “So, where would you put electrical engineers in your classification of humankind? You know, people like Steinmetz?” 

Any professor teaching in an American college in the 1930s would have known of Charles Proteus Steinmetz, even though he’s no longer a household name the way Edison and Tesla are. The “Wizard of Schenectady” was as responsible for the electrification of America as anyone else (arguably more than Edison, who had stubbornly insisted on trying to transmit direct current along power lines until Steinmetz and Tesla and their allies proved the superiority of alternating current). The sociology professor was undoubtedly teaching in a classroom that had artificial light in the ceiling run by electrical generators miles away, thanks to Steinmetz.

“Steinmetz built things. He was a realist,” the professor would have said.

“Oh?” Asimov could have replied. “Then why was he an evangelist for the square root of minus one?”

THE WIZARD OF SCHENECTADY

Steinmetz was in many ways Edison’s opposite, and not just because of their different ideas about how power should be transmitted across long distances. Edison was a commanding five foot ten; Steinmetz was only four feet tall. Edison gave his assistants (“muckers”, they were called) puny salaries; Steinmetz once refused to take a raise from his employer because he felt that his assistants weren’t paid enough. Edison had three biological children; Steinmetz never had any because he was determined not to pass on his genes for kyphosis and hip dysplasia, opting instead to adopt a younger colleague as his son and become a loving grandfather to the colleague’s children. But, like Edison, Steinmetz was a workaholic whose success in solving technological problems came in part from the fact that he devoted his life to them.

As a young man in Bismarck’s Prussia, Karl August Rudolph Steinmetz joined a fraternity that bestowed on him the nickname Proteus after the shape-shifting Greek sea-god from whose name we get the adjective “protean”. Later, his membership in a socialist student group got him in trouble with the authorities and he was forced to flee the country. As a disabled person with very little command of English, he was almost turned away at Ellis Island until a friend spoke up for him, exaggerating his talents in a way that his past accomplishments didn’t justify (but his future accomplishments would). Young Karl became Charles and adopted his former nickname as his legal middle name. He went to work for a friend, Rudolph Eickemeyer, and stayed at Eickemeyer’s company out of loyalty even after his early achievements got the attention of bigger companies. When General Electric offered him a large salary increase if he’d leave his friend’s company, Steinmetz was puzzled: what did salary have to do with the principle of loyalty? General Electric resolved the impasse by buying Eickemeyer’s company.

The late 19th century was an era of technological promise, much of it bound up in the transforming potential of electricity. The major problem was how to get electricity to all the different places where it could do its magic. Edison favored the straightforward approach of pushing electrons over wires from point A to point B, since electricity derives from the motion of electrons. But others favored the less intuitive idea of rocking electrons back and forth along a wire, alternately pushing and pulling. That kind of current, an alternating current, has many technological advantages (which is why it predominates today), but alternating current is harder to model mathematically because it’s dynamic in a way that direct current isn’t. Let’s take a look at that.

First, picture a simple direct current (DC) circuit containing a battery, a lightbulb, and two wires connecting the battery to the bulb in both directions. Three important quantities are the voltage V (the difference in electrical potential between the two ends of the wire), the current I (how many electrons are traveling along the wire), and the resistance R (how hard the electrons have to work to make the trip), and as long as the bulb doesn’t burn out, the quantities are constant. If you were to plot current and voltage as functions of time (with time on the horizontal axis and current or voltage on the vertical axis), you’d just see boring horizontal lines. Moreover, there’s a simple equation relating these three quantities, called Ohm’s Law: V = I R.

But in alternating current (AC) circuits, voltage and current vary over time, and there’s no such simple linear relationship between them. For instance, if you live in the U.S., an outlet labeled “120V” is actually giving you a voltage that oscillates between +170V and −170V; 120 is just the average over time (the “root-mean-square average“, for those of you who care). In the figure below I give a typical plot, showing voltage (the blue curve) and current (the gold curve) as functions of time in an AC circuit. Sometimes the voltage is increasing and the current is increasing, but sometimes the voltage is increasing and the current is decreasing. Both voltage and current follow the pattern of a sine wave, or “sinusoid”, but typically peak current doesn’t coincide with peak voltage; in most circuits there’s a mismatch, or “phase shift”. (See Endnote #2.) Gone is the simplicity of V = I R.

You can see this sort of phase-shift in action if you watch a kid on a swing (after they stop pumping) and you simultaneously attend to position and speed, or more precisely, deflection from the vertical and angular velocity. When the kid is as far to the right as possible, their speed is (instantaneously) zero. When the kid has swung back down to the lowest point of their trajectory (where the deflection is zero), the motion is leftward and the speed is at its maximum. When the kid is as far to the left as possible, their speed is again zero. When the kid returns to the zero-deflection point, the motion is rightward and the speed is at its maximum. And so on. The instant of maximum rightward deflection comes after the instant of maximum rightward speed, specifically 1/4 of a cycle later.

