no. it merely being infinitely non-repeating is insufficient to say that it contains any particular finite string.
for instance, write out pi in base 2, and reinterpret as base 10.
11.0010010000111111011010101000100010000101...
it is infinitely non-repeating, but nowhere will you find a 2.
i’ve often heard it said that pi, in particular, does contain any finite sequence of digits, but i haven’t seen a proof of that myself, and if it did exist, it would have to depend on more than its irrationality.
it’s not a good example because you’ve only changed the symbolic representation and not the numerical value. the op’s question is identical when you convert to binary. thir is not a counterexample and does not prove anything.
Please read it all again. They didn’t rely on the conversion. It’s just a convenient way to create a counterexample.
Anyway, here’s a simple equivalent. Let’s consider a number like pi except that wherever pi has a 9, this new number has a 1. This new number is infinite and doesn’t repeat. So it also answers the original question.
“please consider a number that isnt pi” so not relevant, gotcha. it does not answer the original question, this new number is not normal, sure, but that has no bearing on if pi is normal.
Since Pi is infinite and non-repeating, would that mean any finite sequence of non-repeating digits from 0-9 should appear somewhere in Pi in base 10?
…and this…
Does any possible string of infinite non-repeating digits contain every possible finite sequence of non repeating digits?
are equivalent statements.
The phrase “since X, would that mean Y” is the same as asking “is X a sufficient condition for Y”. Providing ANY example of X WITHOUT Y is a counter-example which proves X is NOT a sufficient condition.
The 1.010010001… example is literally one that is taught in classes to disprove OPs exact hypothesis. This isn’t a discussion where we’re both offering different perspectives and working towards a truth we don’t both see, thus is a discussion where you’re factually wrong and I’m trying to help you learn why lol.
Is the 1.0010101 just another sequence with similar properties? And this sequence with similar properties just behaves differently than pi.
Others mentioned a zoo and a penguin.
If you say that a zoo will contain a penguin, and then take one that doesn’t, then obviously it will not contain a penguin.
If you take a sequence that only consists of 0 and 1 and it doesn’t contain a 2, then it obviously won’t.
But I find the example confusing to take pi, transform it and then say “yeah, this transformed pi doesn’t have it anymore, so obviously pi doesn’t”
If I take all the 2s out of pi, then it will obviously not contain any 2 anymore, but it will also not be really be pi anymore, but just another sequence of infinite length and non repeating.
So, while it is true that the two properties do not necessarily lead to this behavior. The example of transforming pi to something is more confusing than helping.
The original question was not exactly about pi in base ten. It was about infinite non-repeating numbers. The comment answered the question by providing a counterexample to the proffered claim. It was perfectly good math.
You have switched focus to a different question. And that is fine, but please recognize that you have done so. See other comment threads for more information about pi itself.
I see that the context is a different one and i also understand formal logic (contrary to what the other comment on my post says)
It’s just that if the topic is pi, I find it potentially confusing (and not necessary) to construct a different example which is based on pi (pi in binary and interpreted as base 10) in order to show something, because one might associate this with the original statement.
While this is faulty logic to do so, why not just use an example which doesn’t use pi at all in order to eliminate any potential.
I did realize now that part of my post could be Interpreted in a way, that I did follow this faulty logic -> I didn’t
The explanation is misdirecting because yes they’re removing the penguins from the zoo. But they also interpreted the question as to if the zoo had infinite non-repeating exhibits whether it would NECESSARILY contain penguins. So all they had to show was that the penguins weren’t necessary.
