The Shape of Square Roots

Today is 4/4/16, and do you know what that means? It’s Square Root Day!

 Now, since the last Square Root Day was 3/3/09, and the next one won’t be until 5/5/25, you might not have heard of Square Root Day before. So, let me fill you in! A day is a square root day if the day and month both equal the square root of the year.  In this case, 4 (day) * 4 (month) = 16 (year).  Since 4 times itself is 16, that means we have a Square Root Day on our hands!

How should you celebrate Square Root Day?  I recommend roasted carrots, beets, and other “root” vegetables with a frothy “root” beer pairing (but brush your teeth afterwards; otherwise you’ll need a “root” canal!).  And, of course, no Square Root Day is complete without learning something new about square roots!

Thus, in honor of this auspicious and infrequent occasion, we’re going to explore the shape of a square root. And when I say the shape of a square root, I don’t just mean a plain old square! As it turns out, when we ask ourselves tough questions about square roots, we start to find shapes that are a lot more interesting than squares. For example, this one:

But this picture is closer to the end of the story than the beginning, and before we get to the tough questions, we’d better ask some basic ones.

The geometric definition of a square root.

Here’s one to get us started: what does it mean to square a number? Well, the name says it all – to square a number x, you form a square with side lengths equal to x, and the area of that square is what we call the square of x.  See the example to the right.

But wait a second, what if x is negative?! We can’t make a square with sides of length, say, -5, so what’s the square of -5?

Let’s go back to the case where x was positive for a moment. In that case, we can definitely make a square with side length x, and then write an algebraic formula that expresses the area in terms of x: the area of the square with side length x is equal to x*x, or, x^2 for short. Even though there is no way to make a square with sides of length -5,  the algebraic formula still makes sense, and we know that -5 * -5=25. Thus, using this formula we can still give meaning to the square of a negative number. This is a fundamental idea in mathematics – sometimes it is possible to give the same concept two equivalent definitions in one situation, even though just one of the definitions makes sense in a more general situation.

One way to think about taking square roots is as inverting the operation of squaring a number: if I start with a number x, then x is always a square root of x squared (x^2). But there’s a problem – what if a number has more than one square root? Then I might not be able to pin down x just by knowing that it’s a square root of x^2. So, we’re led to the following question: if I start with a number like 16, how many square roots does this number have?

If you look at today’s date (4/4/16), you’ll find one square root of 16, the number 4.  But notice that -4*-4=16 too, which means that -4 is also a square root of 16. Now, if we were only interested in where square root days fell on the calendar, we wouldn’t have to worry about -4. But, for doing math, it’s important to remember that -4 is also a square root of 16. So, all positive numbers in fact have two square roots.

People often refer to the square root instead of just a square root, and when they do this, what they really mean is the positive square root. For the moment, the fact that we can pick a number to call the square root instead of just a square root might seem like just a linguistic convenience, but later on this will become a crucial point.

We know that positive numbers each have two square roots, but what about the rest of the numbers?  Besides positive numbers, we have negative numbers and zero. To start with, zero only has one square root (since -0 and 0 are equal). Further, a bit of experimentation will convince you that since the square of any number other than zero is positive (for example, the square of -4 is 16), a negative number must not have any square roots at all!

If we think about squaring and square roots using our geometric definition (examining the area of a square), then this comes as no surprise. After all, area is always a positive quantity, so how could a negative number be equal to the area of a square? This logic is sound as long as we’re using the geometric definition of squaring,  but as soon as we start to use the algebraic definition (which defines a square number as the multiplication of a number by itself), this problem goes away. In fact, if you think about it, it starts to look like there’s no algebraic reason a negative number shouldn’t have a square root, even though we can’t seem to find one.

Mathematicians have a great trick in situations like this: if we can’t find something, but we also can’t find a reason it shouldn’t be there, then we just make up a name for it and declare that it exists. So, let’s make a new number, and call it i, with the property that i^2=-1. We usually call i an imaginary number, but let’s not treat it unfairly: it turns out i is extremely useful and important, even when dealing with very real problems. For example, i is fundamental to the study of waves, which arise not just in the ocean but also in the vibration of guitar strings, the behavior of electricity and light, and even in compression algorithms for some of the images in this article.

Now, I said i is a number, which means we should be able to treat it like any other number: we should be able to use it when we add subtract, multiply and divide.  We won’t say exactly how to do this, but here’s where something miraculous happens:just knowing that i^2=-1 gives us enough information to understand how i behaves under any algebraic manipulation.  For example, if we have a real number x then we should be able to multiply it by i to get a new number i*x. Now, we can square i*x to get a new number (i*x)^2. But using the rules of arithmetic, we know

(i*x)^2 = i^2 * x^2= -x^2,

and so we see that when we square i*x, we get a negative number, which means that that negative number has a square root.  In fact, as soon as we have the number i and all of its multiples, we discover that every negative number has a square root!  Even better, since –i, just like i is a square root of -1, every negative number has two square roots, just like positive numbers before.  Moreover, we can consider all numbers constructed from algebraic manipulations using i and real numbers (these are called complex numbers), and it will turn out that every non-zero complex number has exactly two square roots that are also complex numbers!

