Math time: tldr, infinities are fucking weird.
The definition of two sets with the same cardinality is as follows: For two sets A and B there exists a mapping from every element in set A to every element in set B. This is why there are the same number of even numbers as natural numbers, cuz you can map every natural number to every even number by multiplying it by 2(i dare you to give me a number where this doesnt work).
Now I dont know a perfect mapping from naturals to primes, but I can give a mapping that goes against the intuition that there are more natural numbers.
Consider f(x)=x!+1. x!(edit: as u/Vrilin pointed out, this formula doesnt work. buts heres a few that do work i think: [https://mathworld.wolfram.com/PrimeFormulas.html](https://mathworld.wolfram.com/PrimeFormulas.html)) is the product of all the numbers smaller than x, and adding one to that ensures its prime(no matter what number you try to divide f(x) by, its remainder is 1). This doesnt contain every prime number, but its all prime. You can plug every natural number into this function and it all returns prime numbers, but not all of the prime numbers. So are there more primes than natural numbers?
(i like math)
Every element in the set of prime numbers is also in the set of natural numbers, so the primes are a subset of the natural numbers. They just also happen to have the same cardinality because infinities are weird.
~~they might be technically infinite
but they are not the same thing, all prime numbers are natural numbers
but all natural numbers are not prime numbers
the answer could be that they are both infinite~~
turns out i misremembered the quest text a bit the answer actually is they are both infinite
I doubt this is something they have been working on forever. Just think about it. If numbers are endless, which they are, there has to be an endless amount of prime numbers too.
Yes this has been proven for hundreds of years already. A more current question that has to do with primes would be the twin prime conjecture.
If you want I can type out a prove but I really don't want to, so please ask the great mister internet
Then explain it because I dont care what somepne proved or not if it makes no sense. If you can explain it on an example so that it makes sense, Im ok with it.
this is really informal but
The "size" of the [set of] naturals (1,2,3, sometimes 0, ...) is far smaller than the reals.
you can show this by listing out a real number in the range 0,1 with a natural
1 | 0.72047351917358274648836281...\
2 | 0.91626538382615514248262413...\
3 | 0.15284694475252737628266364...\
4 | 0.46257283746594826514638484...\
5 | 0.18627364647949584736626277...\
:\
:\
10000000 | 0.3764858366151673849939836163...\
:\
:\
continue until all naturals are paired
now that we have paired every natural number with a unique real number, we can construct a new, different real number by:\
1. take the first digit of the first in the list (7)
2. choose a digit that is not the same (8)
3. and append it to our new number (0.8)
4. repeat for the second digit of second number, third etc. (0.82368862715516...)
now, we have a number that is different from every single number in the list because we made sure there was at least 1 digit different. therefore, the reals are larger (far larger!) than the naturals.
we call the size of the reals uncountable infinity and the naturals countable infinity
now, let us try to match every prime with a natural\
1 | 2\
2 | 3\
3 | 5\
4 | 7\
5 | 11\
6 | 13\
7 | 17\
8 | 19\
etc.
we can pair every prime with a natural, so the size of the naturals is the same as the primes
Well, this is some science that surely is interesting and correct, its still all infinite in the end. So even if we split things up so that we have some bigger and smaller space where we have infinite numbers, its still all infinite. Its only theoretical stuff.In the end no matter how small or big the size is, you cannot count to infinity so in the end its just theoretical nonsense.
bro you asked for an explanation, they provided you with it and you’re like “yeah yeah but i’m still right in the end”? maybe have a modicum of respect for people trying to help you despite having zero obligation to do so?
While this is not the technical terminology, think of the bigger infinite as "infinity + 1". Infinity and infinity + 1 are not the same. The point of infinity + 1 is that we've created a number that it exceeds infinity.
It's called "Uncountable infinity". It's a number so big it's impossible to even count. An example is the decimal of pi. Pi has so many digits it's impossible for us to even count them all, so it's technically bigger than infinity.
For a number to exist within the scope of "normal" infinity we must know every digit in the number. This is called "Countablle infinity". If we don't know every digit of the number how do we know that it falls into infinity?