If you were to plot, at each instant, a point whose x-coordinate is the deflection of the swing (positive when the swing is on the right, negative when the swing is on the left) and whose y-coordinate is the angular velocity of the swing (positive when the swing is moving rightward, negative when the swing is moving leftward), the moving point would trace out a circle, as shown in the illustration. The circle doesn’t exist in physical space; rather, it exists in a notional “phase space”, in which the vertical axis is for the velocity of the swing, not its position. (Here I’m ignoring some niceties about pendular motion; specifically, its deflection-velocity plot is only approximately circular, and only when the amplitude is small. And as every kid who’s been on a swing knows, it’s the big deflections that are the fun part!)

The mathematics of circular motion is usually described using trigonometric functions, and indeed one can describe current and voltage in alternating-current circuits using sines and cosines, but the formulas can get quite hairy. What Steinmetz realized is that some seemingly pure math he’d learned in his student days could make the formulas much simpler. (See Endnote #3.)

PURE IMAGINATION

Mathematicians had played with imaginary and complex numbers long before their games had any real-world applications. One of the mathematicians who played the hardest was Leonhard Euler, who in 1777 introduced the symbol “i” to signify the square root of minus one. Euler operated on the assumption that whatever “i” might be, it should satisfy the ordinary rules of algebra. So for instance 2i times 3i should be

(2i)×(3i) = (2)×(i)×(3)×(i) = (2)×(3)×(i)×(i) = (2×3)×(i×i) = (6)×(−1) = −6

and 1+i times 1+i should be

(1+i)×(1+i) = 1×1 + 1×i + i×1 + i×i = 1 + i + i + −1 = i + i = 2i

(where the first equality is an application of the distributive law or if you prefer “FOIL“; but see Endnote #4).

Later, the mathematicians Jean-Robert Argand, Caspar Wessel and Carl-Friedrich Gauss independently came up with a visual way to represent complex numbers. You draw a horizontal axis for the real numbers and a vertical axis for the imaginary numbers meeting at a point called the origin, and you depict the complex number a+bi by a point that’s a units to the right of the origin and b units up from the origin, as shown for the complex numbers 2+i, 3+i, and 5+5i. (If a is negative, go left instead of right; if b is negative, go down instead of up.) Note by the way that the origin represents the complex number 0+0i, whch is simultaneously real and imaginary. My daughter, shortly after learning all this, exclaimed “Wait, so zero has been a complex number under my nose this whole time?” Absolutely!

The wonderful thing about the definition of complex number multiplication is the geometry that’s hiding inside it (exactly the kind of shape-shifting geometry Steinmetz needed). Suppose a specific complex number a+bi other than 0+0i is represented by the specific point P in the plane in the manner described above. Let O stand for the origin (where the axes cross, aka 0+0i), and let N be the point on the horizontal axis that corresponds to the complex number 1+0i, aka the real number 1. We define the “magnitude” of the complex number a+bi as the length of segment OP, and we define the “phase” or “angle” of the complex number a+bi as the measure of angle NOP (some people call the phase the “argument” but I won’t). For example, when a=b=1, triangle NOP is an isosceles right triangle with legs of length 1, so the magnitude of 1+i is sqrt(2) and the phase of 1+i is 45°.

The miracle is that if we define multiplication of complex numbers in the way that the ordinary rules of algebra force us to, then magnitudes multiply and angles add! For instance, look at 2i times 3i. 2i has magnitude 2 and phase 90°, and 3i has magnitude 3 and phase 90°; the product of 2i and 3i, namely −6, has magnitude 6 = 2 × 3 and phase 180° = 90° + 90°. Or look at 1+i times 1+i. 1+i has magnitude sqrt(2) and phase 45°; its product with itself, namely 2i, has magnitude 2 = sqrt(2) × sqrt(2) and phase 90° = 45° + 45°. (A puzzle for some of you: can you show that the “multiplication miracle” holds when the two complex numbers being multiplied are 2+i and 3+i? See Endnote #5.)

