I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.
After the amazing response to my post about zero, I thought I’d do one about something that’s fascinated me for a long time: the number i, the square root of -1. Where’d this strange thing come from? Is it real (not in the sense of real numbers, but in the sense of representing something real and meaningful)? What’s it good for?
History
The number i has its earliest roots in some of the work of early arabic mathematicians; the same people who really first understood the number 0. But they weren’t quite as good with i as they were with 0: they didn’t really get it. They had some concept of roots of a cubic equation, where sometimes the tricks for finding the roots of the equation just didn’t work. They knew there was something going on, some way that the equation needed to have roots, but just what that really meant, they didn’t get.
Things stayed that way for quite a while – people could see that there was something wrong, something missing in a variety of mathematical concepts, but no one understood how to solve the problem. What we know as algebra developed for many years, and various scholars, like the Greeks, encountered them in various ways when things didn’t work, but no one really grasped the idea that algebra required numbers that were more than just points on a one-dimensional number-line.
The first real step towards i was in Italy, over 1000 years after the Greeks. During the 16th century, people were searching for solutions to the cubic equations – the same thing that the early arabic scholars were looking at. But getting some of the solutions – even solutions to equations with real roots – required playing with the square root of -1 along the way. It was first really described by Rafael Bombelli in the context of the solutions to the cubic; but Bombello didn’t really think that i was something real or meaningful in the field of numbers; he just viewed it as a peculiar but useful artifact in the process of solving cubic equations. But beyond its use in finding tricks for solving cubic equations, it wasn’t accepted as an actual number.
It got its unfortunate misnomer, the “imaginary number” as a result of a diatribe by Rene Descartes. Descartes was disgusted by it; he believed it was a phony artifact of sloppy algebra. He did not accept that it had any meaning at all: thus he termed it an “imaginary” number, as part of an attempt to discredit the concept.
They finally came into wide acceptance as a result of the work of Euler in the 18th century. Euler was probably the first to really, fully comprehend the complex number system created by the existence of i. And working with that, he discovered one of the most fascinating and bizarre mathematical discoveries ever, known as Euler’s equation. I have no idea how many years it’s been since I was first exposed to this, and I still have a hard time wrapping my head around why it’s true.
And what that really means is:
That’s just astonishing. The fact that there is such a close relationship between i, π, and e is just shocking.
What i does
Once the reality of i as a number was accepted, mathematics was changed irrevocably. Instead of the numbers described by algebraic equations being points on a line, suddenly they become points *on a plane*. Numbers are really two dimensional; and just like the integer “1” is the unit distance on the axis of the “real” numbers, “i” is the unit distance on the axis of the “imaginary” numbers. As a result numbers in general become what we call complex: they have two components, defining their position relative to those two axes. We generally write them as “a + bi” where “a” is the real component, and “b” is the imaginary component.
The addition of i and the resulting addition of complex numbers is a wonderful thing mathematically. It means that every polynomial equation has roots; in particular, a polynomial equation in “x” with maximum exponent “n” will always have exactly “n” complex roots.
But that’s just an effect of what’s really going on. The real numbers are not closed algebraically under multiplication and addition. With the addition of i, multiplicative algebra becomes closed: every operation, every expression in algebra becomes meaningful: nothing escapes the system of the complex numbers.
Of course, it’s not all wonderful joy and happiness once we go from real to complex. Complex numbers aren’t ordered. There is no < comparison for complex numbers. The ability to do meaningful inequalities evaporates when complex numbers enter the system in a real way.
What i means
But what do complex numbers mean in the real world? Do they really represent real phenomena? Or are they just a mathematical abstraction?
They’re very real. There’s one standard example that everyone uses: and the reason that we all use it is because it’s such a perfect example. Take the electrical outlet that’s powering your computer. It’s providing alternating current. What does that mean?
Well, the voltage – which (to oversimplify) can be viewed as the amount of force pushing the current – is complex. In fact, if you’ve got a voltage of 110 volts AC at 60 hz (the standard in the US), what that means is that the voltage is a number of magnitude “110”. If you were to plot the “real” voltage on a graph with time on the X axis and voltage of the Y, you’d see a sine wave:
But that’s not really accurate. That implies that the current is basically shutting on and off really quickly! But it’s not – there’s a certain amount of energy in the alternating current, and the amount of energy is actually constant over time. But it’s a system in motion: it’s constantly changing. The voltage at time t1 on the complex plane is a point at “110” on the real axis. At time t2, the voltage on the “real” axis is zero – but on the imagine axis it’s 110. In fact, the magnitude of the voltage is constant: it’s always 110 volts. But the vector representing that voltage is rotating through the complex plane. When it’s rotated entirely into the imaginary plane, the energy is expressed completely as a magnetic field. This is really typical of how i applies in the real world: it’s a critical part of fundamental relationships in dynamic systems with related but orthogonal aspects, where it often represents a kind of rotation or projection of a moving system in an additional dimension.