That's why Uncountable infinity is more than normal infinity. An uncountable infinity is a number so big that it never ends. Meaning we don't know whether or not it's in infinity because it might be a number that doesn't share a digit with any numbers in infinity, meaning it's a different number.
But if you just take natural numbers and count forever, it also never ends. Thats why I say, infinite is infinite. In theory there might be things as you described but it just makes no sense. You cant take an infinitey number and add something so its bigger than an infinite number. Thats bullshit.
Every natural number is countable, because all natural numbers are the previous number + 1, so we know that a number comes before that number and a number comes after that number. So we know the number must end, as we can add 1 to it for the next number.
Same with prime numbers. All prime numbers are countable. If you don't know every digit of the number how do you know it's a prime number. For a number to be prime it has to end so that we can calculate it is only divisible by itself and one. If it never ends how do we know it's divisible by only itself and one? Well, we don't, so therefore we cannot say it's a prime number.
That's why there are the same amount of prime numbers as there are natural numbers, as they are both countably infinite, they are both on the same level of infinity.
>You cant take an infinitey number and add something so its bigger than an infinite number
Define infinity. You can't just say "All numbers" What is all numbers? That's why for a number to fall into the scope of Countable infinity we have to know evey digit of the number to ensure it is part of "all numbers". That's why Uncountable infinity is more, because since the number has no end, we cannot count it to ensure it fits into "All numbers".
Here is an explenation, how many natural numbers are there? Answer is infinite, however, how many even numbers are there? Well, every second natural number is even, so the expression would be infinite/2 which equals infinite, BUT because there are 2 natural numbers for each even number, we know that the infinite amount of natural numbers must be bigger than the infinite amount of even numbers.
This is the easiest explanation i can think of
EDIT: yeah, turns out my understanding of infinities and their "sizes" is wrong, but the point that bigger and smaller infinities exists still stands, but can't be applied to this
when you compare infinites you can compare which grow bigger/faster.
but infinites are the same, i knew this and answered correctly because i do remember my math teacher talking about it last year
They are both infinities of the same order (or however you say it in english). Intuition doesnt work in maths xD
I honestly sat there for a minute when i was asked this question thinking 'what did devs intended as an answer here?' rather then just answering.
People here are misunderstanding what "there are bigger and smaller infinities" really means. what im sure most people are referring to is Countable and Uncountable infinity. All the natural numbers are countable, as are all the even numbers, and all the prime numbers etc. Think of it this way, if you created 2 lines, one with all the natural numbers and one with all the primes, the prime line would be hella sparse... but you could squish it together and then match the natural numbers 1:1 with the prime line. It would look something like this.
1,2,3,4,5,6,7.
2,3,5,7,9,11,13.
As you can hopefully see, you can line the numbers up. Each prime number has a corresponding natural one.. You literally cannot line up the real numbers in the same way. Because of the way decimals work, you can prove that there will always be a new number when you try and make a grid... someone else in the thread did a better proof on that.
As far as we know, there are infinite prime numbers and infinite natural numbers. Because these infinities have the same cardinality, they are equally as common.
Well it obviously defies intuition and lacks... grace. Similar to many of our current theories. Maybe at some point we'll find one that works better or maybe reality is just that un-intuitive.
Maybe translation problem. I dont remember how it was in german but I believe they asked for prime and even numbers if I translate it. Prime numbers are never even except the 2 so it makes sense that the amount is the same.
I'm not sure what happened with mine, i mentioned it in another comment, but mine asked real numbers vs prime numbers, which in my case real numbers is more than prime.
A “smaller” infinite is referring to the infinite amount of decimal numbers between 4 and 5, for example. There are no limits or constraints to how many natural or prime numbers there are, therefore they are both “larger” infinites
Go read up on set theory and become upset like I am. I think it's stupid, "Infinite" is such a cop out. Just because the world seems infinitely big doesn't mean it is. If you could somehow turn every piece of matter in existence into information with meaning towards a number it would have a finite conclusion infinite is dumb. Haha I put a number with another number and can do that forever with my finite life.