Multiplying a complex number by −1 has the effect of leaving the magnitude alone while rotating the corresponding point halfway around the origin. (In fact, the rule for multiplying complex numbers gives us a new way to understand the rule for determining the sign of the product of two real numbers; see Endnote #6.) In a similar way, multiplying any complex number by i has the effect of rotating the corresponding point a quarter of the way around the origin, in the counterclockwise direction; see Endnote #7. Steinmetz realized that the mathematics of multiplication by i was a very crisp way of representing the physics of a 90 degree phase shift. (See Endnote #8.) He couldn’t use the letter i because electrical engineers were already using I to represent current flow (recall our earlier equation V = I R), so Steinmetz chose to use j instead, and to this day many electrical engineers use j instead of i to signify the square root of −1.

The sociology professor’s mistake (back in Asimov’s story) lay in part in thinking that mathematics is only about static quantities, such as the number of pieces of chalk you’re holding in your hand. But math can also be about things that, like mythical Proteus, keep changing. Take the professor’s chalk and drag it against a blackboard, at an angle that makes a squeaky sound some find painful, and you have an oscillating physical system that could be described by sines and cosines but might better be described through the use of complex numbers. Indeed, the 90 degree phase shift (of which the complex number i is the numerical thumbprint) is ubiquitous in physics. I’ve already mentioned electrical circuits and kids on swings, but there are lots of other examples.

Real numbers have magnitude and sign; analogously, complex numbers have magnitude and phase. That’s why I wish complex numbers had been dubbed “phased numbers”. (See Endnote #9.) Real numbers are phased numbers whose phase is either 0 degrees (for positive real numbers) or 180 degrees (for negative real numbers). Likewise, imaginary numbers are phased numbers whose phase is either 90 degrees or 270 degrees.

I should stress that Steinmetz did not experimentally discover that the flow of electrons had a hitherto unnoticed imaginary component — he merely showed that the mathematical formalism of electrical engineering becomes simpler if we pretend that the current that we measure is but the shadow, along the real line, of a quantity whose true home is the complex plane. The gif Another way of looking at sine and cosine functions (created by Christian Wolff; permission pending) illustrates this in a lovely way. A green point moves in a circle in the x,y plane. Its projection onto the x,z plane gives a blue sinusoid, while its projection onto the y,z plane gives a red sinusoid that is 90 degrees out of phase with the first one. In our analogy, one of these sinuoids is real current (or real voltage) while the other sinuoid is imaginary current (or real voltage). If we apply this point of view to our AC circuits, then we can revive the equation V = I R by reinterpreting all three quantities as complex numbers. V now represents the complex voltage, I now represents the complex current, and R gets replaced by a complex number called “impedance” that extends the concept of resistance. (See also Endnote #10.) The linear relation between current and voltage, so handy in the study of direct current circuits, has been restored! And the only price we had to pay was to graduate from the real number line to the complex number plane.

The mathematics of back-and-forth in one dimension is best expressed in terms of the mathematics of round-and-round in two dimensions. For instance, when you spin this disk clockwise (see Endnote #11), the vertical coordinates of the blue and gold points match up with the behavior of the blue and gold curves I used to illustrate the behavior of voltage and current in an AC circuit. And as the disk spins about its center, the ratio of the gold point to the blue point remains fixed if we view the two points as complex numbers, since the ratio of their magnitudes stays 4-to-3 while their phase-lag stays 90 degrees.

(If any of you reading this are good at creating animations, please let me know; I’d love to be able to include a gif that shows the spinning disk and makes the connection with those sinusoids “pop”!)

Since complex currents and complex voltages are useful fictions, not scientific facts, perhaps the sociologist was right to call this way of thinking about the world “mystical”.  But if so, Steinmetz was an extremely unusual and useful kind of mystic: not the kind who makes occult pronouncements about the spirit plane but the kind who, invoking a different sort of plane, brings about a world in which it’s easier to make toast.

THE GREAT UNIFICATION

If you’re impatient to get back to Asimov’s sociology professor and find out what really happened in that classroom, you might want to skip this section. But I can’t resist giving you a peek into what mathematicians did with Euler’s i, starting with Euler himself. (This is the stuff Steinmetz would have learned as a math major at the University of Breslau.)

