Are there more even numbers than odd numbers?

Very simple 'yes-or-no' question, but I can't find the answer anywhere. My gut feeling says the number of odd and even numbers are equal but I managed to write up something that contradicts my intuition. Although I still think my gut feeling is right, I can't find any logical or mathematical errors with my "proof". Can somebody please look it over and tell me which one of me is right?

Statement 1: For every positive odd integer $o$ there is an even integer $e=o+1$.

Statement 2: For every positive integer $n$ there is an negative integer i.e. $-n$.

Conclusion 1: The number of positive odd integers ($O_{positive}$) is equal to the number of positive even integers ($E_{positive}$). If there is a negative equivalent for every positive integer then the number of negative odd integers ($O_{negative}$) is equal to the number of negative even integers ($E_{negative}$). In short: $$O_{positive}=E_{positive}=O_{negative}=E_{negative}$$

Statement 3: The number zero is "neutral" (neither positive nor negative).

Statement 4: The number zero is an even integer.

Conclusion 2:

\begin{align} O_{total} & = O_{positive} + O_{negative} + O_{neutral} \\ & = O_{positive} + O_{negative} + 0 \\ & = O_{positive} + O_{negative} \end{align}

And:

\begin{align} E_{total} & = E_{positive} + E_{negative} + E_{neutral} \\ & = E_{positive} + E_{negative} + 1 \\ \end{align}

So:

\begin{align} E_{total} & = O_{total} + 1 \\ E_{total} & > O_{total}\\ \end{align}

Right? As an engineer I use math daily, but that doesn't make me a mathematician. So please be gentle :)

• If you want to know how infinities like this "work" in the average mathematician's mind, you should have a look at the story of Hilbert's hotel. – Arthur Jan 9 '14 at 14:10
• possible duplicate of Are half of all numbers odd? – Mark S. Jan 10 '14 at 23:57

First things first - there are an infinite number of both even numbers and odd numbers.

It's important to realize that $\infty$ (infinity) is not a number. Therefore it doesn't really make sense to talk about the "number" of even or odd numbers, or to write statements like $E_{\rm even}+1$, because that's assuming that $E_{\rm even}$ is a number that you can sensibly add $1$ to.

However, perhaps surprisingly it does make sense to ask if there are more even numbers than odd numbers. That is, you can compare two infinite quantities, or compare a finite quantity and an infinite quantity, even if you can't meaningfully add and subtract infinite quantities.

They way we define more, less and the same for infinite quantities is as follows. For two collections $A$ and $B$ (say $A$ are the even numbers and $B$ are the odd numbers) we say that

• If you can associate every item in $A$ with a unique item in $B$, and vice versa, then $A$ and $B$ are the same size.

• If you can associate every item in $A$ with a unique item in $B$, but not vice versa, then $B$ is bigger than $A$.

• If you can associate every item in $B$ with a unique item in $A$, but not vice versa, then $A$ is bigger than $B$.

In your case, you can associate every even number $n$ with the odd number $n+1$, and you can associate every odd number $m$ with the even number $m-1$ (assuming 0 is even) so therefore there are just as many odd numbers as even numbers.

This can lead to seemingly paradoxical results, because e.g. you can associate every whole number $n$ with the even number $2n$, and every even number $m$ with the whole number $m/2$, so there are just as many even numbers as whole numbers, even though the even numbers are a subset of the whole numbers.

• Aha, so if I understand correctly, the mistake I made was thinking that: infinity + 1 > infinity? – Jordy Jan 9 '14 at 14:22
• @Jordy The mistake was in thinking that $\infty$ is a number, and that $\infty+1$ is an expression that makes sense. It's easy to see that $\infty$ isn't a number, for here is a list of all the numbers: $\{0,1,2,3,4,\dots\}$. Where is $\infty$ in that list? You can't say "at the end", because the list doesn't have an end! (You also can't say "it's the ninth element in the list, but it's fallen over.") – Chris Taylor Jan 9 '14 at 14:26
• @Jordy This next bit, you'll have to imagine me saying in a stage whisper. Here it is: mathematicians have come up with a way of treating $\infty$ as a number! Shh, don't tell anyone. If you want the secrets, you'll have to learn a bit more math, and then go and read about transfinite ordinals. The smallest infinite ordinal is normally written $\omega$. Confusingly, $1+\omega=\omega$, but $\omega+1>\omega$. – Chris Taylor Jan 9 '14 at 14:28
• An old question and answer, but I disagree with the oft-repeated dogmatism that "$\infty$ is not a number" -- I think it teaches people the wrong idea. Is $i$ a number? Are the quaternions numbers? What about cardinals or ordinals? Furthermore, to me, all of the OP's reasoning is perfectly valid except the final conclusion, i.e. when from $E_{total} = O_{total} + 1$ he or she derives that $E_{total} > O_{total}$. All the rest is valid arithmetic with cardinalities, or alternatively valid arithmetic with infinity. – 6005 Oct 18 '16 at 18:51
• @6005 I think most people at the level of the OP equate "number" with "natural number" or "integer" or "rational number" or maybe "real number", so it is helpful to point out that $\infty$ is not one of these. Sure, there are various number systems in which $\infty$ is a quantity that you can do arithmetic with - but introducing them when someone isn't clear what is and is not a natural number only confuses, rather than clarifies. – Chris Taylor Oct 18 '16 at 20:04

As there is a bijection $$f(x) = x + 1$$ sending any odd number to an even, this shows that the sets have equal size.

Here, I assumed that the natural numbers start with $1$, if they should start with $0$, simply define the same function on the even numbers.

protected by Asaf Karagila♦Oct 18 '16 at 12:44

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