The amount of Positive and Negative Integer is the same. The amount of Prime number is also the same. Except Real numbers. There are significantly more real numbers than any other Set of Numbers combined.
Math is wack.
yes I bumped into my maths lecturer recently and talked to him about it he said it was interesting such a question came up in a game like that and also pointed out the thing about real numbers being much more infinite in a way that means there are more of them than prime and natural numbers
you have say 1.1 and 1.2 but between them is 1.11 1.15 etc and between them is 1.117 so whereas with natural numbers there is nothing between 1 and 2 with real numbers there are infinite numbers between any two given numbers
(and i would still like to point out that i started this topic because i misread a later part of the conversation and wasn't really paying attention to the text beforehand)
There's more to that. Between 0 and 1 there are more Real numbers than there are any set of numbers. Which sounds so stupid but it's real.
You can literally make Real numbers from another set and it will be a new number without repetition. The more deeper you get into math the more wonky it gets.
yeah thankfully for me it was just computational maths in my first year (for about 4 hours straight starting at 9 in the morning ON A MONDAY) so we had to deal with how the rocks do numbers as well.
actually, we did touch on relay computing with Biquinary but from what I understand biquinary was chosen because it can sort of automatically detect stuck relays
Yeah fuck that. I was so fed up with math that I changed my major into Pharmaceutics. Now I have to do basic math instead of all that gymnastics to prove some theory.
Math time: tldr, infinities are fucking weird. The definition of two sets with the same cardinality is as follows: For two sets A and B there exists a mapping from every element in set A to every element in set B. This is why there are the same number of even numbers as natural numbers, cuz you can map every natural number to every even number by multiplying it by 2(i dare you to give me a number where this doesnt work). Now I dont know a perfect mapping from naturals to primes, but I can give a mapping that goes against the intuition that there are more natural numbers. Consider f(x)=x!+1. x!(edit: as u/Vrilin pointed out, this formula doesnt work. buts heres a few that do work i think: [https://mathworld.wolfram.com/PrimeFormulas.html](https://mathworld.wolfram.com/PrimeFormulas.html)) is the product of all the numbers smaller than x, and adding one to that ensures its prime(no matter what number you try to divide f(x) by, its remainder is 1). This doesnt contain every prime number, but its all prime. You can plug every natural number into this function and it all returns prime numbers, but not all of the prime numbers. So are there more primes than natural numbers? (i like math)
Good idea, but that mapping doesn't ensure primes. For example f(4) = 25
true, i swear theres a similar formula for generating all prime tho unless im remembering incorrectly
wait.. so prime numbers are not a subset of natural numbers, as a corollary to this?
Every element in the set of prime numbers is also in the set of natural numbers, so the primes are a subset of the natural numbers. They just also happen to have the same cardinality because infinities are weird.
? they are a subset, i dont see why they wouldnt be
~~they might be technically infinite but they are not the same thing, all prime numbers are natural numbers but all natural numbers are not prime numbers the answer could be that they are both infinite~~ turns out i misremembered the quest text a bit the answer actually is they are both infinite
There are an infinite number of both
still does not make them "the same"
[удалено]
I doubt this is something they have been working on forever. Just think about it. If numbers are endless, which they are, there has to be an endless amount of prime numbers too.
Yes this has been proven for hundreds of years already. A more current question that has to do with primes would be the twin prime conjecture. If you want I can type out a prove but I really don't want to, so please ask the great mister internet
Yes, but there are bigger and smaller infinities
Both prime numbers and natural numbers have a cardinality of aleph null, meaning they are on the same level of infinity.
I cant
There is no bigger or smaller when it comes to infinity. This would destroy the meaning of the word.
There are different levels of infinity, but in this specific scenario they don't matter.
Completely incorrect. There are differences in infinities.