The best thing Euler did with the number i was discover the equation

e^iθ = cos θ + i sin θ

where e is the constant 2.718… discovered by Jacob Bernoulli but often called Euler’s number, cos is the cosine function, and sin is the sine function. (For more about e, watch the 3Blue1Brown video “What’s so special about Euler’s number e?”, and for more about Euler’s amazing formula, watch the 3Blue1Brown video “What is Euler’s formula actually saying?”.) What makes this equation astonishing is that the left and right sides of this equation come from different worlds. The left side is an exponential function (if we leave aside the suspicious circumstance of the exponent being an imaginary number), and therefore points at phenomena like compound interest, population growth, radioactive decay, and the initial spread of novel pathogens. (It was indeed the application of exponential functions to banking that led Bernoulli to discover e in the first place.) Meanwhile, the right hand side (again ignoring the i) features two functions, sine and cosine, introduced thousands of years ago for surveying land, navigating seas, and plotting the paths of planets. It would seem that the compounding of interest has little to do with the motions of heavenly bodies, yet Euler’s formula tied them together intimately, showing them to be two different aspects of a single mathematical phenomenon. We celebrate Newton’s unification of terrestrial ballistics with the motion of the Moon, and Maxwell’s unification of electricity, magnetism, and light, but we don’t say nearly enough about how Euler’s discovery built a secret passageway that links numerous disciplines within mathematics and far beyond.

The complex number e^iθ always lies on the circle of radius 1 centered at 0. If we want to talk about other kinds of nonzero complex numbers, we use the representation re^iθ. where r is a positive real number. This is called the “polar representation” of the complex number r times cos θ + i sin θ, often abbreviated as r cis θ, where the three letters in “cis” stand for cosine, i, and sine. Handily, r cis θ has magnitude r and phase θ. When you multiply a complex number z by the complex number r cis θ you scale up z by a factor of r while rotating it around the origin by an angle of θ. Addition of complex numbers also has as geometric interpretation, but it’s best expressed not in polar form but rectangular form: when you add a+bi to a complex number, you shift it horizontally by a and vertically by b. The fact that addition and multiplication of complex numbers have geometric interpretations in terms of familiar operation like scaling, rotating, and shifting has a lot to do with the way complex numbers turn out to be useful in many surprising contexts.

One picayune but useful consequence of Euler’s monumental discovery is that you don’t have to memorize many trig formulas once you know how to traverse the passageway between the world of exponential functions and the world of trig functions; see Endnote #12. You can also use complex numbers to get algebraic proofs of certain geometric facts (see Jim Simons’ video “Three pretty geometric theorems, proved by complex numbers”) and to find nice solutions to combinatorial puzzles (see Endnote #13 as well as the 3Blue1Brown video “Olympiad-level counting”) and sometimes to reduce nasty-looking geometric optimization problems to manageable complexity (see Endnote #14).

More profound applications of the complex numbers turned up in 19th century mathematics, especially Bernhard Riemann’s work in number theory, leading French mathematician Paul Painlevé to write “Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.” (The saying was popularized by Jacques Hadamard through his book The Psychology of Invention in the Mathematical Field, in which he prefaced the adage by “It has been written…” without acknowledging Painlevé as the source.)

Even though the advent of complex numbers led to new beginnings in many branches of mathematics, in an important way, it was an ending too.  Earlier math had been full of equations whose solutions seemed impossible but which led to new kinds of numbers.  Want to solve 2x = 1? Invent fractions.  Want to solve x+2 = 1? Invent negative numbers.  Want to solve x² = 2? Invent irrational numbers.  Want to solve x² = −1? Invent imaginary numbers.  You might think we could keep at this game forever, writing down impossible equations and then inventing new numbers to render the impossible possible.

For instance, what about the equation x² = i?  You might think we need to go beyond the complex numbers to solve it.  But we don’t, and if you remember how multiplication of complex numbers works, it’s not hard to figure out where the square root of i is hiding in the complex plane: it has magnitude 1 and phase 45°. That is, it’s r + ir where r = sqrt(2). (There’s also a square root of i with phase 225° on the other side of the origin.) So we can solve x² = i in the complex number system without us having to bring new numbers into the game.

This is just one example of a theorem that’s so important that it’s called the Fundamental Theorem of Algebra: if you write down an equation (more specifically a polynomial equation) involving a single unknown number x, that equation (unless it reduces to something silly like x = x+1) will always have a solution in the system of complex numbers. So you could say that with the advent of complex numbers, the discipline of algebra, after many centuries of wandering and struggle, had finally found its true home.

WHAT IS REAL?

But wait — I sense a presence in the room: it’s the spirit of Asimov’s professor, and he’s downright gleeful. “All you’ve proved is that a spirit of mysticism has infected the world of science, thanks to the closet-mathematician Steinmetz and other traitors to Reality!” And (though it’s galling to have a ghost lecture me about mysticism versus realism) I have to admit he has a point. Steinmetz identified as a mathematician in his younger years, before he came to America and switched to engineering. And the “infection” has continued to spread.

At first the use of complex numbers was confined to branches of physics that studied wavelike phenomena. If you want to understand how light works in classical optics, you need to think of a photon as a kind of self-sustaining feedback loop between an electrical oscillation and a magnetic oscillation, propagating through space. To understand this screwlike motion, you need the twisty mathematics offered by complex numbers.

