Following on from her Pi Day video explaining why you might or might not find all of Shakespeare in the digits of Pi, see Celebrate Pi Day It Contains All Human Knowledge, Vi Hart has another presentation with two examples of non-repeating non-normal numbers.
The whole point of showing you two such numbers are that it is a common human failing that to conclude that non-repeating infinite numbers have to be normal - they don't.
To quote Vi Hart, in verse as before:
It's easy to invent a number that's Irrational, it goes and won't repeat, Yet still won't ever have Macbeth with Cats Or something short and simple as a tweet.
Imagine you put 1, then 2, then 1, Then two 2s, one, three twos, one, dot dot dot... It's non-repeating, and never is done. But is there "123" in here? There's not.
Take zero, then Oh 1, and then repeat The last two parts together. Oh Oh One. And then repeat the last two parts again. 0-1, 0-0-1. Anyway that's neat,
And Pi is still irrational, we're sure. And transcendental. We don't know much more.
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