I have another topic today suggested by Beth, of the I Didn’t Have My Glasses On …. inspiration blog. It overlaps a bit with other essays I’ve posted this A-to-Z sequence, but that’s all right. We get a better understanding of things by considering them from several perspectives. This one will be a bit more historical.
Pop science writer Isaac Asimov told a story he was proud of about his undergraduate days. A friend’s philosophy professor held court after class. One day he declared mathematicians were mystics, believing in things they even admit are “imaginary numbers”. Young Asimov, taking offense, offered to prove the reality of the square root of minus one, if the professor gave him one-half pieces of chalk. The professor snapped a piece of chalk in half and gave one piece to him. Asimov said this is one piece of chalk. The professor answered it was half the length of a piece of chalk and Asimov said that’s not what he asked for. Even if we accept “half the length” is okay, how do we know this isn’t 48 percent the length of a standard piece of chalk? If the professor was that bad on “one-half” how could he have opinions on “imaginary numbers”?
This story is another “STEM undergraduates outwitting the philosophy expert” legend. (Even if it did happen. What we know is the story Asimov spun it into, in which a plucky young science fiction fan out-argued someone whose job is forming arguments.) Richard Feynman tells a similar story, befuddling a philosophy class with the question of how we can prove a brick has a interior. It helps young mathematicians and science majors feel better about their knowledge. But Asimov’s story does get at a couple points. First, that “imaginary” is a terrible name for a class of numbers. The square root of minus one is as “real” as one-half is. Second, we’ve decided that one-half is “real” in some way. What the philosophy professor would have baffled Asimov to explain is: in what way is one-half real? Or minus one?
We’re introduced to imaginary numbers through polynomials. I mean in education. It’s usually right after getting into quadratics, looking for solutions to equations like . That quadratic has two solutions, but it’s possible to have a quadratic with only one, such as . Or to have a quadratic with no solutions, such as, iconically, . We might underscore that by plotting the curve whose x- and y-coordinates makes true the equation . There’s no point on the curve with a y-coordinate of zero, so, there we go.
Having established that has no solutions, the course then asks “what if we go ahead and say there was one”? Two solutions, in fact, and . This is all right for introducing the idea that mathematics is a tool. If it doesn’t do something we need, we can alter it.
But I see trouble in teaching someone how you can’t take square roots of negative numbers and then teaching them how to take square roots of negative numbers. It’s confusing at least. It needs some explanation about what changed. We might do better introducing them in a more historical method.
Historically, imaginary numbers (in the West) come from polynomials, yes. Different polynomials. Cubics, and quartics. Mathematicians still liked finding roots of them. Mathematicians would challenge one another to solve sets of polynomials. This seems hard to believe, but many sources agree on this. I hope we’re not all copying Eric Temple Bell here. (Bell’s Men of Mathematics is an inspiring collection of biographical sketches. But it’s not careful differentiating legends from documented facts.) And there are enough nerd challenges today that I can accept people daring one another to find solutions of .
Quadratics, equations we can write as for some real numbers a, b, and c, we’ve known about forever. Euclid solved these kinds of equations using geometric reasoning. Chinese mathematicians 2200 years ago described rules for how to find roots. The Indian mathematician Brahmagupta, by the early 7th century, described the quadratic formula to find at least one root. Both possible roots were known to Indian mathematicians a thousand years ago. We’ve reduced the formula today to
With that filtering into Western Europe, the search was on for similar formulas for other polynomials. This turns into several interesting threads. One is a tale of intrigue and treachery involving Gerolamo Cardano, Niccolò Tartaglia, and Ludovico Ferrari. I’ll save that for another essay because I have to cut something out, so of course I skip the dramatic thing. Another thread is the search for quadratic-like formulas for other polynomials. They exist for third-power and fourth-power polynomials. Not (generally) for the fifth- or higher-powers. That is, there are individual polynomials you can solve by formulas, like, . But stare at it and you can see where that’s “really” a quadratic pretending to be sixth-power. Finding there was no formula to find, though, lead people to develop group theory. And group theory underlies much of mathematics and modern physics.
