Infinity is my favorite number.

September 15, 2014

I've recently been embroiled in a lovely debate on Numberphile's video, "Infinity is bigger than you think", in which Dr. James Grime starts off:  "We're going to break a rule.  We're breaking one of the rules of Numberphile.  We're talking about something that isn't a number.  We're going to talk about infinity."

I, too, was a longtime believer of what high school students all over are told:  "Infinity is not a number; infinity is a concept."  As my studies of mathematics progressed, however, I began to see that perhaps the things I had always taken for granted were not as black-and-white as they had seemed.  There was a lot more nuance to mathematics than I had ever realized, and learning those nuances opened up an entire new level of understanding, unlocking all sorts of links between concepts that had previously seemed worlds apart.  So it's no wonder that "Infinity Is Not A Number" (which I will occasionally abbreviate as "IINAN") was one of the first claims to which I took a fine-tooth comb.  What I learned changed my stance on infinity and firmly cemented it as my favorite number - not just concept, but honest-to-god number.*

The most common argument made IINAN proponents involves the curious property that \(\infty +1=\infty\).  This, they say, leads to all sorts of contradictions, because all one has to do is simply subtract \(\infty\) from both sides:
\underline{-\infty\ \ \ \ \ \ }&\underline{\ \ -\infty}\ \ \\
\end{align*}\]Oh no!  We know that the statement \(1=0\) is obviously false, so there must be a false assumption somewhere.  Many IINAN defenders claim that the false assumption was that we tried to treat \(\infty\) as a number.  But that's not actually where the problem with infinity lies.

The problem is that we tried to do algebra with it.

For mathematicians, the most convenient place to do algebra is in a structure called a field.  If you're already familiar with what a field is, great, but if not, you can think of a field as a number system in which the age-old operations of addition, subtraction, multiplication, and division — the four operations that my father often notes are the only ones he ever needs when I talk about the kinds of math I teach — work exactly as we'd like them to.  The fields with which we are most familiar are the rational numbers (\(\mathbb{Q}\)), the badly-named so-called "real" numbers (\(\mathbb{R}\)), and often the complex numbers (\(\mathbb{C}\)).  One basic thing about a field is that the subtraction property of equality holds:  For any numbers \(a\), \(b\), and \(c\) in our field, if \(a=b\), then \(a-c=b-c\).

What about \(\infty\) though?  When we attempted to use \(\infty\) in an algebra problem, we got back complete garbage.  And we know that the subtraction property of equality should hold for any numbers in a field.  What this means, then, is that \(\infty\) is not part of that field (or any field as far as I'm aware).  So, when someone says "Infinity is not a number", what they really mean is "Infinity is not a real number."  (It's not a complex number, either, for that matter.)  It doesn't follow the same rules that the real numbers do.

But that doesn't mean it's not a number at all.

We've seen this sort of thing happen before.  The Greek mathematician Diophantus, when faced the equation \(4x+20=0\), called its solution of \(-5\) "absurd" — yet now students learn about negative numbers as early as elementary school, and we barely blink an eye at their use in everyday life.  Square roots of negative numbers seemed equally preposterous to the Italian mathematician Gerolamo Cardano, and the French mathematician René Descartes called them "imaginary", a term that we're unfortunately stuck with today.  But imaginary numbers — and the complex numbers we build from them — are a vital part of physics, from alternating currents to quantum mechanics.

So what makes infinity any different from \(i\)?

Sure, it seems bizarre that a number plus one could equal itself.  But it's equally bizarre that the square of a number could be negative.  And sure, we can get a contradiction if we do certain things to infinity.  But that happens with \(i\) as well!  If we attempt to use the identity \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), we can arrive at a similar contradiction:
i\cdot i&=\sqrt{1}\\
-1&=1\ \]When this equation fails, you don't see mathematicians clamoring that "\(i\) isn't a number"!  Instead, the response is that the original equation doesn't work like we thought it did when we extend our real number system to include the complex numbers — instead, the square root function takes on a new life as a multi-valued function.  There's that nuance again!  For the same reason, infinity makes us look closer at something as simple as subtraction, at which point we find that \(\infty-\infty\) is an indeterminate form, something that we need the tools of calculus to properly deal with.

The truth is, mathematicians have been treating \(\infty\) as a number** for quite some time now.

In real analysis, which was developed to give the techniques of calculus a rigorous footing, points labelled \(+\infty\) and \(-\infty\) can be added to either end of the real number line to give what we call the extended real number line, often denoted \(\overline{\mathbb{R}}\) or \(\left[-\infty,+\infty\right]\).  The extended real number line is useful in describing concepts in measure theory and integration, and it has algebraic rules of its own, though analysts are still careful to mention that these two extra points are not real numbers.  What's more, the extended real line is not a field, because it doesn't satisfy all the nice properties that a field does.  (But that just makes us appreciate working in a field that much more!)

Projective geometry gives us a different sort of infinity, what I like to call an "unsigned infinity", one that is obtained by letting \(-\infty\) and \(+\infty\) overlap and creating what is known as the real projective line.  And complex analysis, which extends calculus to the complex plane, takes it even further, letting all the different infinities in all directions overlap to create a sort of "complex infinity", sometimes written \(\tilde{\infty}\), sitting atop the Riemann sphere.  What I particularly like about these projective infinities is that, using them, you can actually divide by zero! ***

So, since there are actually a number of different kinds of infinity that can be referred to, I would say that, more specifically, complex infinity is my favorite number.

The tough thing about this situation is that the concept of "number" is a very difficult one to precisely and universally define — similar to how linguists still struggle to come up with a universal definition of "word".  By trying to come up with such a description, you end up either including things that you don't want to be numbers (such as matrices) or excluding things that you do want to be numbers (such as complex numbers).

The best we can really do is keep an open mind about what a "number" is.

* I'm not saying that infinity isn't a concept.  When it really comes down to it, every number is a concept.  That's the beauty of having abstracted the number "two" as an adjective, as in "two sheep", to "two" as a noun.

** There's an argument to be made that treating something like a number doesn't mean it is a number.  But at some point, the semantic distinction between these two becomes somewhat blurred.

*** Don't worry, I'll make a post about how to legitimately divide by zero in the near future!