If the digits of Pi really go on infinitely without repeating…

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… then don’t they inevitably repeat after all?

For example, the digit ‘3’ repeats a lot, of course.

The digits ’31’ also repeat.

Then the digits ‘314’ do as well.

And so on… 3141, 31415, 314159, etc etc.

According to [the Pi-Search page](https://www.angio.net/pi/):
>The string 31415926 occurs at position 50366472. This string occurs 3 times in the first 200M digits of Pi.
counting from the first digit after the decimal point. The 3. is not counted.

So, if it truly does continue forever, isn’t it likely that every set of digits must repeat at some point, no matter how long the string may be?

If not, please explain why.

In: 0

Nope, not necessarily.

Imagine a sequence which starts with a 1, then a zero, then a one again. Then *two* zeros and a one. Then *three* zeros and then a one. Then *four* zeros and a one. You can keep going like this forever, and the sequence will never start over and repeat itself. Of course individual sub-sequences will occur many times (we’ll see a lot of 000000 strings, for example), but the full sequence will never start repeating itself.

look at the much much more simple string that consists of:

1

01

001

0001

00001

…..

now if instead of writing these each in their own line, you’d write them one after the other, you’d have an infinite decimal number that never repeats and that only has 0 and 1 in it.

there is an actual proof that Pi cannot be described by two integers a and b where a/b = Pi, but that is far from ELI5, but in order for the decimals in Pi to repeat themselves after a certain point (or even ending), this would be a requirement.

If a number is rational, then its decimal expansion is either finite, or it repeats itself infinitely, e.g. 0.3333333…, 0.16161616… and 14.2857142857142857…

“Infinite non-repeating” doesn’t means there are no repetitions at all. It means that the sequence doesn’t repeat endlessly like the examples above.

In fact, there’s a unproven conjecture that Pi contains every possible finite sequence of digits, so naturally it does contain repetitions.

When we say “without repeating” what we mean is that it starts repeating some finite sequence and *only* that sequence. For example, at some point it was *only* 314314314314314… forever and nothing else.

That is what is meant when people say it doesn’t repeat.

If Pi is, as is believed, a normal number then yes, any finite sequence of digits will appear, and indeed reappear infinitely many times.

But it would not fall into a repeating pattern. Those repeats would occur at different intervals, in a way indistinguishable from randomness.

What it means is that there isnt a pattern.
Eventually every finite sequence of digits will repeat, but since the whole sequence isnt finite it wont be repeated.

What mathematicians mean with “without repeating” is that you won’t ever find a single pattern repeating itself until infinity, not that certain strings of numbers will never be found twice. Using your example of 31415926, an infinitely repeating version of pi would look like this
3.14159263141592631415926314159263141592631415926… In other words, the pattern would immediately start over again the moment it ends. So any string of numbers can repeat itself in pi as long as it doesn’t do so in a pattern.

Statistically speaking yes, all of those sequences will eventually appear an infinite number of times, but being an irrational number doesn’t mean there is no repetition, it just means that it does not repeat indefinitely. So it doesn’t get to a point where it starts repeating the same sequence over and over.

You don’t see the a second version of the number 3 until the 10th digit of pi, you don’t see a repeat of 31 until you get to 140 digits, and you don’t get to 314 until the 2,122nd digit. So sure you start to see larger repeating portions of pi showing up, but they are showing up at a way slower speed than the total sequence is expanding. So by the time you get to that patter you’ve got 1000’s more digits you need to match to fully repeat. There is no reason to believe that you will always repeat in an infinite series like this.

No, a repeat would mean that the a finite number of digits repeat periodically. Yes given infinit digits you can find any finite number that shows up more than once in the digits of pi. But there are no numbers that periodically show up. Like the number 31. Yes it shows up occasionally but you can never predict when it will show up again. Yes numbers do repeat but the lack of a periodic cycle of digits is what makes pi irrational. (Well irrational numbers irrational in general.)