The first great breakthrough solving the general cubic, , came near the end of the 14th century in some manuscripts out of Florence. It’s built on a transformation. Transformations are key to mathematics. The point of a transformation is to turn a problem you don’t know how to do into one you do. As I write this, MathWorld lists 543 pages as matching “transformation”. That’s about half what “polynomial” matches (1,199) and about three times “trigonometric” (184). So that can help you judge importance.
Here, the transformation to make is to write a related polynomial in terms of a new variable. You can call that new variable x’ if you like, or z. I’ll use z so as to not have too many superscript marks flying around. This will be a “depressed polynomial”. “Depressed” here means that at least one of the coefficients in the new polynomial is zero. (Here, for this problem, it means we won’t have a squared term in the new polynomial.) I suspect the term is old-fashioned.
Let z be the new variable, related to x by the equation . And then figure out what and are. Using all that, and the knowledge that , and a lot of arithmetic, you get to one of these three equations:
where p and q are some new coefficients. They’re positive numbers, or possibly zeros. They’re both derived from a, b, c, and d. And so in the 15th Century the search was on to solve one or more of these equations.
From our perspective in the 21st century, our first question is: what three equations? How are these not all the same equation? And today, yes, we would write this as one depressed equation, most likely . We would allow that p or q or both might be negative numbers.
And there is part of the great mysterious historical development. These days we generally learn about negative numbers. Once we are comfortable, our teachers hope, with those we get imaginary numbers. But in the Western tradition mathematicians noticed both, and approached both, at roughly the same time. With roughly similar doubts, too. It’s easy to point to three apples; who can point to “minus three” apples? We can arrange nine apples into a neat square. How big a square can we set “minus nine” apples in?
Hesitation and uncertainty about negative numbers would continue quite a long while. At least among Western mathematicians. Indian mathematicians seem to have been more comfortable with them sooner. And merchants, who could model a negative number as a debt, seem to have gotten the idea better.
But even seemingly simple questions could be challenging. John Wallis, in the 17th century, postulated that negative numbers were larger than infinity. Leonhard Euler seems to have agreed. (The notion may seem odd. It has echoes today, though. Computers store numbers as bit patterns. The normal scheme represents negative numbers by making the first bit in a pattern 1. These bit patterns make the negative numbers look bigger than the biggest positive numbers. And thermodynamics gives us a temperature defined by the relationship of energy to entropy. That definition implies there can be negative temperatures. Those are “hotter” — higher-energy, at least — than infinitely-high positive temperatures.) In the 18th century we see temperature scales designed so that the weather won’t give negative numbers too often. Augustus De Morgan wrote in 1831 that a negative number “occurring as the solution of a problem indicates some inconsistency or absurdity”. De Morgan was not an amateur. He coded the rules for deductive logic so well we still call them De Morgan’s laws. He put induction on a logical footing. And he found negative numbers (and imaginary numbers) a sign of defective work. In 1831. 1831!
But back to cubic equations. Allow that we’ve gotten comfortable enough with negative numbers we only want to solve the one depressed equation of . How to do it? … Another transformation, then. There are a couple you can do. Modern mathematicians would likely define a new variable w, set so that . This turns the depressed equation into
And this, believe it or not, is a disguised quadratic. Multiply everything in it by and move things around a little. You get
From there, quadratic formula to solve for . Then from that, take cube roots and you get three values of z. From that, you get your three values of x.
You see why nobody has taught this in high school algebra since 1959. Also why I am not touching the quartic formula, the equivalent of this for polynomials of degree four.
There are other approaches. And they can work out easier for particular problems. Take, for example, which I introduced in the first act. It’s past the time we set it off.
Rafael Bombelli, in the 1570s, pondered this particular equation. Notice it’s already depressed. A formula developed by Cardano addressed this, in the form . Notice that’s the second of the three sorts of depressed polynomial. Cardano’s formula says that one of the roots will be at
Put to this problem, we get something that looks like a compelling reason to stop:
Bombelli did not stop with that, though. He carried on as though these expressions of the square root of -121 made sense. And, if he did that he found these terms added up. You get an x of 4.
