Most people are probably aware that the square root of -1 isn’t a regular real number: it’s the imaginary number, which we often write as i. Of course, imaginary numbers aren’t any more “imaginary” than many other useful mathematical constructs; my feeling is that, if something in math can be used for something measurable, then it’s real in a meaningful sense.
People often ask the same question when they learn about imaginary numbers: if i is the square root of -1, what’s the square root of i? Is it a new kind of imaginary number, or can you write it in terms of “regular” imaginary numbers, or is it even a meaningful question? As you might expect, the question is not only meaningful, but leads to some interesting geometrical insights (even if most of us won’t find it as exciting as Zach Weiner’s character in his comic strip).
Complex numbers in
Complex numbers are the combination of real and imaginary numbers. In general, we write complex numbers as the sum of the real part and the imaginary part, and often find it useful to plot them on the complex plane, as shown at the left. All real (“ordinary”) numbers lie on the horizontal axis, and all imaginary numbers (which are just multiples of i) lie along the vertical axis.
An equivalent but often more useful way of writing complex numbers uses the exponential function:
where r and φ are both real numbers. Additionally, r is a positive number that represents the magnitude of the complex number: the distance from the origin of the complex plane to the point represented by z. The relative weight to the real and imaginary portions are represented by φ (the Greek letter “phi”, pronounced either “fye” or “fee”, depending on how pedantic you want to be), which is known as the phase. Finally, e is the exponential number, roughly equal to 2.71828 (but going on to infinite number of digits). The exponential number, like π, is a fundamental geometric quantity.
The exponential part of the equation is what we most care about, since r is just a scaling factor. We can write it as a sum of trigonometry functions:
Think of it like this: the complex number is like one point on a right triangle, whose
hippopotamus hypotenuse connects the number to the origin. The legs of the triangle are the real and imaginary parts of the complex number.
Now we can see that the phase φ represents an angle, but we need to use radians instead of degrees. One full circle is 2π radians (360°), a half-circle is π radians (180°), and a quarter circle is π/2 radians (90°). With that, we can see that positive real numbers are just complex numbers where the phase φ = 0, and negative real numbers correspond to complex numbers with φ = π. We also get Euler’s formula, which I wrote about in a previous blog post:This is one of those really interesting formulas, since it takes two irrational numbers (e and π) with the imaginary unit i, and the result is…a negative integer.
The phase is like the hour hand on a clock: there’s a bit of redundancy built in. If you add or subtract 2π from any phase, you get the same complex number!In fact, you can do the same with 4π, 6π, or any other even number multiplying π —they all give you the same complex number. We’ll use that redundancy shortly.
Totally radical, man
Now let’s get back to the original definition of i:The notation in the last expression might not be familiar, but it’s pretty straightforward. The square root reverses the action of squaring something:(Savvy people have noticed that I’m neglecting the negative square root solution, but hold on: we’re getting there.) In fact, that’s one particular version of a general rule: if you raise a number to an exponent, then follow that with another exponent, that’s the same as raising the first number to the multiple of both exponents. For example,
If the exponent is a whole number, then its meaning is fairly obvious: x2 means you multiply the number x by itself, x3 means you multiply the number x by x2, and so forth. If the exponent is a fraction, the meaning is a little less clear, but still perfectly manageable.
So let’s combine the exponent rules with Euler’s formula:That’s a great consistency check, but it doesn’t tell us anything we didn’t already know. Or does it? Let’s go back to the complex plane and see what the square root did in a geometrical sense:
That’s very interesting: the square root acts like a rotation, moving the dot from its location at -1 to a new place at i along the circumference of the circle. (Update: see note at the end of the post.)
Now we’re finally ready to answer the question from the beginning of the post: what is the square root of i?In other words, the square root of i doesn’t need a new mathematical concept: it’s just another complex number. Using the complex plane again, we can see that the square root again just rotates things around.
So, at its root, the square root function is a rotation. (Another name for “root” is “radical”, from “radix”, from which we also get the word “radish”. So the next time you think of political radicals, think of giant walking radishes reciting square roots. Or something.)
One number enters, two numbers leave
Now let’s turn to an easy and obvious question: what’s the square root of 1? One answer is quick:However, we determined earlier that we can add 2π to any phase angle and get back the same complex number, so let’s repeat the process with that knowledge in hand:Obviously 1 and -1 aren’t the same number, so there are two distinct answers for the square root of 1, as we expect. Complex numbers show us another way to see that, and as before, we can think of it as a rotation.
We can do the same trick for any complex number, as you might guess:andThese second square roots are all a rotation by π away from the other solutions we found previously! The square root of every complex number will have two different solutions, similarly separated by π radians. Since positive real numbers and negative real numbers are also separated by π radians in the complex plane, they also obey the same rule.