Now we’re ready to ask a really interesting question: We began by observing that every positive real number has two square roots, one negative and one positive, and that we can pick out one of them to call “the square root” by taking the positive one. We have now seen that every non-zero complex number has exactly two square roots that are also complex numbers. But do we have a way to distinguish one of them from the other? That is, is there a reasonable way to pick, for each complex number, a single square root to call “the square root,” just like we can for the positive real numbers by singling out the positive square root?

The answer, amazingly, is no. To see this, we must first give a definition of “reasonable”: Imagine you have a pencil in each hand and two sheets of graph paper in front of you, one for each hand.  In this example, your left hand will be the lead hand, and your right hand will be the follower hand.  We’ll think of the points on the graph paper as representing complex numbers (the point (a,b) corresponding to the complex number a+b*i), and we’ll call a choice of the square root reasonable if, every time you draw a path with your left (lead) hand on one piece of graph paper without lifting the pencil from the paper, then you can at the same time draw with your right (follower) hand the square root of the corresponding left hand point also without lifting up the pencil. If you have to lift your follower pencil to draw the square root, then this square root is not the square root.Though we don’t describe it here, the process of taking a square root of a complex number has a simple geometric interpretation in this picture which makes it possible to carry out this process. Here’s what it might look like if you drew a squiggle with your left hand and followed along with your right drawing the square root:

Now, imagine drawing a circle around the origin of radius 1 starting at the point (1,0) on the x-axis, which represents the positive real number 1, with your lead hand. Then, your follower hand path must also start at the point (1,0), since we’ve already picked out the positive square root of 1 as the square root of 1. As you start to move your left hand pencil around the circle away from 1, you have no choice about how to move your right hand – at any time there are two possible square roots you could draw with your right hand, but only one of them that you can reach without lifting up the pencil.

In fact, as you trace out the circle with your lead hand, your follower hand will trace out the same circle, but at only half the speed. In particular, by the time your left hand has gone almost all the way around and landed nearly back at 1, your right hand will only have made it nearly halfway around the circle, close to -1. But since we’ve already declared that 1, not -1, is the square root of 1, as your left hand comes back to 1 again you’ll have no choice but to pick up your right hand in order to bring the square root pencil back to 1. Thus, because you are forced to pick up your right hand pencil off the paper, there can be no consistent reasonable choice of the square root for every complex number!

But don’t worry – I’ve already told you mathematicians are crafty, and just as we’re ready to introduce brand new numbers when we need them, we’re also ready at a moment’s notice to introduce a new shape to fix a problem. Now imagine taking your lead sheet of graph paper, and cutting it with a pair of scissors horizontally along the positive x-axis. You could then take another sheet of graph paper, cut it along the x­-axis in the same way, and then staple the two sheets together along the cuts. If you’re using actual graph paper instead of your imagination, it’ll work better if you also cut out a small circle around the origin (This is related to the fact that zero only has one square root!).  It should look something like this:

If you take this new shape made out of graph paper, and you draw your lead path on it, then, when you reach the end of the circle, you won’t yet be back where you started.  This is because during your pencil’s journey, it will have moved onto the second sheet of graph paper that you added. Thus, it won’t be a problem that your right hand has only made it to -1 and not all the way back to 1. In order to get back where you started with your left hand, you’ll have to go around another circle on the second glued piece of graph paper. If you follow along as before, taking the square root with your right hand, then you’ll find that by the time you’ve ended up at the starting point again with the left hand, your right hand has also done a complete circle and is back at 1. This is what it would look like on the graph paper if you hadn’t put in the second set of staples (which makes it pretty hard to drawn on!):

In the end it is possible to define the square root, just not when you only use a single sheet of graph paper – you need to cut and glue to create a new shape where the square root lives. And that is what we mean when we say the shape of a square root!                                                                                          

Now, just by thinking about square roots, we’ve created a new shape.  Even more, the method we’ve used to build it (by tracing paths) lies at the heart of a whole field of mathematics called algebraic topology. Algebraic topology gives us tools for deciding when one shape can be squished and squeezed into another shape – for example, algebraic topology can be used to distinguish between some of the different possible shapes of the entire universe. Not a bad Square Root Day’s work, if you ask me!

Moreover, all of this came from asking questions about square roots, but we can also ask the same questions with more complicated equations. For example, if you take cubic or higher roots (for example, the opposite of x*x*x*x) you’ll need to glue more and more pieces of graph paper together, and you’ll get shapes that circle around longer and longer before coming back to where they started.

And if you try to take a square root of a cubic polynomial, you can even end up with a shape that looks like a donut! And thus, if nothing else, we’ve ended up with a new treat to eat on square root day.

Editor’s Note: This article originally contained several mathematical errors that were inadvertently introduced by the editors (who are terrible at mathematics) during the editing process.  These errors have been corrected since.