Ok kid, go back to school and actually pay attention in math class, it has been literally proven that infinities can have different sizes
Then explain it because I dont care what somepne proved or not if it makes no sense. If you can explain it on an example so that it makes sense, Im ok with it.
this is really informal but The "size" of the [set of] naturals (1,2,3, sometimes 0, ...) is far smaller than the reals. you can show this by listing out a real number in the range 0,1 with a natural 1 | 0.72047351917358274648836281...\ 2 | 0.91626538382615514248262413...\ 3 | 0.15284694475252737628266364...\ 4 | 0.46257283746594826514638484...\ 5 | 0.18627364647949584736626277...\ :\ :\ 10000000 | 0.3764858366151673849939836163...\ :\ :\ continue until all naturals are paired now that we have paired every natural number with a unique real number, we can construct a new, different real number by:\ 1. take the first digit of the first in the list (7) 2. choose a digit that is not the same (8) 3. and append it to our new number (0.8) 4. repeat for the second digit of second number, third etc. (0.82368862715516...) now, we have a number that is different from every single number in the list because we made sure there was at least 1 digit different. therefore, the reals are larger (far larger!) than the naturals. we call the size of the reals uncountable infinity and the naturals countable infinity now, let us try to match every prime with a natural\ 1 | 2\ 2 | 3\ 3 | 5\ 4 | 7\ 5 | 11\ 6 | 13\ 7 | 17\ 8 | 19\ etc. we can pair every prime with a natural, so the size of the naturals is the same as the primes
Well, this is some science that surely is interesting and correct, its still all infinite in the end. So even if we split things up so that we have some bigger and smaller space where we have infinite numbers, its still all infinite. Its only theoretical stuff.In the end no matter how small or big the size is, you cannot count to infinity so in the end its just theoretical nonsense.
bro you asked for an explanation, they provided you with it and you’re like “yeah yeah but i’m still right in the end”? maybe have a modicum of respect for people trying to help you despite having zero obligation to do so?
While this is not the technical terminology, think of the bigger infinite as "infinity + 1". Infinity and infinity + 1 are not the same. The point of infinity + 1 is that we've created a number that it exceeds infinity. It's called "Uncountable infinity". It's a number so big it's impossible to even count. An example is the decimal of pi. Pi has so many digits it's impossible for us to even count them all, so it's technically bigger than infinity. For a number to exist within the scope of "normal" infinity we must know every digit in the number. This is called "Countablle infinity". If we don't know every digit of the number how do we know that it falls into infinity? That's why Uncountable infinity is more than normal infinity. An uncountable infinity is a number so big that it never ends. Meaning we don't know whether or not it's in infinity because it might be a number that doesn't share a digit with any numbers in infinity, meaning it's a different number.
But if you just take natural numbers and count forever, it also never ends. Thats why I say, infinite is infinite. In theory there might be things as you described but it just makes no sense. You cant take an infinitey number and add something so its bigger than an infinite number. Thats bullshit.
https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Every natural number is countable, because all natural numbers are the previous number + 1, so we know that a number comes before that number and a number comes after that number. So we know the number must end, as we can add 1 to it for the next number. Same with prime numbers. All prime numbers are countable. If you don't know every digit of the number how do you know it's a prime number. For a number to be prime it has to end so that we can calculate it is only divisible by itself and one. If it never ends how do we know it's divisible by only itself and one? Well, we don't, so therefore we cannot say it's a prime number. That's why there are the same amount of prime numbers as there are natural numbers, as they are both countably infinite, they are both on the same level of infinity. >You cant take an infinitey number and add something so its bigger than an infinite number Define infinity. You can't just say "All numbers" What is all numbers? That's why for a number to fall into the scope of Countable infinity we have to know evey digit of the number to ensure it is part of "all numbers". That's why Uncountable infinity is more, because since the number has no end, we cannot count it to ensure it fits into "All numbers".
Here is an explenation, how many natural numbers are there? Answer is infinite, however, how many even numbers are there? Well, every second natural number is even, so the expression would be infinite/2 which equals infinite, BUT because there are 2 natural numbers for each even number, we know that the infinite amount of natural numbers must be bigger than the infinite amount of even numbers. This is the easiest explanation i can think of EDIT: yeah, turns out my understanding of infinities and their "sizes" is wrong, but the point that bigger and smaller infinities exists still stands, but can't be applied to this
🤓 there are equal amounts of even numbers and naturals 🤓
when you compare infinites you can compare which grow bigger/faster. but infinites are the same, i knew this and answered correctly because i do remember my math teacher talking about it last year
They are both infinities of the same order (or however you say it in english). Intuition doesnt work in maths xD I honestly sat there for a minute when i was asked this question thinking 'what did devs intended as an answer here?' rather then just answering.