Then in the first half of the 20th century came the quantum revolution. Physicists came to realize that elementary particles (and to some extent objects made up of those particles, even including macroscopic ones like pieces of chalk) had a wavelike aspect, and that certain phenomena could only be understood if you treated complex numbers not just as a useful fiction but as part of the bedrock of reality.

An elementary particle, viewed as a wave, has a phase, and we can experimentally measure how particles’ phases change when they interact. Probabilities don’t just add; sometimes they cancel, interfering destructively the way waves do. (See Endnote #15.) Quantum physics has phase baked into its structure at the smallest scales that our current theories can reach. It’s not just light that behaves in a screwy way; quantum physics asserts that the whole damned universe is screwy, which is why we need twisty mathematics to describe it.

There’s another sense in which the sociology professor was sort of right (though several centuries behind the times): complex numbers did arise from an approach to math that renounced the physical world and even common sense. The 16th century algebraist Gerolamo Cardano, after deriving the complex roots of the “impossible” equation x²−10x+40=0, declared his own analysis to be “as subtle as it is useless”. Rafael Bombelli, building on Cardano’s work, made complex numbers more respectable by giving clear and consistent rules for operating with them, but he never attempted to explain what complex numbers were. (See Endnote #16.)

Hiding in the background of Bombelli’s work was the radical notion (announced more overtly in 19th century England) that if you give clear and consistent rules for operating with fictional quantities, then you can study those fictional quantities on their own terms as elements of a notional number system, deferring or dismissing the question of what those quantities actually mean. This gives license to a sort of “mysticism” in which mathematicians create new number-systems simply by specifying rules of operation, not worrying about whether or how these number-systems correspond to anything in the real world. Maybe there’ll be an application in a hundred years, or a thousand, or never; who knows? In the meantime, there’s plenty of exploring to do.

In Asimov’s anecdote, when the professor challenges Asimov to hand him the square root of minus one pieces of chalk, the brash undergrad says he’ll do it if the professor first gives him one-half of a piece of chalk. When the professor breaks a piece of chalk into two pieces and gives one to Asimov, saying “Now for your end of the bargain,” Asimov points out that what he’s been handed is a single (smaller) piece: one piece of chalk, not one-half. The professor counters that “one-half a piece of chalk” means one half of a standard piece of chalk, and Asimov asks the professor how he can be sure that it’s exactly half, and not, say, 0.48 or 0.52 of a standard piece of chalk.

What I take away from the end of Asimov’s story is that the difference between a “concrete” number like one-half and an “abstract” number like the square root of minus one is a difference in degree, not a difference in kind. Both are useful fictions. The fictional aspect of one-half comes into view when we notice that the professor’s attempt to hand Asimov half a piece of chalk depends on both a societal agreement on what a standard piece of chalk is and a societal agreement about how much error is permitted. The latter is a bit hazy; where do we draw the line between dividing something in half and dividing it into two unequal pieces? Come to think of it, I’m sure there are measurable differences between the different pieces of chalk that come out of a chalk factory. Quality control doesn’t require that the differences be indiscernible.  So the definition of a “standard” piece of chalk is a bit fuzzy too.

Of course I’m splitting hairs here, and ordinary conversation demands adherence to a largely unspoken agreement about which hairs to leave unsplit.  And that indeed is my point.  Even a seemingly simple mathematical concept like one-half is a collaboration between the universe and a society of minds observing the universe — just like the square root of minus one.

Or to put it more succinctly: Real numbers are more imaginary than most people realize, and imaginary numbers are more real than most people imagine.

Thanks to Richard Amster, Sidney Cahn, Jeremy Cote, Sandi Gubin, Henri Picciotto, Tzula Propp, and Paul Zeitz.

REFERENCES

Titu Andrescu and Zuming Feng, 102 Combinatorial Problems From the Training of the USA IMO Team, 2003.  

Isaac Asimov, Adding A Dimension, 1964.

Floyd Miller, The Man Who Tamed Lightning: Charles Proteus Steinmetz, 1965.

Paul Nahin, An Imaginary Tale: The Story of sqrt(−1), 1998.

David Richeson, The Scandalous History of the Cubic Formula, Quanta Magazine, 2022.

Danny Augusto Vieira Tonidandel, Steinmetz and the Concept of Phasor: A Forgotten Story, 2013.

Paul Zeitz, The Art and Craft of Problem Solving, 2006.

