Are half of all numbers odd?

Question

Plato puts the following words in Socrates' mouth in the Phaedo dialogue:

I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called odd, which is not the same as three? Yet the number three and the number five and half of numbers in general are so constituted, that each of them is odd though not identified with the idea of odd. And in the same way two and four and all the other series of numbers are even, each of them, though not identical with evenness. (104a-b)

The philosophical point is that there exists an Idea (or Form) called Odd and odd numbers are merely specific instances of Odd. The are not, themselves, identical to the concept of Oddness.

But I bolded a throwaway phrase that prompts my question: Are half of all integers odd?

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Plato likely did not consider one to be odd nor did he likely consider either zero or negative integers among his set of "numbers in general". I don't see his proof (if he had or was aware of one), but I would imagine it be something along the lines that for every even number N there is an odd number N+1. Therefore half of all numbers greater than 1 are odd.

I'm aware that zero is even, which makes me think there is one extra even number than odd numbers. My thinking is that if half of all positive numbers are odd and half of all negative numbers are odd, than leaving out zero, half of all integers are odd. But when you add in the only non-positive, non-negative number, which is even, you have an extra even number. (The programmer in me wants to add the concepts of -0 and +0 for symmetry. The rest of me thinks that's nuts!)

However, I think my imagined platonic proof still works as long as zero is definitely even and won't work if it's either odd or neither parity.

Are either of these proofs valid?

Answer 1

The basic problem with your argument is that the concept of half of an infinite set is not well-defined. It’s possible to pair up the even integers with the odd integers with none left over in either set, and if we were talking about finite sets, that would be a demonstration that each was half of the whole set of integers. However, it’s also possible to pair up the multiples of $100$, say, with all the rest of the integers with none left over in either set, and the multiples of $100$ are obviously only part of the set of even numbers. Clearly, then, this kind of pairing argument cannot lead to any very useful notion of half of the set of integers.

There is a notion of asymptotic density of a set of positive integers that does a pretty good job of capturing many people’s intuitive sense of what half (or any other fraction) of the set of positive integers should mean. Let $A\subseteq\mathbb{Z}^+$; for each $n\in\mathbb{Z}^+$, $$\frac{|A\cap\{1,2,\dots,n\}|}n$$ is the fraction of the first $n$ positive integers that are in the set $A$. If this fraction approaches a limit $\alpha$ as $n$ increases, i.e., if $$\lim_{n\to\infty}\frac{|A\cap\{1,2,\dots,n\}|}n=\alpha\;,$$ the set $A$ is said to have asymptotic density $\alpha$. (Note that the limit need not exist: not all sets of positive integers have asymptotic densities. It’s not hard to show that if $A$ is the set of even integers or the set of odd integers, its asymptotic density is $1/2$, so it’s meaningful to say that each of these sets comprises half of the positive integers in terms of asymptotic density. Similarly, for each positive integer $n$, the set of multiples of $n$ can easily be shown to have density $1/n$, exactly as one would wish.

Added: In the comments Srivatsan points out an excellent example: the perfect squares can be paired up one-for-one with the integers, but their asymptotic density is $0$: they get sparser and sparser as you get further away from $0$. Thus, in one sense there are just as many perfect squares as there are integers, and yet in another sense almost every positive integers is a non-square.

Answer 2

While they are both countable, there is a more applicable mechanism for measuring how many natural numbers have a certain property. If $A \subset \mathbb{N}$, let $$A_n = \text{number of elements in A less or equal to n}$$ then we can define density of $A$ as $$\operatorname{density}(A) = \lim_{n\to\infty} \frac{A_n}{n}$$Then the odd numbers have density $\frac{1}{2}$.

Answer 3

It's already mentioned in the comments that you will encounter some problems when talking about half of the elements of an infinite set. There are ways to compare the size of infinite sets.

See: http://en.wikipedia.org/wiki/Cardinality

Two sets have the same cardinality (=size), if there exists a bijection between them. The natural numbers and all other sets with the same cardinality are called countably infinite. That means roughly that we can find a way of enumerating them (and additionally they are infinite). It is easy to give a bijection from the natural numbers to the even numbers. Just associate to a number $n$ the even number $2n$, so we conclude that the cardinality of the even numbers is the same as the cardinality of all natural numbers! The same is true for the odd numbers, so in set theory the two sets you want to compare are considered equally large!

