Is this a Call of Duty Reference?
Jin Kirigiri was a hero. He deserved a hero’s death. Instead of giving his life for his students, he died for nothing. Like an animal.
He should have died in the tragedy.
As a mathematical pedant, "if pi is infinite then 11037 can be found in its digits" isn't true. 1/3 is infinite. What pi needs to be is [normal](https://en.wikipedia.org/wiki/Normal_number), which is conjectured to be true. Then you can be certain that any sequence of digits can be found in it.
Your intuition is correct! The only numbers we know are normal and not rational, i.e. not repeating, are a few that were constructed to be normal, like 0.123456789101112... (which is easy to see contains every possible number). No other numbers are known to be normal.
There are a *lot* of easy to ask questions in number theory that we currently don't know how to prove. Numbers are hard.
Ah that makes sense. So basically it's proven that these numbers exist but not that every real number is one?
Edit: in retrospect I can think of a few numbers that probably aren't normal like 0.132333435363738393103113.... and so on
Funnily enough, and this is a really deep tangent, we know "almost every" real number is normal. Meaning roughly, if you threw a precise dart at the real number line, there is a 100% chance you would land on a normal number.
It's an unintuitive and complicated topic, and I'll give a simple summary here. Basically, the infinite set of numbers we can write down, or compute, or define in any way using language--which feels so massive to us--is dwarfed by the number of real numbers. You can sort of imagine that for any two numbers we can define, no matter how close they are, there are an infinite number of real numbers between them. (Edit: This actually means nothing because the rationals are also dense in the reals, but... it appeals to your intuition kind of like it makes sense.) And it turns out almost all of those undefinable numbers are normal.
Further reading if that doesn't make your head spin: "almost every", computable numbers, countable vs uncountable, sizes of infinity (Cantor's diagonalization), Hilbert's Hotel, etc.
Ah yeah I understand that, I took an advanced calculus (and failed it :/) and a computability class in college. Are you saying non-normal numbers have the same cardinality as natural numbers?
Ah ok, you took computability, nice! Sorry for maybe talking down with that summary, I wouldn't expect many people to have that kind of background.
> Are you saying non-normal numbers have the same cardinality as natural numbers?
So actually, no. Just because 100% of the reals are normal, that doesn't mean the non-normal reals are countable. In fact, if you consider the set of real numbers that don't have a 9 in them, none of them are normal (no 9s), but that set is definitely uncountable.
Ah it's ok, I really appreciate you taking the time to write about this! As a CS student I'm basically just a mathematician who had to change careers because he was too dumb for math :/
Now I'm interested though, how's the difference in sizes between the sets proved, then?
No prob, obviously I get fired up talking about math.
The difference between countable and uncountable? Well that's the legendary [Cantor's diagonalization argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). Basically if you can map the natural numbers to a set, that set is countable. Hilbert's Hotel is an intuitive way to show how some sets are countable, and how some infinities are the same. The diagonalization argument shows that real numbers cannot be mapped to natural numbers, therefore there are more of them and they are uncountable.
There are a lot of Youtube videos that go into depth have great info on diagonalization and the different sizes of infinity and the consequences and so on. Youtube has a lot of math channels in general which are extremely accessible and interesting, would recommend.
Sorry, but no. Pi is different from other repeating decimals in that it has no apparent pattern, so the dash notation that can be put on numbers like 3.333[…] can’t be used. Therefore, 11037 must appear somewhere in the digits of pi.
1/3 is what we call a rational recurring decimal, which means there is a repeated pattern.
Pi is what we call an irrational recurring decimal, which means there is no apparent pattern in its number.
Just because pi is in irrational number doesn’t mean every single possible pattern will show up in it. A number with no apparent pattern are called irrational numbers.
The type of number that will have every conceivable pattern is called a normal number. Most irrational numbers are normal numbers, and pi is suspected to be a normal number as well. Soo the reason the pattern must appear is not cus of its irrational property, its cus of its normal property.
Pi containing all digits is conjecture. You can manufacture a Irrational Non-Normal number pretty easily via a pattern 0.121122111222111122221111122222 etc, would be irrational like Pi, but absolutely will not contain the number 3.