ENDNOTES

#1. Specifically, one method for solving the equation x³ = 15x + 4 involves writing the solution in the form

before rewriting it as

cancelling the two impossible terms of opposite sign, and concluding (correctly) that x=4 solves the problem. See the Veritasium video “How Imaginary Numbers Were Invented” as well as David Richeson’s Quanta Magazine article listed in the References for more on this.

#2. This picture is potentially misleading since current and voltage are measured in different units; superimposing them has no physical meaning. However, it’s still a helpful way to compare the phases of the two quantities. In the illustration, current lags behind voltage by one-quarter of a cycle, which is what happens when your only circuit elements are capacitors. The phase shift when your only circuit elements are inductors is also one-quarter of a cycle, but in the opposite direction, with voltage lagging behind current. For circuits that contain both capacitors and inductors, things get complicated; more specifically, as Steinmetz noticed, they get complex!

#3. The two functions I chose for the voltage and current in my figure depicting alternating current were 4 cos t and 3 sin t, two sine waves that are 90 degrees out of phase. Ignoring the fact that one of them represents a voltage and the other represents a current, let’s add them. Here’s a graph of the function 4 cos t + 3 sin t:

Notice that we get another sine-wave, but it’s in phase with neither 4 cos t nor 3 sin t. Interestingly, if you were to measure the amplitude of this function — the sum of a sine wave of amplitude 4 and a sine wave of amplitude 3 — you’d find that it’s exactly 5. And if you suspect that this equality has something to do with the 3-4-5 right triangle, then (just like the triangle) you are right! The crucial fact is that the two sine waves being combined were exactly 90 degrees out of phase with each other. If we’d added two sine waves that were in phase with each other, one of amplitude 4 and the other of amplitude 3, we’d get a sine wave of amplitude 4+3=7 because of constructive interference. In the opposite case, where the waves are 180 degrees out of phase with each other, we’d get a sine wave of amplitude 4−3=1 because of destructive interference. And in the intermediate case, where the waves are 90 degrees out of phase with each other, we get a sine wave of amplitude 4+3i or rather sqrt(4²+3²) = 5 (the magnitude of 4+3i). Sine waves, unlike pieces of chalk in a classroom, can interfere with each other constructively or destructively or in an intermediate manner.

#4. Some computer programmers who implement complex number arithmetic use a slight variant of the formula. On computers, multiplications are more time-consuming (or as one says “expensive”) than additions, so one often focuses on reducing the number of multiplications even if the number of additions is increased. Let E = ac, F = bd, and G = (a+b)(c+d). Then clearly E−F is ac−bd and you can check that G−E−F is ad+bc. So a computer can calculate the real and imaginary parts of the product of two complex numbers using just three real multiplications rather than the obvious four real multiplications.

#5. The magnitudes of 2+i and 3+i are respectively sqrt(2²+1²) = sqrt(5) and sqrt(3²+1²) = sqrt(10), whose product is sqrt(50), which is the magnitude of 5+5i, which is the product of 2+i and 3+i. Likewise, using some trigonometry you can show that if 𝛼 is the phase of 2+i and 𝛽 is the phase of 3+i, then 𝛼 plus 𝛽 is exactly 45 degrees, which is the phase of 5+5i. One way to prove this is to use the tangent addition formula: we know that tan 𝛼 is 1/2 and tan 𝛽 is 1/3, so tan 𝛼+𝛽 is

implying that 𝛼+𝛽 is 45°. A slicker way to prove the formula is to draw a picture in which a right angle (angle AOD) is decomposed into an angle of measure 𝛼 (angle AOB), an angle of measure 45 degrees (angle BOC), and an angle of measure 𝛽 (angle COD), as in the picture below:      

#6. Positive numbers have phase 0 degrees and negative numbers have phase 180 degrees.  So the rule for the sign of a product of real numbers as embodied in the table

is essentially the same as the rule for adding angles that are multiples of 180 degrees as embodied in the table

#7. I’ve heard of an intriguing bit of kinesthetic pedagogy that Michael Pershan and Max Ray-Riek developed as a way of informally introducing middle-schoolers to complex numbers. Kids arranged on a number line can be led to invent 90 degree rotation as a choreograpic enactment of multiplication by the square root of −1! Visit Henri Picciotto’s page Kinesthetic Intro to Complex Numbers to learn more. Teachers and students may also find other things of interest at Picciotto’s more advanced page for complex number pedagogy.

#8. Like most stories that shine a spotlight on a single pioneering innovator, my story leaves out a lot. Steinmetz wasn’t the first or only person to suggest using complex numbers to understand electrical circuits involving alternating current; several people had the idea independently at about the same time. But Steinmetz was the chief proponent of this method in the U.S., and in his writings he compellingly demonstrated its virtues.

