The Curious Case Of The Interesting Number Paradox

Every single person I know well in my life has a favourite number. I have always found this topic intriguing. My personal favourite is the number 5. If you asked me why it had to be 5 though, my answer would be irrational (pun intended).

Something about the way 5 slices tens draws me in. It helps that I have five digits in each of my hands and legs. Furthermore, 5 has a certain curvature about it. But at the same time, it features straight lines as well. And to top it all off, 5 is prime!

I am pretty sure that you have your own “irrational” reasoning to justify your favourite number. Is it a single-digit number? Is it a double-digit number? I wonder if anyone out there has 26879542 as their favourite number. So many questions! And yet, every single one of them leads to irrational answers.

Along the same lines, here is an interesting question:

Can we classify every natural number (1, 2, 3…) as “interesting” or “uninteresting”?

It turns out that an attempt to do this would lead us to a funny paradox.

The Interesting Number Paradox — Explained

Before I explain the paradox, I would like to recall a famous story in mathematical circles.

During a discussion among mathematicians, G.H. Hardy remarked that the number 1729 on the plate of the taxicab he had just ridden seemed “rather a dull one”.

To this, Ramanujan immediately pushed back by stating that 1729 is interesting because it is the smallest number that is the sum of two cubes (and in two different ways too).

Although this was just casual banter among two impressive mathematicians, we can hardly call this a paradox. But still, I thought that it provides a nice starting point.

In 1945, Edwin F. Beckenbach published a short letter in The American Mathematical Monthly. In this letter, Beckenbach conjectured that every natural number is interesting and not a single one of them is uninteresting.

He humorously went on to prove this by induction/contradiction. The proof goes along the following lines:

1 -> This number is a factor of every positive integer.

2 -> This number is the smallest prime.

3 -> This number is the smallest odd prime.

4 -> This is Bieberbach’s number.

…

Suppose that there exists a non-empty set of uninteresting natural numbers. By its very definition, this set must have one smallest number, ’i’. But the fact that ‘i’ is the smallest uninteresting number makes it interesting!

Therefore, ‘i’ gets removed from the uninteresting set and is added to the interesting set. But then, a new number ‘j’ takes its place as the smallest uninteresting number of the set of all uninteresting numbers. This makes it interesting, and it is also removed from the set and added to the set of all interesting numbers.

‘i’ retains its place in the set of all interesting numbers as it is the former smallest number from the set of all uninteresting numbers. We continue this process until the set of all uninteresting numbers becomes an empty set.

And that, in short, is the interesting number paradox — when we try to define a set of all uninteresting numbers, it appears to lead to a contradiction.

History and Controversy

Of course, when Beckenbach published his original “proof” in 1945, he was not necessarily serious about it. He ended the proof with:

“Is this proof valid?”

Edwin F. Beckenbach

I assume that when mathematicians talk about the interesting number paradox, they are not serious about it either and are just humoured by the subjectivity in play.

This essay is supported by Generatebg

Martin Gardner presented this paradox as a “fallacy” in this Scientific American column in 1958. For David Wells, 39 appeared to be the first uninteresting number, which made it especially interesting (a contradiction).

My Thoughts

If you ask me, the interesting number paradox is just humour shared by mathematicians. When I think about it logically, I feel that it is challenging to apply binary classification to potentially infinite subjective outcomes. Which brings me back to my original point.

Why is 5 my favourite number? It is because I like it more than any other number! Any explanation I come up with is likely to be subjective and irrational to a vastly different person with different tastes.

Where do you slot in here? Do you have more than one favourite number?

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Further reading that might interest you: 

• Fermi Problems: How To Deal With Huge Numbers?
• The Story Of The Banned Numbers
• The Strong Law Of The Small Numbers

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