How to Divide by Zero: An Interview with Bill Shillito

For this post, I interviewed a colleague about a new project he is working on: a website where he encourages his readers to consider the possibility of dividing by zero.  Bill Shillito has a Master’s degree in Secondary Mathematics Education and currently works as both an academic tutor and an independent curriculum designer. In this interview, we discussed his “How to Divide by Zero” website, which can be found at:  https://www.1dividedby0.com/

Start by briefly describing your 1dividedby0 project.

Well the basic idea is that it’s a website devoted to how to actually divide by zero.  Not just in the limit, not getting around it by using calculus, but legitimately actually do it.

What is better about your approach rather than merely taking the limit?

It’s not so much an instance of “better”.  It’s more that division by zero is something that I remember learning is supposedly impossible and that never sat right with me.  The general belief amongst teachers and the general public is that you can sort-of-kind-of do it if you use limits, or if you get around it in some way, but that actually dividing by zero is impossible.

Ever since learned about imaginary numbers being invented to allow another supposedly “impossible” operation (taking the square root of a negative number), I couldn’t help but think, why can’t you come up with some other kind of way to allow division by zero?  Some other number that we just don’t have yet?

When you think of division by zero in terms of limits, you get two possible answers of positive and negative infinity.  But if you could somehow find a way to join the two ends of the number line — as if it wrapped around somehow — and make it so that positive infinity and negative infinity are connected at some new number you didn’t have before, then that new number could be a legitimate answer to division by zero.

As it turns out, that’s actually not that farfetched a concept, and it’s made possible using a branch of math called projective geometry,  It’s actually a pretty well-known thing among mathematicians, but you never see it mentioned in any textbook below the undergraduate level.

So the purpose of the 1dividedby0 website is to lay down the foundations of how it works, in as simple and visual a way as possible, so that any curious high schooler (or maybe even younger in some cases!) could understand it.  The site also shows how understanding division by zero helps make the rest of high school algebra make sense, but it also tries to explain why mathematicians would be reluctant to divide by zero in the first place.

How did you come up with the idea to create this website?

Every summer at Georgia’s Governor’s Honors Program (GHP), I teach a course called “To Infinity and Beyond”, where we look at topics like cardinal and ordinal numbers, p-adic numbers, superreal and hyperreal numbers, surreal numbers, infinite series, set theory, measure theory, logical paradoxes, inversive geometry, and even a little bit of algebraic geometry… it’s so much fun and the students love it.

One of the lessons I’ve always taught as part of that course was in fact division by zero, and lots of students said that was their favorite part of the course.  That’s what got me thinking, what if I made a website where I could share this with a larger audience?

Michael Hartl’s website www.tauday.com was a huge part of that inspiration as well.  In his Tau Manifesto, Hartl takes a so-far-universally-accepted idea of $\pi$ being the fundamental circle constant, and challenges it little by little, leading the reader through the thought proces, until the conclusion is inevitable.  That was the sort of feel I tried to convey with my own website about division by zero.

Say more about how you used these ideas with high school students.

Well, it started when I was working at a tutoring center, and I found out about the whole “unsigned infinity” thing myself.  I started showing some of my students who were struggling with things like trigonometry and rational functions how to think of some of the things they had trouble with using unsigned infinity, and that “light bulb moment” happened with them.

One moment that I remember that really spurred me to try this with a student was when one of my students had trouble with a specific problem: finding the cotangent of 90 degrees.

They tried evaluating it as $\frac{1}{\tan(90^{\circ})} = \frac{1}{\text{undefined}}$, so they thought it was undefined.  I explained the line that I’d always been told, that in that one special case you couldn’t say that $\cot\theta = \frac{1}{\tan\theta}$, and that instead you had to say

\[\cot(90^{\circ}) = \frac{\cos(90^{\circ})}{ \sin(90^{\circ})} = \frac{0}{1} = 0.\]

The reaction I got was “Why does math have to be full of all these stupid rules and exceptions?  Why can’t things just work and make sense?”

And at that moment, I decided “you know what, let’s look at this another way.”  And I showed them how to think of $\tan (90^\circ)$ as unsigned infinity, so that \[\frac{1}{\tan(90^\circ)} = \frac{1}{\infty} = 0,\] and suddenly it all made much more logical sense.  They never missed that question again.

So from then on, when I taught students in Precalculus and Calculus, I figured, why hold that knowledge back from them, when instead I could have them see that it does make the rest of their mathematics make sense?

I also was excited to be teaching something supposedly “controversial”!  But a huge part of that was to emphasize to them the idea that math is something you can play with and ponder about, and break the rules and see what cool stuff happens when you do.

You’ve mentioned that these ideas are fairly well known by mathematicians, even though K-12 teachers are not generally aware of them?  Are they written about in university level textbooks?

Well, besides obviously showing up in books about projective geometry, in real analysis and topology books, you’ll often see it described as the “one-point compactification of the real line”, and the complex analogue involving the Riemann sphere is a pretty central thing to find in a complex analysis book.  At the graduate level, once you start getting into algebraic geometry (which I’m really interested in!), it’s pretty common to include points or lines at infinity.

Going back to mathematics educators not being aware of these things: A colleague of mine recently said that her son came home one day from school asking how to add infinity plus infinity and she told him infinity was not a number but a concept.  I know for a lot of teachers, that is what they tell their students when they ask about these sort of things. What would you say to a student who is trying to add infinity plus infinity?

If a young student asked me how to add infinity plus infinity, I would most likely ask them what they meant by that.  How do they understand infinity? What do they think infinity plus one should be? What about infinity minus one? Infinity times three?  What do they see in their head when they think of all this? Let them hash things out, and give validation to the mathematical system that they’re building in their head, and also give them some things to think about that could lead them down new mathematical rabbit holes.

I vehemently disagree with the notion that “infinity isn’t a number, it’s a concept”.  Two is a number and also a concept. If you mean “infinity isn’t a badly-named so-called ‘real’ number” (thanks Vi Hart for that phrase: https://www.youtube.com/watch?v=23I5GS4JiDg ), then that’s fine.  Neither is the square root of negative one, but that doesn’t keep it from being its own kind of number.

Another great example there is with the $0.999… = 1$ debate.  Some kids will accept the various proofs out there (like multiplying both sides by 10 and subtracting), but others have this idea in their head that $0.999…$ is just the tiniest bit away from 1, infinitely close to it, but not quite there.  Instead of “What part of ‘that’s how it is’ don’t you understand?”, take the opportunity for the student to explore that idea of things being infinitely close together. Maybe they’ll end up coming up with something interesting and not unlike the hyperreal numbers.

This reminds me of Robert Ely’s article (https://www.jstor.org/stable/20720128) about nonstandard student conceptions, where he discusses a student who believed strongly in the idea of infinitely small numbers, and had an internally consistent logic for how those numbers worked.

He probably calls it that to link it with nonstandard analysis by Abraham Robinson, who came up with the whole hyperreal numbers thing!

What kind of lessons can curriculum designers take away from your project?  How might we be able to design curricula that encourage breaking the rules?

Well, for one, I’d love to see “you can’t do such-and-such” replaced with “you can’t do such-and-such YET”.  Let students know that there is more to math, that there’s always another way of thinking about things, and maybe even hint at what it might be like.  Or even phrase it as a “What if you could? What might that be like?”

Infinity is too rich and beautiful a concept to deny our students the fun of playing with it.  It already captures their attention, why shut that curiosity down?

If you have questions for Bill Shillito, feel free to post them in the comments.  You can also follow him on Twitter at: https://twitter.com/solidangles