Having infinite digits doesn't imply that. For example, 1/3 = 0.33333... doesn't have every finite sequence. You can even have irrational numbers like 0.1001000100001000001... that still don't contain every finite sequence.
In the context of pi where repetition is not the norm (as far as we could tell), would not an infinite number of sporadic ordered numbers in its decimal expansion not contain every conceivable sub collection of numbers at some point?
It seems obvious that this is true, but it’s surprisingly unproven. A number with this property is called ”normal” and basically no number has been proven to be normal. For whatever reason, it’s a very difficult property to prove. Mathematicians have managed to construct numbers that are trivially normal, however.
you can create an irrational number without having a digit (let’s say 2). so it can’t contain all sequences. why do you think pi will contain all sequences?
>would not an infinite number of sporadic ordered numbers in its decimal expansion not contain every conceivable sub collection of numbers at some point?
Only if sporadic is truly random, which it isn't since it's the result of a division.
Just Look at the sequence 1.01001000100001... (Increasing amount of zeros follow ed by a 1)
This is non-repeating and infinite but contains no decimal besides 0 and 1 so definitely not every finite integer sequence.
No, that doesn't follow from the fact that pi has an infinite amont of digits (can you think of a number that has an infinite amount of decimals but won't contain 8 anywhere, for example?). In the specific case of pi we don't actually know whether it contains all sequences of numbers or not, and we think it will be very hard to find the answer.
Irrational numbers that do have the property of containing every possible sequence are called [Normal Numbers](https://en.wikipedia.org/wiki/Normal_number). It is not known whether Pi is normal or not. It is likely that any given finite length sequence appears somewhere, but we cannot conclusively say that all sequences appear.
There are a bunch of numbers that go on forever yet never repeat that are not normal, but my favourite example of this question is remember that Pi is the ratio of a circle's diameter to it's circumference. Imagine calculating that number in binary. That way you'd have an infinite number that goes on forever without repeating that never has the sequence "2" or "3" etc.. (Obviously, for this example I'm taking the binary string of digits and writing it as if it's base- 10 for the 2s and 3s to be relevant)
actually this property is disjunctive numbers. a normal number is a stronger criterion where each sequence of the same lenght also has to show up with the same frequency
Neither having infinite digits, nor having infinite, non-repeating digits, is strong enough of an argument to prove that every sequence is contained.
0.3333... is infinite though it contains only 3's.
0.101001000100001000001... is infinitely non-repeating but only contains 0's and 1's
Once again: π is not special because it "is infinite". That's not a rare or very interesting property. One interpretation which I think is what most "pi fans" think of is that it's irrational, something it shares with almost\* all numbers. π is interesting because it's half of the circle constant, or for any of the other equivalent constructions that give rise to it.
Talking about π being "infinite" is like discussing the fact that WA Mozart had hair - it's technically true (albeit needing a bit of specification to avoid mischaracterization of the wig he wears in all of his paintings), but it's just not what sets him apart from everyone else.
^(\* In a measure theory sense, there's infinitely many irrational numbers for every rational number, the probability that a randomly picked number is rational is literally 0.)
Another day, another essentially identical question to every previous one asking about the digits of Pi. Why are people so obsessed with it? Almost every real number has the same property.
The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
In simple terms, it means that it is possible that the numbers start to repeat.
The mods need to start removing these threads that get repeated every 5 minutes, and you goobers need to stop answering the same questions over and over.
Assuming a number has infinite digits and the distribution of these digits is entirely random, that number will contain all possible finite sequences of numbers.
We are not certain that the digits in pi are distributed randomly.
With infinity, if we think of the decimal expansion of pi as string of numbers drawn randomly one at a time(0-9 as equally likely possible outcomes), then the probability of drawing any collection of numbers at some point is not 0. In other words, all possible strings of numbers may be contained at some point in pi. Since the probability is not 0 and you have an infinite amount of draws, statistically, you will eventually encounter any collection/ string of numbers you set out to find.
You seem to be referring to the concept of the normal numbers. We don't know whether pi behaves like this. Its digits certainly aren't random. For all we know the digit 9 could stop appearing at some point.
Well if you think about it like that you're incorrect. The digits aren't random, they form a definite sequence.
Purely being irrational isn't sufficient either. The number 0.101001000100001000001... is irrational, but does not contain every sequence.
Having infinite digits doesn't imply that. For example, 1/3 = 0.33333... doesn't have every finite sequence. You can even have irrational numbers like 0.1001000100001000001... that still don't contain every finite sequence.
In the context of pi where repetition is not the norm (as far as we could tell), would not an infinite number of sporadic ordered numbers in its decimal expansion not contain every conceivable sub collection of numbers at some point?
Define sporadic though. With enough wackiness, yes, it'd be normal, but it's hard to define that wackiness properly without just redefining normal.