You also see it in the Fourier transform: when we analyze sound using a computer, one of the tricks we use is decomposing a complex waveform (like a human voice speaking) into a collection of basic sine waves, where the sine waves added up equal the wave at a given point in time. The process by which we do that decomposition is intimately tied with complex numbers: the fourier transform, and all of the analyses and transformations built on it are dependent on the reality of complex numbers (and in particular on the magnificent Euler’s equation up above).
I highly recommend the intuitive and informal presentation on the complex numbers in Richard Feynman’s Lectures on Physics (volume 1).
I think you’ve mashed up several unrelated concepts.
If you solve Maxwell’s equations in free space you get a Laplacian of electric and magnetic fields — a second order differential equation. Solve that and you get the classic sinusoid equation. That usually gives you the exchange between electric and magnetic fields, but I seem to recall that they’re out of phase in ‘circularly polarized’ light. Of course quantization plays havoc on all of this.
If you look at power from a wall socket, the magnetic field is strongest when the electric flow is highest, and zero when it’s zero. There’s energy in the changing magnetic field that will retard the change in the electric field, but it’s trivial compared to the power coming from the utility.
However everyone but residences and offices uses ‘three phase’ power. That’s three wires containing alternating current at 120 degrees from each other. (That is, you have current with phase 0, 120′ and 240′). The power from an alternating electric current is sin(theta)^2 (where theta is the phase) and if you work the numbers with three phase current you’ll find it’s a constant value. It’s akin to the well-known sin(theta)^2 + cos(theta)^2 = 1. That’s why generators produce three phase power and industrial motors and other processes use it.
Aside: that’s why power lines usually have three lines. You only see exceptions at the ends where residences get two wires and neighborhoods get four (for the ‘common’ line). You get household power by connecting to one of the three lines plus a common line shared by all households in the neighborhood. The common line is being pulled by the power usage of hundreds of houses so should be close to zero. But not quite, so you can still get a shock if you short the common and ground lines. Hence ‘ground fault’ interruptors in wet areas where you could accidently create a short.
Diversity mathetmatics,
(pi)^4 + (pi)^5 = e^6
31, 331, 3331… are primes
n^2 + n + 41 generates primes for n = 1,2,3…
Um, the real numbers are a complete ordered field, so they very much are closed under addition and multiplication. What you’re thinking of is that they’re not closed under taking roots, or in more generality, extracting roots of arbitrary polynomials.
Peter Klausler: Asimov also wrote several very good descriptions of imaginary numbers, including one in either Realm of Numbers or Realm of Algebra
Uncle Al: Maybe I missed the point of your comment, but:
pi^4 + pi^5 seems to differ from e^6 by about four-millionths of a percent; eerily close, but not equal.
33331, 333331, 3333331, and 33333331 are also primes, but 333333331 is not–no pattern such as this will indefinitely generate primes; although it’s self-evident why this pattern does so well for 3s in base-10.
Your (well, Euler’s) prime number formula breaks for n=40 (obviously)–no polynomial generates only primes.
Mark: You tantalizingly mention being amazed by the relationship between e, i, and pi, and why it is true. I agree that it’s an amazing relation, but could you elaborate for those of us less math savvy why it’s true? I’ve always been amazed by it, but never known the underlying reason.
@#3:
First:
pi^4 = 97.4090910…
pi^5 = 306.0196848…
e^6 = 403.4287935…
(e^6)-(pi^4+pi^5) = 1.76734512… × 10^(-5)
Second:
333,333,331 = 17 × 19,607,843
Third:
Obviously for n=40 and n=41, n^2 + n + 41 is not prime.
For n=40, n^2 + n + 41 = 1681 = 41^2
For n=41, n^2 + n + 41 = 1763 = 41×43
Fine post in general but the example is a little strange. While we often use imaginary numbers in describing the steady state behavior of AC systems (although they are not necessary just a more convenient way to represent the numbers) the magnitude at any particular moment is not constant and the energy may or may not be stored depending on the nature of the load.
The amplitude of the sinusoidal wave maybe constant and over time the average magnitude is constant but at any particular moment the outlet does have a particular value. For instance you could sample the value with a fast device and would most assuredly measure a different value.
Also the energy is not constant unless you are powering a constant power device. If the load is purely resistive the power really and truly would be less or more at any given time. If you had a capacitive or inductive load the energy would be stored for a while but it does not “move” into the complex plane. If this is the typical example of the physical meaning of complex numbers I think you need another one.
I know it’s not really at the level of this blog post, but it’s certainly at the level of some previous blog posts, so why not do a post on contour integration? The way the theorems of contour integration just work, and the beautiful structure embodied in the cauchy-reimann equations would probably surprise someone who has never been exposed to it before.
