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.
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.
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.