The Paw Print

By Elena Whitford, Features Editor

Hello everyone! I’m Elena, a junior and the Features Editor for the Paw Print. Welcome to my new column, Payton Pop Math, where I turn the tables on the idea that math is boring and pointless. Each month, I’ll break down a new math concept in a fun and accessible way. Even if you’re struggling to stay awake in A2AB, this column will help you see that math isn’t just rote memorization but rather a way of being creative, just like writing or art can be.

To ease into the column, today I’m going to examine a lighthearted proof called the Interesting Number Paradox. Not only is it almost like a joke, it also works as an introduction to basic mathematical proof techniques. 

This paradox revolves around the question: what if we were to classify all natural numbers as either “interesting” or “uninteresting”? 

The proof goes something like this: We are trying to classify the numbers 1, 2, 3, … by whether or not they are interesting. Clearly, the first few are interesting. For example, 1 is interesting because it is the first number, 2 is the first even number, 3 is the first odd prime, and so forth. 

Suppose that not all numbers are interesting. Then, there would exist some number n such that n is the first uninteresting number. However, this would mean that n is interesting, with the factor that makes it interesting being that it is the first uninteresting number. 

Clearly, we have a contradiction on our hands. This means that every number must be interesting!

Apart from being silly, this argument demonstrates the mathematical techniques of proof by contradiction and induction. Proof by contradiction relies on creating an assumption, then following it to a clearly wrong answer to show that it cannot be the case. The assumption that at least one number is not interesting leads us to a contradiction, and the conclusion that every number is interesting.

Induction, on the other hand, is a structured way of writing a proof that appears at first to be complicated, but breaks down to a simple method. First, we establish that our argument is true for a basic case, and then we prove that if the first case is true, our argument can extend to all the cases we want. For this reason, induction is often used for problems like the Interesting Number Paradox where we have to prove that a particular quality applies to all numbers, not just some.

But where, you may ask, does induction appear in the Interesting Number Paradox? While this method isn’t explicitly used here in the way proof by contradiction is, we can see that this proof has its roots in induction. 

Inductive proofs have two basic steps: the base case, in which we show that what we want to prove is true for some small number, and the inductive step, in which we show that if our proof is true for one number, it must be true for them all. Both of these steps are present in our proof, as we will now see. 

The base case appears when we notice that there clearly exist some interesting numbers, namely 1, 2, and 3. The inductive step is slightly harder to see, but because we have shown that there are some interesting numbers, the idea that there is any number that is uninteresting would distinguish that number from the others and make it interesting. This means every number is interesting. 

Induction and proof by contradiction are two of the most useful methods in higher mathematics, so it’s interesting to see that they both make an appearance in a seemingly unserious proof. 

Finally, a famous discussion about interesting numbers between 20th-century mathematicians G. H. Hardy and Srinivasa Ramanujan led to the idea of taxicab numbers — I’ll discuss what those are in a later column!