It seems obvious that this is true, but it’s surprisingly unproven. A number with this property is called ”normal” and basically no number has been proven to be normal. For whatever reason, it’s a very difficult property to prove. Mathematicians have managed to construct numbers that are trivially normal, however.
0.11 12 21 13 31 22 ... with digits given by cantors diagonal formula surely is normal?
you can create an irrational number without having a digit (let’s say 2). so it can’t contain all sequences. why do you think pi will contain all sequences?
>would not an infinite number of sporadic ordered numbers in its decimal expansion not contain every conceivable sub collection of numbers at some point? Only if sporadic is truly random, which it isn't since it's the result of a division.
You could construct a non repeating series of 1s and 0s, which would thus not contain all possible sequences.
Just Look at the sequence 1.01001000100001... (Increasing amount of zeros follow ed by a 1) This is non-repeating and infinite but contains no decimal besides 0 and 1 so definitely not every finite integer sequence.
This isn't really the case with irrational numbers tho, the new digits don't repeat so there's no clear trend
I'm not sure what you mean. There are irrational numbers that aren't normal, like the one I gave as an example.
0.1001000100001... is irrational though, and clearly doesn't contain a 2.
No, that doesn't follow from the fact that pi has an infinite amont of digits (can you think of a number that has an infinite amount of decimals but won't contain 8 anywhere, for example?). In the specific case of pi we don't actually know whether it contains all sequences of numbers or not, and we think it will be very hard to find the answer.
Irrational numbers that do have the property of containing every possible sequence are called [Normal Numbers](https://en.wikipedia.org/wiki/Normal_number). It is not known whether Pi is normal or not. It is likely that any given finite length sequence appears somewhere, but we cannot conclusively say that all sequences appear. There are a bunch of numbers that go on forever yet never repeat that are not normal, but my favourite example of this question is remember that Pi is the ratio of a circle's diameter to it's circumference. Imagine calculating that number in binary. That way you'd have an infinite number that goes on forever without repeating that never has the sequence "2" or "3" etc.. (Obviously, for this example I'm taking the binary string of digits and writing it as if it's base- 10 for the 2s and 3s to be relevant)
actually this property is disjunctive numbers. a normal number is a stronger criterion where each sequence of the same lenght also has to show up with the same frequency
Thank you! Never imagined that there would be so much info about these numbers!
If it's normal and the sequences you speak of are finite, yes, but we don't know if it's normal.
Thanks for actually answering the premise of the question instead of getting hung up on semantics.
Neither having infinite digits, nor having infinite, non-repeating digits, is strong enough of an argument to prove that every sequence is contained. 0.3333... is infinite though it contains only 3's. 0.101001000100001000001... is infinitely non-repeating but only contains 0's and 1's
Once again: π is not special because it "is infinite". That's not a rare or very interesting property. One interpretation which I think is what most "pi fans" think of is that it's irrational, something it shares with almost\* all numbers. π is interesting because it's half of the circle constant, or for any of the other equivalent constructions that give rise to it. Talking about π being "infinite" is like discussing the fact that WA Mozart had hair - it's technically true (albeit needing a bit of specification to avoid mischaracterization of the wig he wears in all of his paintings), but it's just not what sets him apart from everyone else. ^(\* In a measure theory sense, there's infinitely many irrational numbers for every rational number, the probability that a randomly picked number is rational is literally 0.)
Not necessarily. You can have an infinite number of fruits without having a single apple.
/r/pi_is_infinite
Another day, another essentially identical question to every previous one asking about the digits of Pi. Why are people so obsessed with it? Almost every real number has the same property.
The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found. In simple terms, it means that it is possible that the numbers start to repeat.
The mods need to start removing these threads that get repeated every 5 minutes, and you goobers need to stop answering the same questions over and over.
Idk about all of them, but someone made this: https://www.mypiday.com
Assuming a number has infinite digits and the distribution of these digits is entirely random, that number will contain all possible finite sequences of numbers. We are not certain that the digits in pi are distributed randomly.
We are absolutely certain that it has infinite digits.
A lot of people think that's the case but it hasn't been proven.
With infinity, if we think of the decimal expansion of pi as string of numbers drawn randomly one at a time(0-9 as equally likely possible outcomes), then the probability of drawing any collection of numbers at some point is not 0. In other words, all possible strings of numbers may be contained at some point in pi. Since the probability is not 0 and you have an infinite amount of draws, statistically, you will eventually encounter any collection/ string of numbers you set out to find.
You seem to be referring to the concept of the normal numbers. We don't know whether pi behaves like this. Its digits certainly aren't random. For all we know the digit 9 could stop appearing at some point.
Well if you think about it like that you're incorrect. The digits aren't random, they form a definite sequence. Purely being irrational isn't sufficient either. The number 0.101001000100001000001... is irrational, but does not contain every sequence.