By tying the example to pi they seemed to be trying to show something about pi. I don’t think that was the intention.
this is correct but i think op is asking the wrong question.
at least from a mathematical perspective, the claim that pi contains any finite string is only a half-baked version of the conjecture with that implication. the property tied to this is the normality of pi which is actually about whether the digits present in pi are uniformly distributed or not.
from this angle, the given example only shows that a base 2 string contains no digits greater than 1 but the question of whether the 1s and 0s present are uniformly distributed remains unanswered. if they are uniformly distributed (which is unknown) the implication does follow that every possible finite string containing only 1s and 0s is contained within, even if interpreted as a base 10 string while still base 2. base 3 pi would similarly contain every possible finite string containing only the digits 0-2, even when interpreted in base 10 etc. if it is true in any one base it is true in all bases for their corresponding digits
They also say “and reinterpret in base 10”. I.e. interpret the base 2 number as a base 10 number (which could theoretically contain 2,3,4,etc). So 10 in that number represents decimal 10 and not binary 10
It was just an example of an infinite, non-repeating number that still does not contain every other finite number
Another example could be 0.10100100010000100000… with the number of 0’s increasing by one every time. It never repeats, but it still doesn’t contain every other finite number.
I don’t think the example given above is an apples-to-apples comparison though. This new example of “an infinite non-repeating string” is actually “an infinite non-repeating string of only 0s and 1s”. Of course it’s not going to contain a “2”, just like pi doesn’t contain a “Y”. Wouldn’t a more appropriate reframing of the original question to go with this new example be “would any finite string consisting of only 0s and 1s be present in it?”
They just proved that “X is irrational and non-repeating digits -> can find any sequence in X”, as the original question implied, is false. Maybe pi does in fact contain any sequence, but that wouldn’t be because of its irrationality or the fact that it’s non-repeating, it would be some other property
no. it merely being infinitely non-repeating is insufficient to say that it contains any particular finite string.
for instance, write out pi in base 2, and reinterpret as base 10.
it is infinitely non-repeating, but nowhere will you find a 2.
i’ve often heard it said that pi, in particular, does contain any finite sequence of digits, but i haven’t seen a proof of that myself, and if it did exist, it would have to depend on more than its irrationality.
Isnt this a stupid example though, because obviously if you remove all penguins from the zoo, you’re not going to see any penguins
Its not stupid. To disprove a claim that states “All X have Y” then you only need ONE example. So, as pick a really obvious example.
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it’s not a good example because you’ve only changed the symbolic representation and not the numerical value. the op’s question is identical when you convert to binary. thir is not a counterexample and does not prove anything.
Please read it all again. They didn’t rely on the conversion. It’s just a convenient way to create a counterexample.
Anyway, here’s a simple equivalent. Let’s consider a number like pi except that wherever pi has a 9, this new number has a 1. This new number is infinite and doesn’t repeat. So it also answers the original question.
“please consider a number that isnt pi” so not relevant, gotcha. it does not answer the original question, this new number is not normal, sure, but that has no bearing on if pi is normal.
OK, fine. Imagine that in pi after the quadrillionth digit, all 1s are replaced with 9. It still holds
“ok fine consider a number that still isn’t pi, it still holds.” ??
They didn’t convert anything to anything, and the 1.010010001… number isn’t binary
then it’s not relevant to the question as it is not pi.
The question is
Since pi is infinite and non-repeating, would it mean…
Then the answer is mathematically, no. If X is infinite and non-repeating it doesn’t.
If a number is normal, infinite, and non-repeating, then yes.
To answer the real question “Does any finite sequence of non-repeating numbers appear somewhere in Pi?”
The answer depends on if Pi is normal or not, but not necessarily
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In terms of formal logic, this…
…and this…
are equivalent statements.
The phrase “since X, would that mean Y” is the same as asking “is X a sufficient condition for Y”. Providing ANY example of X WITHOUT Y is a counter-example which proves X is NOT a sufficient condition.
The 1.010010001… example is literally one that is taught in classes to disprove OPs exact hypothesis. This isn’t a discussion where we’re both offering different perspectives and working towards a truth we don’t both see, thus is a discussion where you’re factually wrong and I’m trying to help you learn why lol.
Is the 1.0010101 just another sequence with similar properties? And this sequence with similar properties just behaves differently than pi.
Others mentioned a zoo and a penguin. If you say that a zoo will contain a penguin, and then take one that doesn’t, then obviously it will not contain a penguin. If you take a sequence that only consists of 0 and 1 and it doesn’t contain a 2, then it obviously won’t.