Extending everything we’ve learned to an arbitrary complex number is really easy: Since r is always a positive real number, we can calculate its square root using a calculator (or using the binomial series, which is how I learned to do it in geometry class back in 1933). The act of taking the square root ends up rescaling and rotating any complex number. If I remember to get around to it in my copious spare time, there’s some deep stuff going on there, relating to something known as conformal geometry. We can also keep going with cube roots (found by using 1/3 instead of 1/2, and getting three solutions instead of 2), or any exponent we want.
Now let’s look at the mathematical expression that got Zach Weiner’s protagonist all hot and bothered: raising i to the power of i. As his ladyfriend pointed out, it’s a real number:to three decimal places (it’s an irrational number). That won’t be true for just any complex number, since the real number r raised to the power of i will be complex. However, that’s a story for another day. You wouldn’t want me to use up all the mathy talk in one post, would you?
As a commenter noted and as I said above, the operation of taking a power is not a rotation in the usual sense. For hints on how to define rotations in a more rigorous way, see my earlier posts on quaternions, Clifford algebras, and complex numbers. I was attempting to use an analogy to clarify how we can understand the phase of a complex number, and as a lead-in to a possible future post on conformal geometry.
17 responses to “Talk mathy to me: what’s the square root of i?”
You said: “..the square root again just rotates things around.”
It is true that, with appropriate definitions, the square root has angle equal to half the angle of what you started with. (Why did you not just say that? Strictly, every angle corresponds to an infinity of numbers, differing by adding or subtracting a number of 2\pi’s . And halving all those give the numbers measuring two different angles, those being the angles of the two different square roots of the given number.This also makes it easy to understand cube roots, etc. )
But what you say above is hard to accept as correct:
A rotation is supposed to be a rotation by a fixed angle, and if it is taken correctly to be that, then the square root function is in no way a rotation.
On the other hand, if you really want a rotation to allow changes of angle of all sorts of amounts (in fact every possible amount, depending on which number you are ‘square rooting’), then absolutely every complex valued function of a complex variable is a rotation in that unorthodox sense, and that way of defining rotation is useless, perhaps worse than useless. I think you have a weak sense of even the most basic idea of such a function.
And that is not to say that looking at things geometrically is bad; no, it’s good, and I commend your effort there.
No, it’s not useful to think of every function of a complex variable as a rotation, and I wouldn’t claim it is. I’m restricting myself to powers and (as I noted in the post) even those rescale complex numbers unless r = 1, so they aren’t simply rotations when that isn’t the case.
Now that I’ve said that, I really wish you wouldn’t call me an idiot in the comments of my own blog. That’s a bit offensive, no?Correction: the commenter did not call me an idiot. I was a little offended by the tone of the sentence “I think you have a weak sense of even the most basic idea of such a function.” We writers are only human, and sometimes let comments get under our skins. My apologies for reacting as strongly as I did.
Firstly, the only person so far to call you an “idiot” is you, and I certainly did not.
Thank you for the clarification. The more accurate you can be, the better it is for those learning something new.
Pardon for pointing out another misconception which would cause confusion if any of your readers were to go further to learn math elsewhere because of your inspiration: But in another response below, saying that “…are orthogonal in the sense that they’re linearly independent” is again false with respect to the way mathematicians use those words. Being orthogonal implies linear independence, but not conversely, and that is one of the first important lessons to be learned in linear algebra.
To go back to the top paragraph, I am now aware of your level of sensitivity to being told that you are wrong on some point which the responder happens to know about. I shall avoid doing so in future, by the simple expedient of not bothering to read what you write. A second motivation for that is my earlier hope of learning new things here about physics, a subject where I need lots of study to get anything like satisfactory knowledge. But now, seeing the superficial knowledge (or at least unfortunate flippancy) of elementary math exhibited, I fear the same may be true of physics, and getting knowledge from other sources will be better for me.
Thanks for your efforts.
An excellent blog post–well done, Matthew Francis! Thank you for sharing!
Excellent and clear blog post (as usual) Dr. Francis. Thank you.
Question: The artifice of the imaginary axis being perpendicular to the real axis. Is that a convention or is there a deeper mathematical logic to it? If it is just a convention then the idea of a square root being a rotation is the consequence of an artifice used to represent a complex number.
The real and imaginary axes are
orthogonal in the sense that they’relinearly independent: if you have x + i y = 0, that means both x and y have to be zero for the equation to be true. (Correction from my earlier statement: linear independence doesn’t necessarily imply the axes are orthogonal, as another commenter pointed out. This is what happens when I try to write something coherent before 6 AM.) To define orthogonality, you need to have a “metric”: a way to multiply to numbers together in a way that cancels out the “non-parallel” bits. I don’t think such a metric exists for ordinary complex numbers (though you can write something like a metric using Clifford algebras and representing i within that algebra). For most purposes, it’s enough that the phase angle is π/2 (90°) between the axes, something derived from the series expansion of the exponential function.