Was Descartes ever right about anything? It seems like every thing I’ve read by him, or seen referenced about him, was him making an incredibly specious argument and then denouncing (often preemptively) anyone who disagreed with it.
@#9
He developed Cartesian Coordinates (which is where they got the name “Cartesion”) and did a bunch with curves and such. Stephen Hawking has a chapter on him in his book God Created the Integers
Hi Mark,
Paul J Nahin [EE PhD], ‘An Imaginary Tale: The Story of [the Square Root of Minus One]’ is an excellent read of the history of the imaginary unit.
This book touches upon Charles Proteus Steinmetz [GE, chief EE] using Grassman Algebra to associate “i” with phasor equations used in alternating electricity.
About 30 years after Steinmetz, Heisenberg and Schrodinger began using “i” in quantum mechanics. The PhD mathematician with a BS in EE, but no physics degree, PAM Dirac almost certainly had to be aware of the value of the imaginary unit and its association with electromagnetism.
The easiest way to make sense of the formula
exp(iθ) = cos(θ) + i sin(θ)
in a fairly rigorous way is to regard the functions exp, sin, and cos as shorthands for their Maclaurin series. Just prove (or take it on authority, if you’re lazy like me) that the series are absolutely convergent everywhere on the complex plane, and Bob’s your uncle.
I believe that voltage standards refer to root-mean-square (RMS) voltage, not peak voltage. So the US standard of 120v at source, peaks at about 170v, not at 120v. And when you say, “the magnitude of the voltage is constant,” is it possible that you mean the RMS voltage is constant? RMS voltage, being effectively a type of average, could be referred to as “constant,” I suppose, but I don’t believe the instantaneous voltage could be called “constant” over time.
The complex plane is a 2-dimensional surface that differs from regular 2-d euclidean space because you conjugate rather than multiply.
We know we can generalise euclidean spaces to many different dimensions so why not generalise to complex spaces as well?
This has been tried by, IIRC Hamilton, who came up with quaternions. For this system, conjugation is between 1, i, j and k with i^2 = j^2 = k^2 = -1. This has been further generalised to octonions and beyond and John Baez has used such in his quantum gravity research.
I am on-and-off playing with a variation of quaternions in which conjugation is by 1, i and j, in which;
1i = ij = j1 = -i1 = -ji = -i1 = -1.
I know, it violates 1*x = x but only in conjugation and it’s fun to mess with it.
eddie has underspecified quaternions. To his definitional identities he needs to add ijk = -1.
I’m all in favor of fun with Math, but it seems to me that he “fun” is contradictory, and in any case is not quaternions if it is somehow consistent.
Compare to known results on noncommutative division algebra, or on the definition of division algebras in general, and why quaternions are such a special case.
Thanks for a marvelous post. I never really groked the relationship between electrical and magnetic fields, although I could do the math just fine. Today I got it.
Chris @#2: the electrical and magnetic fields are out of phase. That’s exactly what MarkCC is saying. And, it’s exactly what I just understood.
I’m going to go read up on Maxwell’s equations again. Perhaps this time I’ll see the real beauty in them that’s always escaped me.
Your comment about
eiπ = -1
is dead on. It makes no immediate intuitive sense, yet it bind together (when stated in the form eiπ + 1 = 0) what are arguably the five fundamental constants to mathematics.
I have a sweatshirt and a t-shirt that say “eiπ + 1 = 0”, and I’ve gotten literally dozens of people asking me about it. “Is that really true?” “What does that… um… *mean*?” “So does eiπ + 1 equal negative 1, then?” etc. When asked about my peculiar obsession with the equation, my response is simple: It’s elegant, true, and binds fundamental math concepts together. It’s practically the definition of fascinating.
I love the proof, too. I can start from basic algebra, get to deriving the necessary Taylor series, and combine them as necessary. It’s truly fascinating. But then, I’m somewhat of a nerd.
Descartes is indeed considered to be one of the co-founders of analytical geometry along with Pierre Fermat, he of the famous “last theory”, but if you read his Geometry you want find any mention of rectangular co-ordinates, he doesn’t use them. Rectangular co-ordinates are in fact much older and were first introduced in the cartography of Ptolemaeus in the 2nd century CE. The first person to use them in analytical geometry was the Dutch mathematician Jan de Witt (1625 – 1672) in his Elementa curvarum linearum which was published as an appendix to Frans van Scooten’s (1615 – 1650) Geometria a Renato Des Cartes his improved and expanded Latin edition of Descartes’ Geometry, the original was in French. Newton, who introduced polar co-ordinates, used van Schooten’s book to teach himself the “new” mathematics.