But I find the example confusing to take pi, transform it and then say “yeah, this transformed pi doesn’t have it anymore, so obviously pi doesn’t” If I take all the 2s out of pi, then it will obviously not contain any 2 anymore, but it will also not be really be pi anymore, but just another sequence of infinite length and non repeating.
So, while it is true that the two properties do not necessarily lead to this behavior. The example of transforming pi to something is more confusing than helping.
We need to start teaching formal logic in grade schools I’m going insane.
The original question was not exactly about pi in base ten. It was about infinite non-repeating numbers. The comment answered the question by providing a counterexample to the proffered claim. It was perfectly good math.
You have switched focus to a different question. And that is fine, but please recognize that you have done so. See other comment threads for more information about pi itself.
I see that the context is a different one and i also understand formal logic (contrary to what the other comment on my post says)
It’s just that if the topic is pi, I find it potentially confusing (and not necessary) to construct a different example which is based on pi (pi in binary and interpreted as base 10) in order to show something, because one might associate this with the original statement.
While this is faulty logic to do so, why not just use an example which doesn’t use pi at all in order to eliminate any potential.
I did realize now that part of my post could be Interpreted in a way, that I did follow this faulty logic -> I didn’t
Let’s abstract this.
S = an arbitrary string of numbers
X = is infinite
Y = is non-repeating
Z = contains every possible sequence of finite digits
Now your statements become:
The explanation is misdirecting because yes they’re removing the penguins from the zoo. But they also interpreted the question as to if the zoo had infinite non-repeating exhibits whether it would NECESSARILY contain penguins. So all they had to show was that the penguins weren’t necessary.
By tying the example to pi they seemed to be trying to show something about pi. I don’t think that was the intention.
i just figured using pi was an easy way to acquire a known irrational number, not trying to make any special point about it.
this is correct but i think op is asking the wrong question.
at least from a mathematical perspective, the claim that pi contains any finite string is only a half-baked version of the conjecture with that implication. the property tied to this is the normality of pi which is actually about whether the digits present in pi are uniformly distributed or not.
from this angle, the given example only shows that a base 2 string contains no digits greater than 1 but the question of whether the 1s and 0s present are uniformly distributed remains unanswered. if they are uniformly distributed (which is unknown) the implication does follow that every possible finite string containing only 1s and 0s is contained within, even if interpreted as a base 10 string while still base 2. base 3 pi would similarly contain every possible finite string containing only the digits 0-2, even when interpreted in base 10 etc. if it is true in any one base it is true in all bases for their corresponding digits
It does contain a 2 though? Binary ‘10’ is 2, which this sequence contains?
They also say “and reinterpret in base 10”. I.e. interpret the base 2 number as a base 10 number (which could theoretically contain 2,3,4,etc). So 10 in that number represents decimal 10 and not binary 10
that number is no longer pi… this is like answering the question “does the number “3548” contain 35?” by answering “no, 6925 doesnthave 35. qed”
It was just an example of an infinite, non-repeating number that still does not contain every other finite number
Another example could be 0.10100100010000100000… with the number of 0’s increasing by one every time. It never repeats, but it still doesn’t contain every other finite number.
op’s question was focued very clearly on pi, but sure.
I don’t think the example given above is an apples-to-apples comparison though. This new example of “an infinite non-repeating string” is actually “an infinite non-repeating string of only 0s and 1s”. Of course it’s not going to contain a “2”, just like pi doesn’t contain a “Y”. Wouldn’t a more appropriate reframing of the original question to go with this new example be “would any finite string consisting of only 0s and 1s be present in it?”
Then put 23456789 at the start. Doesn’t contain 22 then but all digits in base 10.
They just proved that “X is irrational and non-repeating digits -> can find any sequence in X”, as the original question implied, is false. Maybe pi does in fact contain any sequence, but that wouldn’t be because of its irrationality or the fact that it’s non-repeating, it would be some other property
Like the other commenter said its meant to be interpreted in base10.
You could also just take 0.01001100011100001111… as an example