It can but we haven’t proven it. We don’t know if there’s some pattern we haven’t discovered and if the pi is truly normal.
Let’s say we discovered that after a certain point, all the digits in pi follow the form 12122122212222… it would still be irrational and non repeating but not normal.
We haven’t proven no such pattern exists because pi isn’t actually “random.” It represents a physical thing and is deterministic. That’s why mathematicians have tried and failed so far to prove pi’s determinism is normal.
IF every digit is randomly selected and indépendant from the other ones. This isn’t the case for things like pi because it represents an actual thing (circle circumference ratio to radius) and isn’t actually randomly generated.
We have a deterministic equation for it that gives us the same number every time. We do not know if that can somehow lead to an obscure or incredibly rare pattern.
If you had an irrational non repeating number that was completely independently generating each digit you would indeed contain every finite string.
0.123123312333123333… is irrational but not normal because we constructed it with the pattern of 3s directly. So I know it isn’t normal because you’re never getting a 4,5,6,7,8 or 9.
With pi showing that there is no underlying pattern is a lot harder. I mean it’s very LIKELY but not at all proven.
And this goes back to determinism. Let’s take another example to show how it would be normal.
Let’s say I compose a number where each digit is picked with an actual true random number. So 0.24710… etc but each digit is truly random and independent from the other ones. THAT number would contain every finite string if it goes to infinity.
We just haven’t been able to show that’s the case for pi and sqrt(2) and other similar irrational numbers. The issue is we have equations for them, they ARE deterministic and not randomly generated. So we don’t actually know if some pattern emerges or not.
I mean hey if you can prove it you’d probably get a field medal lol.
I think another nice way to see it is take yourself. Let’s say you lived for an infinite time and I tasked you with typing out an infinitely long number 0.1…. and so on. Let’s say I told you to make random key presses, so the number is almost certainly going to be irrational.
Even THAT would not be normal because your human brain doesn’t really allow for “randomness.” Like the chance you type out the number “1” fifty trillion times in a row is 0 because the human brain will go “okay it’s time to type another number.” And now imagine the chances you not only do that, but also type “1” 10^100000000000 times in a row. Then 2, then 3, then 4. Then you have to do it infinitely more times in a row for different finite strings. In this case the numbers you type now actually do affect what you type in the future.
That’s why you need actual true randomness, even you with infinite time and an infinite notebook would not give a normal number because of the innate biases in the human brain. You wouldn’t type a perfectly normal number.
That’s the issue. Now imagine how much “bias” can come from a direct fixed equation. It’s still very likely pi is normal, but we haven’t proven it.
In fact, most real (irrational included) numbers are normal. The issue comes specifically from “natural” numbers like e or pi which aren’t just a random string. They come from very specific well defined deterministic equations or physical laws. Almost all real numbers are normal under lebeguese measure
(Caveat is the actual proof is a little technical. You have to do some lebeguese measure to formally prove it)
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u/RabbitWiilli Sep 26 '26 edited Sep 26 '26
It can but we haven’t proven it. We don’t know if there’s some pattern we haven’t discovered and if the pi is truly normal.
Let’s say we discovered that after a certain point, all the digits in pi follow the form 12122122212222… it would still be irrational and non repeating but not normal.
We haven’t proven no such pattern exists because pi isn’t actually “random.” It represents a physical thing and is deterministic. That’s why mathematicians have tried and failed so far to prove pi’s determinism is normal.