7 ms·

It's because there simply are more odd numbers than even numbers... (j/k!)

raverbashing 11y ago
I think if you consider Z rather than N the opposite is true Proof: For ever number k there exists -k which has the same parity as k (that is, if k is even, -k is even as well) For every k there is (k + 1) with opposite parity (same thing with -k and -(k + 1) Now, if we count the numbers, from 1 to infinite, the number of even and odd numbers is the same from 1 to -infinite the number of even and odd numbers is the same as well And you get an extra zero, an even number
- seanyeh 11y ago
  Technically, you can create a bijection between all natural numbers and all even numbers. For all natural numbers k, there is an even number 2k (and for every even number 2k, there is a natural number k). So, the number of even numbers and natural numbers is the same. That's what happens when you try to reason with infinity ;)
  - raverbashing 11y ago
    Yes, with Natural numbers Curious what is possible in Integer numbers > That's what happens when you try to reason with infinity ;) Ah yes! Fascinating subject
    - greiskul 11y ago
      Cardinality of natural numbers and integers is also the same. 0 -1 1 -2 2 -3 3... 0 1 2 3 4 5 6...
- greydius 11y ago
  Count from 2 to infinity and 0 to -infinity and you will get the opposite result: you'll have an extra odd number, 1.
  - alphabetam 11y ago
    The function n -> n + 1 maps every even number into an odd, and it has an inverse (obviously n -> n - 1). Thus, the two sets have the same cardinality.
    - bpicolo 11y ago
      Into a unique odd (which is key) : P
- dragonwriter 11y ago
  Intuitions based on assuming that you can treat "infinity" the way you would a finite number are often, as in this case, wrong. The "infinity" referenced here, aleph-null, is provably both the cardinality of the set of even integers, and the set of odd integers. (And also the set of integers, and any other countably infinite set.) So, unintuitive as it might be when thinking about finite subsets of the integers, there are not only as many positive integers as negative integers, but as many of either as there are integers. (There are, however, bigger infinite sets, there are, for instance, more reals than integers.) See, https://en.wikipedia.org/wiki/Aleph_number https://en.wikipedia.org/wiki/Aleph_number
  - mikeash 11y ago
    Some fun examples about infinity: There are as many prime numbers as there are integers. There are as many fractions between 10 and 11 as there are integers. There are more real numbers between 41.00001 and 41.0002 than there are integers from -infinity to infinity. Even though both are "infinite."
  - noir_lord 11y ago
    Hilbert Hotel is a great thought experiment on this, I saw a BBC open university show on this one night after the pub and it properly bent my mind, I wish I had more mathematical aptitude as this stuff is utterly fascinating.

I think if you consider Z rather than N the opposite is true Proof: For ever number k there exists -k which has the same parity as k (that is, if k is even, -k is even as well) For every k there is (k + 1) with opposite parity (same thing with -k and -(k + 1) Now, if we count the numbers, from 1 to infinite, the number of even and odd numbers is the same from 1 to -infinite the number of even and odd numbers is the same as well And you get an extra zero, an even number

seanyeh 11y ago
Technically, you can create a bijection between all natural numbers and all even numbers. For all natural numbers k, there is an even number 2k (and for every even number 2k, there is a natural number k). So, the number of even numbers and natural numbers is the same. That's what happens when you try to reason with infinity ;)
- raverbashing 11y ago
  Yes, with Natural numbers Curious what is possible in Integer numbers > That's what happens when you try to reason with infinity ;) Ah yes! Fascinating subject
  - greiskul 11y ago
    Cardinality of natural numbers and integers is also the same. 0 -1 1 -2 2 -3 3... 0 1 2 3 4 5 6...
greydius 11y ago
Count from 2 to infinity and 0 to -infinity and you will get the opposite result: you'll have an extra odd number, 1.
- alphabetam 11y ago
  The function n -> n + 1 maps every even number into an odd, and it has an inverse (obviously n -> n - 1). Thus, the two sets have the same cardinality.
  - bpicolo 11y ago
    Into a unique odd (which is key) : P
dragonwriter 11y ago
Intuitions based on assuming that you can treat "infinity" the way you would a finite number are often, as in this case, wrong. The "infinity" referenced here, aleph-null, is provably both the cardinality of the set of even integers, and the set of odd integers. (And also the set of integers, and any other countably infinite set.) So, unintuitive as it might be when thinking about finite subsets of the integers, there are not only as many positive integers as negative integers, but as many of either as there are integers. (There are, however, bigger infinite sets, there are, for instance, more reals than integers.) See, https://en.wikipedia.org/wiki/Aleph_number https://en.wikipedia.org/wiki/Aleph_number
- mikeash 11y ago
  Some fun examples about infinity: There are as many prime numbers as there are integers. There are as many fractions between 10 and 11 as there are integers. There are more real numbers between 41.00001 and 41.0002 than there are integers from -infinity to infinity. Even though both are "infinite."
- noir_lord 11y ago
  Hilbert Hotel is a great thought experiment on this, I saw a BBC open university show on this one night after the pub and it properly bent my mind, I wish I had more mathematical aptitude as this stuff is utterly fascinating.