...in the decimal expansions of irrational numbers.

Thinking about π – 3.141592653589793... you can see these pairs and triplets in the decimal expansion:

• 14, 41, 15, 59, 92, 26, ...
• 141, 415, 159, 592, 926, ...

That’s an interesting topic in itself – and you might (or might not) find your first copy of Shakespeare sooner if you do things that way than the way I’m about to explain, but I leave you to explore that topic yourself.

What I want to do is take my pairs and triplets like this instead:

• (14), (15), (92), (65), ...
• (141), (592), (653), (589), ...

that is, not overlapping each pair or triplet with the previous one. This way, you can think of the pairs or triplets as composite digits in base 100 or base 1000, rather than two or three times as many digits base ten.

I’ve made tables of the occurrences of these “digits” like the tables I made of the occurrences of decimal digits before – but of course they’re enormous (a hundred or a thousand rows, not just the ten rows the tables on the previous page have) and it’s the δ values that are really interesting, so that’s all you’re getting here.

To make these tables, I downloaded the first 1,000,000 decimal digits of √2, π, e, √3, √5 and √7 from the internet (so E&OE). This means I only had 500,000 base 100 “digits”, and only 333,333 base 1000 “digits”.

The main thing to notice – and to realize how unsurprising it is – is how with a hundred digits to choose from rather than ten, you need a lot more “random” digits before the least frequently occurring ones are cropping up nearly as often as the average. And even more so with a thousand.

(Are the digits of irrational numbers random? What exactly does random mean?)