The Debunker: Is Infinity a Number?

by Ken Jennings

April has been Mathematics Awareness Month since 1999, making this the nineteenth time it's been observed. Wait, that's not right. The twentieth time. To celebrate our mathematical awareness, or lack thereof, Ken Jennings of Jeopardy! will be with us all month debunking all the popular misinformation about numbers that you thought once you could count on.

The Debunker: Is Infinity a Number?

Many an elementary school argument can be won—or escalated into violence—by introducing the mathematical concept of infinity. Comebacks like "I dare you times infinity" or "You want to kiss [classmate X] infinity times" are hard to trump. "I dare you times a million" can always be defused with a little "I dare you times a million and one." But how can you beat infinity? What number comes after it? Obviously nothing—unless you're Buzz Lightyear and believe in going beyond infinity.

Then later, in higher math classes like calculus, we actually do see infinity, in the form of that little sideways 8 thing, acting like a number. It might be the value approached by an input in a limit, where you'd normally see a 1 or a 0. It might have a sign: +∞ or −∞. It might even show up informally in the denominator of a fraction that a professor is about to reduce to zero. All this might reinforce the idea that infinity is a very, very big number. And that's not really the case.

Treating infinity like it's a number can get you in arithmetic trouble very quickly. If adding one to infinity gets you another infinite quantity (∞ + 1 = ∞) then you might be tempted to reduce that equation algebraically and find that ∞ - ∞ = 1, which is problematic. Most of the properties of "numbers" that you're thinking of are properties of integers or real numbers, and those don't hold for infinity, which is more usefully treated as a general concept of endlessness. Among the (possibly infinite!) counterintuitive properties of infinity: some infinite sets can contain more members than other infinite sets, and infinite sets can even have new members added to them without getting larger.

One other common mistake that many students make regarding infinity is to believe that 1/0 = ∞. (The thinking probably goes, "How many times could I fit zero things into one? An infinite number, I could keep doing it forever.") But think about the problem this way: Dividing one by zero asks you to find the number you could multiply by 0 to produce 1. Infinity won't do it: an infinite amount of nothing won’t get you one. That's why the true result of dividing by zero is said to be "indeterminate" or "undefined." Dividing by zero gives no constructive answer that doesn't make math break.

Quick Quiz: The different cardinalities (or sizes) of infinite sets are often expressed using what first letter of the Hebrew alphabet?

Ken Jennings is the author of eleven books, most recently his Junior Genius Guides, Because I Said So!, and Maphead. He's also the proud owner of an underwhelming Bag o' Crap. Follow him at ken-jennings.com or on Twitter as @KenJennings.