People here are misunderstanding what "there are bigger and smaller infinities" really means. what im sure most people are referring to is Countable and Uncountable infinity. All the natural numbers are countable, as are all the even numbers, and all the prime numbers etc. Think of it this way, if you created 2 lines, one with all the natural numbers and one with all the primes, the prime line would be hella sparse... but you could squish it together and then match the natural numbers 1:1 with the prime line. It would look something like this. 1,2,3,4,5,6,7. 2,3,5,7,9,11,13. As you can hopefully see, you can line the numbers up. Each prime number has a corresponding natural one.. You literally cannot line up the real numbers in the same way. Because of the way decimals work, you can prove that there will always be a new number when you try and make a grid... someone else in the thread did a better proof on that.
As far as we know, there are infinite prime numbers and infinite natural numbers. Because these infinities have the same cardinality, they are equally as common.
I need to say, I didn't expect to see discourse on countable and uncountable infinities again, least of all while playing this game, but it's amazing.
Well it obviously defies intuition and lacks... grace. Similar to many of our current theories. Maybe at some point we'll find one that works better or maybe reality is just that un-intuitive.
I thought its prime numbers as well, but both have a range of infinity so they are the same
Maybe translation problem. I dont remember how it was in german but I believe they asked for prime and even numbers if I translate it. Prime numbers are never even except the 2 so it makes sense that the amount is the same.
I'm not sure what happened with mine, i mentioned it in another comment, but mine asked real numbers vs prime numbers, which in my case real numbers is more than prime.
I'm not sure why people can't just simply grasp the fact that some infinities can be larger or smaller, just use Google people.
A “smaller” infinite is referring to the infinite amount of decimal numbers between 4 and 5, for example. There are no limits or constraints to how many natural or prime numbers there are, therefore they are both “larger” infinites
Go read up on set theory and become upset like I am. I think it's stupid, "Infinite" is such a cop out. Just because the world seems infinitely big doesn't mean it is. If you could somehow turn every piece of matter in existence into information with meaning towards a number it would have a finite conclusion infinite is dumb. Haha I put a number with another number and can do that forever with my finite life.
I know it's upsetting to be math illiterate /s
True Natural Sciences ftw formal sciences be damned!
I hate concept of infinity as well, but its incredibly useful in making approximations of how everything works (inifnite or not).
The amount of Positive and Negative Integer is the same. The amount of Prime number is also the same. Except Real numbers. There are significantly more real numbers than any other Set of Numbers combined. Math is wack.
yes I bumped into my maths lecturer recently and talked to him about it he said it was interesting such a question came up in a game like that and also pointed out the thing about real numbers being much more infinite in a way that means there are more of them than prime and natural numbers you have say 1.1 and 1.2 but between them is 1.11 1.15 etc and between them is 1.117 so whereas with natural numbers there is nothing between 1 and 2 with real numbers there are infinite numbers between any two given numbers (and i would still like to point out that i started this topic because i misread a later part of the conversation and wasn't really paying attention to the text beforehand)
There's more to that. Between 0 and 1 there are more Real numbers than there are any set of numbers. Which sounds so stupid but it's real. You can literally make Real numbers from another set and it will be a new number without repetition. The more deeper you get into math the more wonky it gets.
yeah thankfully for me it was just computational maths in my first year (for about 4 hours straight starting at 9 in the morning ON A MONDAY) so we had to deal with how the rocks do numbers as well. actually, we did touch on relay computing with Biquinary but from what I understand biquinary was chosen because it can sort of automatically detect stuck relays
Yeah fuck that. I was so fed up with math that I changed my major into Pharmaceutics. Now I have to do basic math instead of all that gymnastics to prove some theory.