However, since roots (and other powers) rescale arbitrary complex numbers, I’m not claiming they’re simple rotations for any complex number. It works that way when r = 1, so it’s an easy way to show that at least some operations for complex numbers have a nice geometrical interpretation.
Firstly “…the idea of a square root being a rotation…” is a perfect illustration of a reader being mislead by the blogger’s error. It is not a rotation any more than absolutely every complex valued function of a complex variable is a rotation, and that would be a misleading use of the word ‘rotation’, one which does not agree with what a mathematician calls a rotation. (Sorry to just repeat what I said earlier, but the blogger’s response to that earlier correction was unsatisfactory.)
Secondly, the reader asks a good question: “Is (the artifice …) a convention or is there a deeper mathematical logic to it? “. The blogger did not answer it. The answer is the following. You want to have beforehand a nice classical geometrical plane with rotations, orthogonality etc., and to take the real numbers as the horizontal axis. You want to take all the real multiples of ‘i’ as going along a line. All that is good and necessary. The question is: which line should the imaginary axis be; in particular, should it be perpendicular to the real axis? The answer is that
1) first and pretty obviously it cannot be the same line as the real axis. That seems trivially obvious, and would definitely lead to an easy logical contradiction if you took the same line; but more to the point,
2) it can in fact be any other line. So, yes, the fact that the square root function has a nice geometric description (though not a rotation), including describing the angle of the answer, is an artifice of choosing the imaginary axis perpendicular.
On the other hand, why not choose that geometric description of the axis, since it leads to such a simple geometric description of many other important things, including that of square roots.
My statement “it cannot be the same line as the real axis” should have read ‘it cannot be parallel to the real axis’ more generally. Taking it parallel but not equal to the real axis would also give a contradiction. It’s good exercises to verify what I have left out, but anyone wanting to see those things I’d be glad to provide (and will continue to read this thread of this blog).
Now this is the kind of post I was waiting for, one that answered a deep question about the nature of complex numbers. Huzzah!
I edited my two previous comments above to hopefully clarify matters.
Thank you Dr. Francis for taking the time, to write the blog post in the first place, and to answer my question – I understand complex numbers better now.
Thanks also to commenter Peter for replying to my question.
“…if something in math can be used for something measurable, then it’s real in a meaningful sense.”
Can you give an example of something in math…which is NOT real in a meaningful sense?
I would guess that by Francis definition, surreals would be an example of math that isn’t meaningful for physics. Maybe “mathy”, but not “real-y”.
For myself, I think Deutsch has a useful, so meaningful, definition of realism, constrained reaction on constrained action. (“If you hit a stone, it hits back.”) In that sense all mechanics tests for realism (and so a robust, interdependent, and measurable with uncertainty, system) from the get go. (Action-reaction, observation-observables.) They would have to, or observation isn’t consistent with that it works.
While that doesn’t get to what is real, except that observations (experiments) are themselves examples of systems attributed with the same realism as the systems we observe, it says that mathematics is games we play in order to make observations and else for fun.
Why anyone would try to attribute a characteristic of “real” for mathematical objects is curious and an effort apriori likely to fail. Any such reification would make them magical objects I think, as they are not under observation. Maybe we should invent a category of “magical numbers” for such? ;-)
The surreals is a good example I think, right now anyway. And I assume the reals are “real”, in the sense we are trying to discuss. But I can see at least three problems that could, but won’t, make this a very long discussion:
1) What if the surreals suddenly did become almost indispensable in physics (or at least some set of which the reals formed a proper subset)? Surely it would be better that any division of mathematical beasts into the “real” and the others should be independent of time. Are the quaternions now real, but not before? The Mandelbrot set? The tangent bundle to spacetime? (Horrible name-dropper I am!)
2) Surely also many people would regard as ‘realer than the reals in general’ the computable real numbers, a very small set despite the fact that the rest of the reals are essentially undescribable, even un-namable, individually.
3) Is the set of all real numbers real? Or is it just some (all) individual ones? Is it the reals from a set theory where the continuum hypothesis holds, or one where it doesn’t?
I’d be inclined to think that the whole exercise of having some distinction like this to be a mistake.
[…] I tell this story as a gentle way to raise the topic of imaginary (and complex) numbers, which I’ve been musing on lately. I think my latest bout of musings were triggered by reading this, which I read via a chain of links that I can’t remember now, but would have included this. […]
Y is square of I is -1 but not 1?
That’s the definition of i: the square root of -1. That’s why i doesn’t correspond to a regular number.