#9. Gauss called numbers like 2+3i “complex” because of the way they are compounded of a real part (2) and an imaginary part (3i). This terminology stresses the additive side of complex numbers, that is, the way you can build them up by adding simpler components together. But that doesn’t tell us anything interesting about what complex numbers are or what they’re good for. Vectors (which we’ll meet in a later essay) are also built by adding simpler components together. For that matter, I could introduce “fruity numbers” like “2-apples-and-3-bananas”, and I could say things like “2-apples-and-3-bananas plus 5-apples-and-7-bananas equals 7-apples-and-10-bananas”, and I could represent fruity numbers using two-dimensional diagrams; then fruity numbers would look a lot like complex numbers and they’d behave the same way vis-a-vis addition, but they’d be very different from complex numbers. What’s distinctive about the complex numbers (as opposed to the fruity numbers) is the specific, meaningful way in which one can multiply them: when you multiply two complex numbers, the phases get added. That’s why I think “phased numbers” is a better name for them.

#10. Going back to the example with V(t) = 4 cos t and I(t) = 3 sin t (the first picture in the essay), we find that the complex voltage 4 e^it is equal to the complex current 3i e^it times the complex impedance 4/(3i). In contrast 4 cos t divided by 3 sin t is not a constant.

#11. The convention of trigonometry is that counterclockwise is the positive direction and clockwise is the negative direction. I suppose we mathematicians could try to convince the world to use clocks that go the other way, but it’s a hard sell; I expect we’ll have to wait a very long time before this happens.

#12. See for instance the video “Double Angle Identities Using Euler’s Formula“. Some may say “That’s a lot of algebra; isn’t it easier just to look it up?”, but I’m not the only mathematician I know who’d rather re-derive a trig identity via complex exponentials.

#13. Here’s a puzzle of mine that Paul Zeitz used in his book of contest problems.  Given a circle of n lightbulbs, exactly one of which is initially on, you’re allowed to change the state of a bulb (on versus off) provided you also change the state of every dth bulb after it (where d is a divisor of n other than n itself), provided that all n/d of the bulbs were originally in the same state as one another (that is, all on or all off).  For what values of n is is possible to turn all the lights on by making a sequence of moves of this kind?

For example, take n=12. We have 12 lights in a circle, one of which is on. You’re allowed to toggle 2, 3, 4, 6, or 12 bulbs from off to on (provided that they’re evenly spaced around the circle), and you’re also allowed to toggle 2, 3, 4, 6, or 12 bulbs from on to off (provided that they’re evenly spaced around the circle).  Taking as many moves as you need, can you get all the lights to be on?  If that’s too hard, can you get all the lights to be off?  Or if that’s still too hard, can you get there to be exactly one light on, but it’s a different light than the one that was on at the start?

If you like puzzles, this may be a good time to stop reading and think for a bit.

All of these tasks are impossible, and not just for n=12.  And we can prove it with complex numbers, provided we know one key fact: if you have two or more evenly-spaced points on the circle of radius 1 in the complex plane, their sum is 0.  I won’t prove the fact here, but let’s see how it shows that the lights puzzle can’t be solved.  The trick is to look at the sum of the positions of the bulbs that are on, using complex number addition.  The sum starts out being nonzero because exactly one light is on and (in the original version) the sum is supposed to end up being zero because all the lights should end up being off when we’re done.  But anytime you turn a bunch of lights on, they’re evenly spaced, so the sum of their positions is zero, which means that when you turn those lights on you’re not affecting the sum of the turned-on lights.  Likewise, when you turn a bunch of lights off, they’re evenly spaced, so the sum of their positions is again zero, and when you subtract zero from a complex number, you don’t change it.

#14. Here is a special case of a new problem I call the “repelling propellers problem”. I place blue dots at the 12 o’clock, 3 o’clock, 6 o’clock, and 9 o’clock positions on a circle. I want to place three red dots on the circle as well, 120 degrees apart from one another, in such a way as to maximize the product of all twelve of the red-point-to-blue-point distances. How do I do it? It looks nasty; there are twelve point-to-point distances to be multiplied, and each of them will be something like a trig function or involve a square root if we adopt a straightforward approach. But complex numbers yield a nice solution. Here again, you might want to stop reading and think on your own for a bit (though you’ll need to know some things about complex numbers that aren’t explained in my essay).

Let the circle in question be the unit circle in the complex plane, so that the blue points are at 1, i, −1, and −i. Let ω be cis 120° so that ω² is cis 240°; if z represents the position of one red point, the other red points are at ωz and ω²z. (You can think of the seven points as the tips of propellers that rotate around 0, with a repelling force between the red propeller tips and the blue propeller tips.)