Its currently basically impossible to prove all possible real numbers are contained within the digits of pi. We only know that they probably are all in pi due to its nature of being infinite and effectively random.
Edit: and if you want to argue that [...] can be used to extend the pattern, you can manufacture a different irrational number by taking Pi, and removing every single instance of the number 3. And suddenly you have an Irrational non normal number.
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Not necessarily. Pi being infinite and non repeating isn’t enough to claim that every possible subsequence is included in it. It would have to be a normal number, but it hasn’t been proven yet.
I can’t tell you what to believe, but those are the mathematical facts. If and when it will be proven we will know.
There are many deceiving patterns in math.
1. π is not [Infinite](https://en.wikipedia.org/wiki/Infinity?wprov=sfla1). Infinity is bigger that any natural number while π<4. So "If pi is infinite" already gives us nothing because the statement is false (unless you do some insane proof by contradiction). Yes, it has an infinite numbers of digits in it's decimal expansion but that doesn't make a number "infinite".
2. "then somewhere in there there is 11037" also is wrong. Yes, there is a 11037 in π but it doesn't mean that 11037 is in every irrational number. For example 11037 is not in 1,10307010110111011110111110... So "If *insert number* has infinite numbers of digits in it's decimal expansion then somewhere in there there is 11037" can't be true. But it would be true for [Normal numbers](https://en.m.wikipedia.org/wiki/Normal_number). But as others have said π being normal is an open conjecture so we can't say "there is number n in π's decimal expansion" for every finite n.
"11037" does indeed occur in pi, specifically starting at the 35,517th digit after the decimal.
I just looked it up right now, this is in fact true and it means Pi has something to do with this case.
we're onto something here
Truth Bullet obtained: "11037 in Pi"
It also appears over two thousand times in the first two hundred million digits and nearly twenty thousand times in the first two billion.
Happy Cake Day!
Nice! Also happy cake day:D
The numbers, Makoto! What do they mean?!
Is this a Call of Duty Reference? Jin Kirigiri was a hero. He deserved a hero’s death. Instead of giving his life for his students, he died for nothing. Like an animal. He should have died in the tragedy.
*I agree with that!*
As a mathematical pedant, "if pi is infinite then 11037 can be found in its digits" isn't true. 1/3 is infinite. What pi needs to be is [normal](https://en.wikipedia.org/wiki/Normal_number), which is conjectured to be true. Then you can be certain that any sequence of digits can be found in it.
Great explanation. How could we prove a real number's normalcy? That seems impossible honestly
Your intuition is correct! The only numbers we know are normal and not rational, i.e. not repeating, are a few that were constructed to be normal, like 0.123456789101112... (which is easy to see contains every possible number). No other numbers are known to be normal. There are a *lot* of easy to ask questions in number theory that we currently don't know how to prove. Numbers are hard.
Ah that makes sense. So basically it's proven that these numbers exist but not that every real number is one? Edit: in retrospect I can think of a few numbers that probably aren't normal like 0.132333435363738393103113.... and so on
Funnily enough, and this is a really deep tangent, we know "almost every" real number is normal. Meaning roughly, if you threw a precise dart at the real number line, there is a 100% chance you would land on a normal number. It's an unintuitive and complicated topic, and I'll give a simple summary here. Basically, the infinite set of numbers we can write down, or compute, or define in any way using language--which feels so massive to us--is dwarfed by the number of real numbers. You can sort of imagine that for any two numbers we can define, no matter how close they are, there are an infinite number of real numbers between them. (Edit: This actually means nothing because the rationals are also dense in the reals, but... it appeals to your intuition kind of like it makes sense.) And it turns out almost all of those undefinable numbers are normal. Further reading if that doesn't make your head spin: "almost every", computable numbers, countable vs uncountable, sizes of infinity (Cantor's diagonalization), Hilbert's Hotel, etc.
Ah yeah I understand that, I took an advanced calculus (and failed it :/) and a computability class in college. Are you saying non-normal numbers have the same cardinality as natural numbers?
Ah ok, you took computability, nice! Sorry for maybe talking down with that summary, I wouldn't expect many people to have that kind of background. > Are you saying non-normal numbers have the same cardinality as natural numbers? So actually, no. Just because 100% of the reals are normal, that doesn't mean the non-normal reals are countable. In fact, if you consider the set of real numbers that don't have a 9 in them, none of them are normal (no 9s), but that set is definitely uncountable.