That being said, there are other notions of size for sets of natural numbers. One of them is natural density: http://en.wikipedia.org/wiki/Natural_density

Instead of "counting" all even numbers at once you restrict the set of numbers you look at to natural numbers $\leq n$ and calculate the proportion $x_n$ of even numbers in this set. $$x_n =\frac{\text{even numbers below }n}{n}$$ The density is then defined as the limit of the $x_n$ and you will find that it is indeed $\frac{1}{2}$, just as the natural density of the odd numbers.

There are many interesting theorems about the densities of certain subsets of the natural numbers like Dirichlet's theorem on primes in arithmetic progressions. One instance of the theorem tells us for example that "half of the primes are congruent to 1 modulo 4 and the other half is congruent to 3".

Answer 4

There is an infinite amount of even numbers.

There is an infinite amount of odd numbers.

How do we compare infinities? Is one infinity larger than the other?

We look at the set of even numbers, say A, and the set of odd numbers B. Two sets are said to have the same cardinality if there is a bijection between them. We can find such a bijection, so these "infinities" are the same.

It might also be worth mentioning that there are "the same number of" fractions as there are whole numbers.

Answer 5

[The other answers have more than answered my question, so my goal in self-answering is to make sure I understand what's been said so far. I would appreciate corrective comments if I make any mathematical mistakes. I'm heavily indebted to JDH's excellent answer to the question "Is infinity an odd or even number?"]

For numbers greater than one, which is the set Socrates refers to, even numbers are those that can be divided on the two sides of a scale so that the scale balances. If I have 10 identical coins (an even number), I can put 5 on each side. But if I add (or subtract) one coin from the total, the scale will not balance. (Something is "odd" about the number.)

Moving up a level of abstraction, we notice that an even number of any identical object will balance: 10 coins, 10 jars of olive oil, 10 grains of sand, etc. So depending on your point of view, evenness is a property of 10 or 10 is an instance of Even. Even when we consider things that we can't reasonably weigh on a scale, such as imaginary objects, can still be labeled as even if there are an even number of them.

So what happens if you take numbers and metaphorically divide them between the two pans of a scale: even numbers in one and odd numbers in the other? It depends on the range of numbers you consider. [2...3] will balance, but [2...4] won't. But you can always add one more number to the range and make the scale balance again, so [2...5] does balance.

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Now Socrates didn't have a concept of infinity, so he probably was thinking of arbitrarily large numbers. If you happen to pick an odd number of numbers, than one or the other pan on the hypothetical scale will be tip. But since the range is arbitrary, you can always pick just one more number and restore the balance. Nothing about zero or infinity or negative numbers changes this: if you want half of all numbers to be odd, just make sure your range includes an even number of numbers.

When you start talking about infinity or an infinite number of numbers and so on, things get weird and you can't really apply everyday intuition. There are any number of paradoxical-seeming results that you can arrive at if you start messing around with infinite series. For instance, you can have a hotel with an infinite number of rooms, an infinite number of guests, and plenty of room for an infinite number of new guests. More to the point, you can match an infinite number of odd numbers to an infinite number of even numbers at any ratio you like without running out of either type of number. So saying that half of all numbers is odd is just as true as saying that 1 in a thousand numbers are odd.

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However, we can still look back at our scale and for large ranges of numbers, the scale either balances or very-nearly balances for any particular range you pick. Sure there is one extra even number in the range [2...1,000], but it would take a pretty accurate scale to determine which pan is weighed down more. And the larger the range, the smaller the difference. Mathematically, this way of thinking corresponds to the concept of natural density. Thankfully, the natural density of the odd numbers turns out to be half when measured this way.

As an aside, Wikipedia mentions that:

This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory.

Summary

As long as we steer away from the (very intriguing) notion of infinity, it seems that Socrates was right to say that half of the numbers are odd.

Answer 6

I think the OP has the idea of two infinite sets A and B having the "same number of elements" if $card(n, A)/card(n, B) \to 1$ as $n \to \infty$, where $card(n, C)$ is the number of elements of $C$ that are less than $n$. In this sense, the evens and odds have the same number of elements.

In this sense, there are more evens than squares, and this is an equivalence relation.

Of course, this leads to the idea of the density of sequences (Schnirliman (sp?), natural, logarithmic, ...) which has many interesting results.

