Exploring Pi: Is Every Possible Number Hidden Within?

Pi's decimal expansion goes on without ever ending or repeating, but that's simply not the same as saying that pi *must* contain every possible sequence of digits.

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This is, admittedly, a tough one to wrap your head around. Because of the nature of infinity, it can be just as hard to imagine pi containing every possible string of digits, and in that way encoding any information you want somewhere within its decimal expansion, as it is to imagine that it does NOT contain every possible string of digits. And, to be sure, it is totally possible that pi DOES contain every possible string of digits—not just once, but infinitely many times! Most mathematicians believe that's true, and they've even given that a name: normality. But no one has been able to prove that pi is normal, and just because it's an irrational number, that's not enough to guarantee it.

Consider the weird number I construct in this video: the digits 1 through 9 repeating over and over again in that order, but with a different number of zeros between them every time. This is an irrational number. Because the zeros are changing, it would be impossible to represent this number with a ratio between two integers. Its decimal expansion also contains every possible digit an infinite number of times. And yet, there are tons and tons of sequences that will never be in this number. For example, 10 will never be in this number.

Now, you might object that pi is just a typical number occuring all on its own out there in the world, and so it won't obey the weird rules this constructed number will. Maybe! I think that's probably true. But so far, no one can prove that it's true, and even more importantly, there's nothing about the nature of irrationality or any particular use of digits that *requires* it to be true.

#PiMystery #InfiniteNumbers #IrrationalNumbers #NumberTheory

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[ibidthewriter]

For anyone curious, a number with this property would be called "Normal". Statistical analysis suggests pi is "normal, " but of course that doesn't prove it.

[matthewholloway]

Good point i never thought about it that way. I definitely thought pi had every possible combination before watching.

[wkromhout]

Even better, the same counts for a number like 0, 101001000100001 etc. So pi doesn't even have to use all digits (after the ones we know of), it can end up in a non-repeating pattern of just ones and zeros.

[davidgillies]

One can show that almost all number (in the set theoretic sense) are normal, but we do not have constructive proofs of a number's normality in the general case. It's believed pi is normal (thus simply normal in base 10) but we can't show this is true,

[martinwulf]

Just like an infinite universe doesn’t mean that the exact arrangement of atoms that is you and all your surroundings must exist out there somewhere.

[Badixon]

We think pi is a normal number but can not prove it.

[matthewdodd]

In very basic terms, a number with the property described at the start has a name. It is called a Normal number. Its more indepth, but its an easy way to understand it.

Pi is not classified as Normal. We as humanity can not classify Pi, e, root 2 or any other non-constructed-for-purpose numbers as normal since we have no way of proving any given number is Normal, outside of very niche cases.

[diegonino]

The people not understanding the point and therefore disregarding what it tries to say baffle me. It's a clear example that infinite does not mean that it contains all possible combinations, but that it's a neverending defined set (sorry if i dont use the correct terminology, im not mathematician nor i speak english on my daily life) in the case of pi we dont know if its set definition allows for all combinations to be contained in it or not, we don't have proof nor a way to prove it right now . Be free to tell me if I made a mistake in anything i said or if i missed any important detail

[extrams]

Easy: Keep the irrational number as written, but increase the base.

11, 0010 0100 ... as it uses the digits of pi-base2, doesn't have the number 2-9, but also doesn't have repeating patterns.

OR, the hexadecimal number 3.1415... doesn't have the pattern FF

[rebeuhsin]

That first part is true, in theory pi might not be normal. Meaning you may not be able to find any string. But I never heard anyone claim this. It is well known that it might not be true. So far it has been shown that odds of finding any string of 2- 11 digits is fairly close to what is perdicted for a truly randiom sequence up to 200 million digits of pi. And pi appears to have no pattern. It never reoeats, but might be "normal". There are numbers that do not look random (like pi), but are proven normal, like concatenating counting numbers 0.1234567891011121314...
It is nornal, and every possible finite string of digits are contained by definition. Point is, it is not something claimed, rather it is known that it may not be true.

[IRanOutOfPhrases]

Could also think of an irrational number with only even digits (or only odd).

[mimzim]

Opposite to what many say the property mentionned is NOT being a normal number. A normal number has this property but it is a too strong condition. In french the property is called "nombre univers".

[somerandomlamer]

This seems more like a fundamental misunderstanding of what "infinite" means and its relationship to all or everything; There are infinitely many numbers between 0 and 1, none of which are 2.

[noodlesthest]

If it does have every possible combination of numbers without repeating then I'd like to boast that I can recite the string of numbers in pi that is 13000 0s in a row. That's right, 13000 digits of pi memorized. Just not sure where in pi it appears.

[smor]

So wait is the claim false (or not proven true) or is the reasoning just incorrect?

[waroftheworlds]

I feel like you just helped me refute binary exclusion principle to show there are more number between 0 and 1 than there are integers.

[mp_rho]

i think i get what you mean, since there’s a difference between a string and a number, but is 0.200 not still just 1/5? how would it become irrational?

[perceptionascending]

Hey I'm new to your channel, but I love the simplicity! So this might be a dumb question but

What if we used a base 12 system. ( Greek)

Instead of a base 10 system ( Arabic)

I think the extra divisible options, would help with the system of quantum computing .

It would more directly translate to a trinary 3 system instead of binary.

Thoughts?

[orterves]

It would be so cool if pi "repeated" in a similar way

[lucasbachmann]

Is there a name for the irrational counter example? Personally i call them constructable irrationals because there's an algorithm you are following.