Mark, your description of the history of ‘i’ is somewhat incorrect. The Arab mathematicians knew and used the general solution for quadratic equations and had individual solution for some forms of cubic equations but they only considered positive solutions. They were in no way aware of anything being missing or of that there are other roots to the equations. The first person to use complex solutions to polynomials was Girolamo Cardano (1501 – 1576) in his Ars Magna (Nuremberg, 1545) who used them in the intermediate stages of algebraic proofs when they cancelled out to deliver a real solution. He regarded this process as legitimate but the complex numbers themselves as ridiculous. Rafael Bombelli (1526 – 1572) in his Algebra (1572) gave the complete rules for addition, subtraction and multiplication of complex numbers and used them in the solution of polynomials much more extensively than Cardano although he also could not accept them as numbers. Albert Girard (1595 – 1632) first formulated the fundamental theory of algebra, that a polynomial has as many roots as its degree, thereby leading to the gradual acceptance of both negative and imaginary numbers.
Of course, Jonathan, in #14 I gave a much simplified acount, both of quaternions and my own musings. I am not redefining unity to make a contradiction. More precicely, I am using an epsilon matrix with indices 1, i and j.
Quaternions use epsilonijk to give chiral relationship between 3 imaginary directions, with real part still separate. My epsilon1ij gives chirality between real and 2 imaginary axes.
I will look up division algebras as you advise.
Of course, you mean j not i which every one knows refers to the current in an electrical circuit.
This is a very interesting and well written piece overall, but there are a couple of points of note.
First, the real numbers are closed under addition and multiplication. Not every polynomial has a real root(and by extension, not all of them are reducible in the real numbers).
You also seem to imply that the complex numbers are the only number system. When it is often useful to speak of number of subsets such as only the integers, only the rationals, etc or in some cases supersets such as the surreals which are useful for some analysis in game theory. In other situations even more exotic number systems are useful. I suspect that is not what you meant, but it is how the text seems to read.
I second John Armstrong, I think you meant to say that the reals are not algebraically closed. As John pointed out, they’re certainly a field.
There are some facts in mathematics whose proof entirely depends on complex numbers, even though they are not about complex numbers at all. If you continue to maintain that i doesn’t exist, then you have to accept that those facts are mere coincidences.
I am aware of no way to prove the following fact without complex numbers (you can use some very complicated trig identities, but those are derived with complex numbers, I think): make n equally spaced marks around a unit circle; then draw the n-1 chords from any one of those marks to all the others; amazingly, the product of the lengths of all of those chords always equals n.
http://polymathematics.typepad.com/polymath/proof_of_the_coolest_math.html
Re #4,#23:
Yes, you’re right – I did mean algebraically closed.
George might like this.
“One of the most profound jokes of nature is the square root of minus one that the physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926…. 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. According to Schrödinger, his fourteen-year-old girl friend Itha Junger said to him at the time, ‘Hey, you never even thought when you began that so much sensible stuff would come out of it.’ All through the nineteenth century, mathematicians from Abel to Riemann and Weierstrass had been creating a magnificent theory of functions of complex variables. They had discovered that the theory of functions became far deeper and more powerful when it was extended from real to complex numbers. But they always thought of complex numbers as an artificial construction, invented by human mathematicians as a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first.”
Birds and Frogs
Freeman Dyson, Notices of the American Mathematical Society, February 2009, Volume 56, Issue 02
The author calls mathematicians who take a lofty conceptual view of their subject birds and those who work in details and solve their problems consecutively frogs. The author shows, by examples from the past and his personal acquaintance, how both types have advanced mathematics.
I’m always a bit annoyed when people use the phrase “the square root of -1”.
i is defined as a number such that i^2 = -1. That makes it a square root of -1. However, it isn’t the square root.
When we say “the square root”, there either is only one square root (e.g. the number 0, modular arithmetic), or we’re talking about the principal square root. However, -1 doesn’t have a principal square root.
This isn’t just semantics, it actually goes quite deep. Complex conjugation is an R-automorphism of C. Un other words, i and -i are completely interchangable, as seen from R.
Lastly, a few random points:
1. The field of complex numbers is representable using matrices; a+bi can be thought of as the matrix [ [ a b ] [ -b a ] ]. So if “i doesn’t exist”, then matrices don’t exist.
2. According to David Hestenes, every instance of complex numbers in physics actually has a geometric interpretation in Clifford algebra. So in that sense, i doesn’t actually exist; it’s a simplification for geometric concepts.
thanks
In the case of voltage, I think you’re confusing a complex representation that is convenient for certain calculations with the thing being represented.
AC voltage is real valued.
It has an amplitude, frequency, and phase, all measurable.
Euler’s formula allows you to to represent a real valued sinusoid as the sum of two complex valued functions. But you can always convert back to the the actual sinusoid with:
cos(wt+p)=Real{e^(j(wt+p))}
Imagine if a creationist had made the errors you’ve made. Scienceblogs would have a field day with it.
i before e except after c?