Which is true. It’s easy to check that it’s right. And here is the great surprising thing. Start from the respectable enough equation. It has nothing suspicious in it, not even negative numbers. Follow it through and you need to use negative numbers. Worse, you need to use the square roots of negative numbers. But keep going, as though you were confident in this, and you get a correct answer. And a real number.
We can get the other roots. Divide out of . What’s left is . You can use the quadratic formula for this. The other two roots are , about -0.268, and , about -3.732.
So here we have good reasons to work with negative numbers, and with imaginary numbers. We may not trust them. But they get us to correct answers. And this brings up another little secret of mathematics. If all you care about is an answer, then it’s all right to use a dubious method to get there.
There is a logical rigor missing in “we got away with it, I guess”. The name “imaginary numbers” tells of the disapproval of its users. We get the name from René Descartes, who was more generally discussing complex numbers. He wrote something like “in many cases no quantity exists which corresponds to what one imagines”.
John Wallis, taking a break from negative numbers and his other projects and quarrels, thought of how to represent imaginary numbers as branches off a number line. It’s a good scheme that nobody noticed at the time. Leonhard Euler envisioned matching complex numbers with points on the plane, but didn’t work out a logical basis for this. In 1797 Caspar Wessel presented a paper that described using vectors to represent complex numbers. It’s a good approach. Unfortunately that paper too sank without a trace, undiscovered for a century.
In 1806 Jean-Robert Argand wrote an “Essay on the Geometrical Interpretation of Imaginary Quantities”. Jacques Français got a copy, and published a paper describing the basics of complex numbers. He credited the essay, but noted that there was no author on the title page and asked the author to identify himself. Argand did. We started to get some good rigor behind the concept.
In 1831 William Rowan Hamilton, of Hamiltonian fame, described complex numbers using ordered pairs. Once we can define their arithmetic using the arithmetic of real numbers we have a second solid basis. More reason to trust them. Augustin-Louis Cauchy, who proved about four billion theorems of complex analysis, published a new construction of them. This used a group theory approach, a polynomial ring we denote as . I don’t have the strength to explain all that today. Matrices give us another approach. This matches complex numbers with particular two-row, two-column matrices. This turns the addition and multiplication of numbers into what Hamilton described.
And here we have some idea why mathematicians use negative numbers, and trust imaginary numbers. We are pushed toward them by convenience. Negative numbers let us work with one equation, , rather than three. (Or more than three equations, if we have to work with an x we know to be negative.) Imaginary numbers we can start with, and find answers we know to be true. And this encourages us to find reasons to trust the results. Having one line of reasoning is good. Having several lines — Argand’s geometric, Hamilton’s coordinates, Cauchy’s rings — is reassuring. We may not be able to point to an imaginary number of anything. But if we can trust our arithmetic on real numbers we can trust our arithmetic on imaginary numbers.
As I mentioned Descartes gave the name “imaginary number” to all of what we would now call “complex numbers”. Gauss published a geometric interpretation of complex numbers in 1831. And gave us the term “complex number”. Along the way he complained about the terminology, though. He noted “had +1, -1, and , instead of being called positive, negative, and imaginary (or worse still, impossible) unity, been given the names say, of direct, inverse, and lateral unity, there would hardly have been any scope for such obscurity”. I’ve never heard them term “impossible numbers”, except as an adjective.
The name of a thing doesn’t affect what it is. It can affect how we think about it, though. We can ask whether Asimov’s professor would dismiss “lateral numbers” as mysticism. Or at least as more mystical than “three” is. We can, in context, understand why Descartes thought of these as “imaginary numbers”. He saw them as something to use for the length of a calculation, and that would disappear once its use was done. We still have such concepts, things like “dummy variables” in a calculus problem. We can’t think of a use for dummy variables except to let a calculation proceed. But perhaps we’ll see things differently in four hundred years. Shall have to come back and check.
Thank you for reading through all that. Once again a topic I figured would be a tight 1200 words spilled into twice that. This and the other A-to-Z topics for 2020 should be at this link. And all my A-to-Z essays, this year and past years, should be at this link.
I’m still looking for J, K, and L topics for coming weeks. I’m grateful for any subject nominations you’d care to offer.