Infinity is not a Real number (or an Imaginary one, either)

One of the mistakes students often make is assuming (justifiably) that infinity is a number. It’s a natural assumption. Numbers are the things we talk about in math, after all, and we treat infinity as a number in some ways. We say things like, “You have an infinite number of toothbrushes.” (It’s clearly none of my business what you are doing with that many toothbrushes.)

What does infinity mean? Let’s go back to the playground for a minute. Double dares turn into triple dares, which eventually turns into cries of, “I infinity dare you!” It is the numerical trump card. Something larger than any other number. In Division by Zero I made the assertion that infinity is either a number or it’s not. That’s how properties work. If it is both a number and not a number that would be a contradiction, which we cannot accept.

So, let’s assume that infinity is a real number. If we couple this with our schoolyard knowledge, then we must say it is the largest real number. (If it weren’t the largest number, then we would just take some larger number to be infinity instead — just like our childish predecessors.)

Since we are assuming infinity is a real number, then ordinary operations like addition are defined for it. This means that infinity + 1 is also a real number. Adding one to a number makes it larger. So infinity + 1 larger than infinity. This is a contradiction. So, we must conclude that our assumption was wrong. Infinity is not a real number.

So, if it isn’t a number, what is infinity? It is a concept — the notion of going on forever. I cringe with Woody every time I hear Buzz’s catchphrase.

Favorite Numbers

What’s your favorite number? No really, I’m curious. I’ve been asking students that every semester for eight years now, and I’ve learned quite a bit about people from that simple question. Single digit numbers are incredibly popular. Likely, if you fit in that crowd, you chose your number at such an early age you can’t remember having any other. You are decisive and have not lost your inner child. Unless, perhaps, your number is 7. Seven is the classic response, and is therefore the default choice for someone without a strong feeling for any other number.

Thirteen is nearly as common. If 13 is your number then you like to be different, but not too different. You probably enjoy pushing people’s buttons. The next most common category is dates of the month and then jersey numbers. Usually a date is an anniversary or a child’s birthday. I find these to be most common in women with at most one child. Jersey numbers are common in the 18-25 year old students with a fond recollection for high school athletics.

23 and 42 have their own cult followings, but identifying with numbers like this mainly serves as a shibboleth — connecting members of the same cultural tribe.

So, what about you? What does your favorite number say about you? My generalizations aside, numbers are something that resonate deep within us. Why is that? We live in a society that decries mathematics, but almost everyone you ask has a favorite number. It’s deeper than that, though. Imagine you are sitting in a classroom with 20 other students and I ask the question. What is your response to hearing that Mike’s favorite number is zero. Did you just sigh a little?

What if Mike said his favorite number was π? Did I hear someone mutter “nerd” under their breath? Have you ever met someone with a negative favorite number? Or a fraction? Admit it. Most of us are experiencing numerical prejudice at the thought of it. How is it that numbers elicit a response that visceral? Is it innate, or has the sum of your frustrations and pains in learning mathematics influenced your opinions about something as innocuous as a number?

I don’t know. What is a number anyway? It is possible that they are just tools for measuring and counting. I believe otherwise. I think that numbers are an alphabet to a universal language. That language is mathematics.