The distance between two complex numbers 𝛼 and 𝛽 is the magnitude of their difference, namely |𝛼−𝛽|. Thus the function of z that we are trying to maximize (subject to the constraint that z must have magnitude 1) can be written as

Since the magnitude of a product of complex numbers equals the product of the magnitudes and vice versa, we can rewrite this expression as the magnitude of the product

But this product (call it P(z)) can be written in a much simpler way. To see how, consider the values of z that make the product equal to 0. Go back to the geometrical problem but consider the reverse desideratum: if you want to make the product of the twelve distances as small as possible, the best places to put those red dots are at 12 o’clock, 1 o’clock, 2 o’clock, …, 10 o’clock, and 11 o’clock, because for all of those positions, there’ll be a red dot and a blue dot that coincide and so are at distance zero from each other, which makes the product of the twelve distances all equal to zero as well. That means that the twelve complex numbers cis 0°, cis 30°, cis 60°, …, cis 300°, and cis 330° are all roots of the degree-twelve polynomial P(z). But those complex numbers are just the roots of the equation z¹² − 1 = 0. So P(z) must be divisible by z¹² − 1. But wait a minute: those two polynomials have the same degree! So P(z) must be equal to z¹² − 1 times some constant C.

We’re making progress here, and though you may be worried about the fact that I haven’t worked out what C is, you’ll soon see that we don’t need to know it. |P(z)|, the quantity we need to maximize, is |C (z¹² − 1)|, which equals |C| |z¹² − 1|, and |C| is constant, so all we’re really trying to do is maximize |z¹² − 1|, which is the distance between z¹² and 1. That is, we want to find a point z on the circle of radius 1 that makes z¹² as far from 1 as possible. But as z varies over the circle of radius 1, so does z¹², and the point on this circle that’s as far from 1 as possible is the point −1. So we need z¹² = −1, which we can achieve (for instance) with z = cis 15°, placing red dots at 2:30, 10:30, and 6:30. In fact, if you place a red dot halfway between any two consecutive hour-marks, and place the other two dots accordingly, you’ll get one of the four dot-configurations that maximizes the product of the red-to-blue distances. (If you’re curious, placing the dots in this way makes the product of all twelve of those distances equal to exactly 2. But my problem didn’t ask you to figure that out.)

In a more general version of the problem, there are two propellers, one with p evenly spaced blades and one with q evenly spaced blades. If you can solve the p=3, q=4 case, you shouldn’t find the general case much harder. An even more general version of the problem features more than two propellers; I don’t know a general solution.

#15. Freeman Dyson, in his article “Birds and Frogs”, published in 2009 in the Notices of the American Mathematical Society (volume 56, pages 212–223) wrote: “Schrödinger … started from the idea of unifying mechanics with optics. A hundred years earlier, Hamilton had unified classical mechanics with ray optics, using the same mathematics to describe optical rays and classical particle trajectories. Schrödinger’s idea was to extend this unification to wave optics and wave mechanics. Wave optics already existed, but wave mechanics did not. Schrödinger had to invent wave mechanics to complete the unification. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrödinger’s idea seemed to be going nowhere. But then came the surprise. Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And Schrödinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom. It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers. This discovery came as a complete surprise, to Schrödinger as well as to everybody else.”

#16. Bombelli’s Algebra wasn’t just the first text to explain the rules governing complex numbers; it was also the first clear European treatment of the rules governing negative numbers. Of course, Chinese and Indian mathematicians already knew about negative numbers and how to work with them, but they hadn’t tried taking square roots of negative numbers as far as I’m aware. Then again, the Indian mathematician Brahmagupta came up with a formula that in some ways foreshadows the discovery of complex numbers. Remember when I said that when you multiply two complex numbers, their magnitudes get multiplied? Write those two complex numbers as a+bi and c+di, so that their product is (ac−bd)+(ad+bc)i. The magnitudes of these three complex numbers are sqrt(a²+b²), sqrt(c²+d²), and sqrt((ac−bd)²+(ad+bc)²). So my assertion about how magnitudes multiply becomes the formula sqrt(a²+b²) sqrt(c²+d²) = sqrt((ac−bd)²+(ad+bc)²), which if you square both sides becomes the simpler but still surprising formula (a²+b²)(c²+d²) = (ac−bd)²+(ad+bc)², true for all real numbers a,b,c,d. This formula tells us that if two positive integers can each be written as a sum of two perfect squares, so can their product. Brahmagupta knew this formula (and others like it), but he couldn’t have known that a thousand years later it would play a role in the study of complex numbers!