Sequences in pi

Within the first thousand digits of pi there are six nines in a row. This should not be a surprise.

Dontworry / CC BY-SA

I hope you didn’t think I was done with pi after my post on pi poetry a couple of weeks ago, because I have a lot more to say about this curious constant.

Pi begins with the following numbers:


Keep on going for about 750 digits and you will find this:


Six nines in a row. Now, the odds of this happening are very low, but pi is an irrational number – the digits just keep going and going and going. Odd sequences and patterns will always appear, given that you keep looking long enough. It’s a little like the law of truly large numbers, or the million monkeys working on a million typewriters.

Six ones appear beginning at the 255945th digit. Six twos don’t appear until close to the millionth digit (963024th, to be precise). My birth date appears in pi – at least, if I follow the American month-day-year convention – and there’s a good chance yours does too. My credit card pin appears thousands of times. No, I’m not saying where.

The first eight digits of the square root of two, another irrational number, appear in pi a little more than fifty thousand digits in. The first eight digits of the golden ratio (also irrational) don’t appear in sequence until nearly 20 million digits in. But they both appear.

You can search for sequences yourself at the link below, which has indexed the first two hundred million digits of pi. Just don’t be surprised if you find a sequence that seems unlikely – two hundred million offers a lot of chances for the unlikely.

Going back to the six nines, Douglas Hofstadter joked that he should memorise pi up until that point so that he could end with the unlikely run of repeated digits plus “and so on.” Heh.

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