(Added a bit later: I see that Lovre beat me by 3 minutes.)

Answer 7

Your question brings up your confusion around the problem of "0" and suggests that this might lead one to think that there is one extra even number.

While this is erroneous and can be proven mathematically as others have above, there is also a simpler explanation which stems from a philosophical question: Why do you assume that the "center" of the natural numberline is "0"?

The concept of "center" is something that works with finite groups. But the numberline is infinite. (What is the midpoint of an infinite ordered set?)

If we had randomly placed the number "-1" as the "center" (ie. there are infinite numbers less than it and larger than it on either side), would that have led you to think that there is one extra odd number?

The point is to make you realize that, in reality, mathematicians' choice of "0" being at the center of every numberline we draw is just a tradition and is just an agreed upon matter. It has no "scientific" basis and should not affect the way you look at things

Now, from a mathematical perspective: $\lim_{x\to\infty}\frac{x}{x+1} =\lim_{x\to\infty}\frac{x}{x} =1$

So even if you have one extra value, it wont affect the overall result because we're dealing with non-finite sets!

So I guess the summary of what I'm trying to tell you is that your initial problem of how to deal with the extra "0" is not actually a problem after all :). You can safely (mathematically) say that half of all numbers are odd.

Answer 8

I actually think that there are good reasons that show that half of all integers are odd. If you ask a lot of mathematicians this question, they'd give you some silly example showing you that, for example, only $\frac{1}{1000}$ of all numbers are odd. They'd do this by constructing a bijection like the following mapping every odd number to a distinct sequence of $999$ even numbers:

$$ \phi(1)=(2,4,6,8,\dots,1998) \\ \phi(3)=(2000,2002,\dots,3998) \\ \dots $$

However, this approach only uses the combinatorial structure of the integers, and ignores the algebraic structure. If we require $\phi$ to be not just a bijection from the odd numbers to some sequence of even numbers, but instead to be a group homomorphism whose kernel is the even numbers, we have only one choice:

$$ \phi(n)=n\mod 2 $$

For finite groups, we know that $|G|/|\ker\phi| = |\phi(G)|$. If we pretend that this applies to infinite groups as well, then that means that there are twice as many integers as there are even numbers, since the image of our homomorphism is the two-element group $\mathbb{Z}_2$. Therefore, half of all integers are odd.

Another approach we could take is to use the orbit-stabilizer theorem, which states that for a finite group $G$, a set $X$ and an action $\psi:G\times X\to X$, we have $|G|=|\mathrm{orb}_\psi(x)||\mathrm{stab}_\psi(x)|$ for any $x\in X$, where $\mathrm{orb}_\psi(x)$ is the orbit of $x$ under $\psi$ and $\mathrm{stab}_\psi(x)$ is the stabilizer of $x$ under $\psi$. If we pretend this applies to infinite groups, we can take $G=\mathbb{Z}$, $X=\{\{\textrm{odd numbers}\}, \{\textrm{even numbers}\}\}$, and let $G$ act on $X$ by addition. Then the orbit of the set of odd numbers is the whole of $X$, which has size $2$, and the stabilizer of that set is precisely the set of even numbers. So we get

$$ |\mathbb{Z}| = 2\left|\{\textrm{even numbers}\}\right| $$

and therefore, half of all integers are odd.

Another, more combinatorial approach, is as follows. We might not want to take full advantage of the algebraic structure of $\mathbb{Z}$, but the example at the top showing that the odd numbers occupy only $\frac{1}{1000}$ of the integers seems silly for a nother reason: the images of the odd numbers $\phi(n)$ are too far away from the odd numbers themselves! One way of using this is to define the notion of density, which has been done above. Consider instead the following approach. Let the integers be the vertices of a graph, and join two integers if they are adjacent. So $2$ and $3$ are joined by an edge, while $2$ and $4$ are not. (This makes sense, because adjacent numbers are in some sense close together. It would be silly to connect the number $3$ to the number $2000$.)

Then this graph is bipartite, and its vertex sets are precisely the set of odd numbers and the set of even numbers. Moreover, the graph is regular, as each vertex has precisely two neighbours. A simple counting argument shows that all finite regular bipartite graphs have vertex sets that are the same size. If we pretend that this applies to infinite regular bipartite graphs (notice a theme here?) then we see that the set of odd numbers and the set of even numbers are the same size, so half of all integers are odd.