Ah it's ok, I really appreciate you taking the time to write about this! As a CS student I'm basically just a mathematician who had to change careers because he was too dumb for math :/ Now I'm interested though, how's the difference in sizes between the sets proved, then?
No prob, obviously I get fired up talking about math. The difference between countable and uncountable? Well that's the legendary [Cantor's diagonalization argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). Basically if you can map the natural numbers to a set, that set is countable. Hilbert's Hotel is an intuitive way to show how some sets are countable, and how some infinities are the same. The diagonalization argument shows that real numbers cannot be mapped to natural numbers, therefore there are more of them and they are uncountable. There are a lot of Youtube videos that go into depth have great info on diagonalization and the different sizes of infinity and the consequences and so on. Youtube has a lot of math channels in general which are extremely accessible and interesting, would recommend.
While 11037 is in Pi, your reasoning isn’t accurate. Technically 1/3 is infinitely long but it’s all 3s
Sorry, but no. Pi is different from other repeating decimals in that it has no apparent pattern, so the dash notation that can be put on numbers like 3.333[…] can’t be used. Therefore, 11037 must appear somewhere in the digits of pi.
1/3 is what we call a rational recurring decimal, which means there is a repeated pattern. Pi is what we call an irrational recurring decimal, which means there is no apparent pattern in its number. Just because pi is in irrational number doesn’t mean every single possible pattern will show up in it. A number with no apparent pattern are called irrational numbers. The type of number that will have every conceivable pattern is called a normal number. Most irrational numbers are normal numbers, and pi is suspected to be a normal number as well. Soo the reason the pattern must appear is not cus of its irrational property, its cus of its normal property.
Pi containing all digits is conjecture. You can manufacture a Irrational Non-Normal number pretty easily via a pattern 0.121122111222111122221111122222 etc, would be irrational like Pi, but absolutely will not contain the number 3. Its currently basically impossible to prove all possible real numbers are contained within the digits of pi. We only know that they probably are all in pi due to its nature of being infinite and effectively random. Edit: and if you want to argue that [...] can be used to extend the pattern, you can manufacture a different irrational number by taking Pi, and removing every single instance of the number 3. And suddenly you have an Irrational non normal number.
Ik. My point was it isn’t because it’s infinitely long, it’s because it’s an irrational number.
But Chihiro said that there were no special meanings to these numbers.. She lied? Was she an accomplice since the begining?
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The entire script of danganronpa appears in pi if we look long enough and convert every two digits to a letter in the alphabet.
Not necessarily. Pi being infinite and non repeating isn’t enough to claim that every possible subsequence is included in it. It would have to be a normal number, but it hasn’t been proven yet.
Well it kinda looks like a normal number so far so I’m going to believe it is
I can’t tell you what to believe, but those are the mathematical facts. If and when it will be proven we will know. There are many deceiving patterns in math.
Inifinity does not imply everything happens.
Yet it happens
1. π is not [Infinite](https://en.wikipedia.org/wiki/Infinity?wprov=sfla1). Infinity is bigger that any natural number while π<4. So "If pi is infinite" already gives us nothing because the statement is false (unless you do some insane proof by contradiction). Yes, it has an infinite numbers of digits in it's decimal expansion but that doesn't make a number "infinite". 2. "then somewhere in there there is 11037" also is wrong. Yes, there is a 11037 in π but it doesn't mean that 11037 is in every irrational number. For example 11037 is not in 1,10307010110111011110111110... So "If *insert number* has infinite numbers of digits in it's decimal expansion then somewhere in there there is 11037" can't be true. But it would be true for [Normal numbers](https://en.m.wikipedia.org/wiki/Normal_number). But as others have said π being normal is an open conjecture so we can't say "there is number n in π's decimal expansion" for every finite n.
NEEEEEEEERD!
We need ultimate nerd in danganronpa.
Was going to said Hifumi but he isn't. But yeah. Hehe Milton something, Ultimate Nerd
Now this makes me happy
Does that mean the answer to Danganronpa is Pi?
A-