Are there more even or odd numbers?

Which do you think there are more of: even numbers or odd numbers? Or are there the same number of each? How could we decide? And what about other kinds of numbers? Are there more even numbers or positive integers? More positive integers or rational numbers? It isn’t entirely simple, because there are infinitely many numbers of every kind mentioned. Even so, we are able to compare them and find out which kind has more and which has fewer.

Let’s try to show this with a small set of numbers. Imagine two sets of numbers: blue numbers and red numbers:

Can we say which there are more of? Red or blue? We can clearly see that… What? You’re colour-blind and can’t see it clearly? All right, from now on we’ll also show the red numbers as squares:

Good, now we can clearly see that there are more blue numbers than red ones. That was easy, because we are working with finite sets of numbers, so we are able to count them. We can see that we have six blue numbers and only three red ones. Infinite sets won’t be so easy, because we can’t count them. So let’s answer the question in the title: are there more even or odd numbers? To keep things simple, let’s stick to positive numbers. Imagine a bag full of all the odd numbers and a bag full of all the even numbers. Which bag is bigger?

Well, imagining infinitely many numbers is not exactly easy, and counting them is even harder. Is there some other way we could compare the sizes of sets?

Comparing two finite sets

Imagine we have a bag full of grooms and a bag full of brides. If we have one bride for every groom and one groom for every bride, we can say that we have the same number of grooms and brides:

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See? Every bride in the first row has her own husband in the second row, and vice versa – every husband has his own wife. But another case could also occur: every bride would have her husband, but not every husband would have a bride:

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Every wife has a husband, but one husband is left sad and alone without a wife. Since one husband is left over, we can say that we have more husbands than wives. Similarly, if one bride is left over

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then we can say that we have more brides than husbands.

Comparing two infinite sets

Can we use this method to compare two infinite sets? Can we pair every odd number with some even number and every even number with some odd number, without using any number more than once? We can! We can pair every odd number n with the even number n + 1. So we pair the number 5 with the number 5 + 1, the number 149 with the number 150, and so on. The following animation shows the pairing:

For every red odd number we have a corresponding unique blue even number, and for every blue even number we have a corresponding unique red odd number. We managed to pair all the odd numbers with all the even numbers. Which means that there are the same number of even and odd numbers!

All right, that wasn’t exactly surprising, was it? Let’s try comparing other sets of numbers: all the positive integers versus the even numbers. Here the result will be more shocking.

Are there more positive integers or even numbers?

Which are there more of? Positive integers or even numbers?

We can clearly see that the positive integers contain all the even numbers. On top of that, the positive integers contain some extra numbers – the odd numbers. So it’s clear that there must be more positive integers – they contain twice as many numbers!

Oh, if only life were that simple!

We don’t care whether one set of numbers contains some extra numbers. What we care about is whether we can somehow pair up the numbers from two different sets. Are we able to pair every even number with a positive integer? Even though there are more positive integers? Well, yes, we can.

We can simply pair every positive integer p with the even number 2 · p, and vice versa – every even number n can be paired with the positive integer n / 2. So we pair the positive integer 5 with the even number 10, and the even number 14 with the positive integer 7, and so on. The following animation shows the pairing for the first few numbers:

Suddenly we have a unique even number for every positive integer and a unique positive integer for every even number! Never mind that there are more positive integers. There are infinitely many of both, and we will never run out. From our point of view, then, the set of even numbers is just as big as the set of positive integers! 😱

It’s a little counter-intuitive, but that’s how it is.

Are there more positive integers or rational numbers?

Rational numbers are fractions. All the numbers that can be written in the form p/q, where p and q are integers and q ≠ 0, are rational numbers. For example ½, ¾ or 8 are rational numbers, but π is not. There are far more rational numbers than positive integers. For example, between the numbers 2 and 4 there is exactly one integer, the number 3, but infinitely many rational numbers. There are even infinitely many rational numbers between 0.001 and 0.002. That is how many rational numbers there are!

So you would expect that the set of rational numbers must be bigger than the set of positive integers. But at the same time we all sort of suspect there’s a catch. All right, let’s try to pair the rational numbers with the positive integers. We start by writing the rational numbers in a table like this:

The pattern should be clear: all the fractions in the first row have 1 as the numerator (above the fraction line) and the steadily increasing numbers 1, 2, 3 and so on as the denominator. In the columns it’s the other way round. If we carried on to infinity, this table would contain all the fractions there are. For example, the fraction 1478/517 is in row 1478 and column 517. Now let’s try to build our pairing function. We could build it like this:

• We pair the positive integer 1 with the first fraction in the first row, i.e. 1/1
• We pair the positive integer 2 with the second fraction in the first row, i.e. 1/2
• We pair the positive integer 3 with the third fraction in the first row, i.e. 1/3
• We pair the positive integer 4 with the fourth fraction in the first row, i.e. 1/4
• …

Did we manage to pair up all the positive integers this way? Yes, we did. Every positive integer has its counterpart. The positive integer 174 has the fraction 1/174 as its counterpart. But does it work the other way round too? Does every fraction have a counterpart among the positive integers? What counterpart does the fraction 2/3 have? None, because with our pairing function we never got to the second row. We only managed to pair up the first row of fractions.

Does this mean that there are more rational numbers than positive integers? Unfortunately not – the fact that one pairing strategy doesn’t work doesn’t mean another won’t. This time let’s go along the diagonals:

• We pair the positive integer 1 with the fraction 1/1 (the first diagonal),
• the positive integer 2 with the fraction 1/2 (the second diagonal),
• the positive integer 3 with the fraction 2/1 (the second diagonal),
• the positive integer 4 with the fraction 1/3 (the third diagonal),
• the positive integer 5 with the fraction 2/2 (the third diagonal),
• the positive integer 6 with the fraction 3/1 (the third diagonal),
• and so on.

You can watch the pairing strategy in the animation:

This way we can cover all the fractions. No diagonal is infinite, so we never get stuck in any diagonal “forever”. In the end every fraction will have some positive integer assigned to it, and of course every positive integer will have a unique fraction assigned to it.

Well… actually, not quite. If we look at our table of fractions again, we find that many numbers appear more than once. For example, the fractions 1/1, 2/2, 3/3 and so on represent the same number – the number 1. Likewise, the fractions 1/2, 2/4 and 3/6 represent one half. In fact, the positive integers 1, 5, 13 and so on are mapped to the same rational number 1! We have several grooms marrying the same bride. That won’t do!

Fortunately, we can fix this easily. We simply cross out of our table all the fractions that are not in their simplest form, i.e. all the fractions that can be cancelled down. After crossing them out, only unique rational numbers remain in the table, and our trick with the diagonals will work. Every positive integer will be paired with a unique rational number and vice versa.

Which means that the set of positive integers is just as big as the set of rational numbers!

So far it looks as if all infinite sets are the same size. Is there an infinite set that is bigger than the set of positive integers?

Are there more positive integers or real numbers?

Here comes our last resort: the real numbers. We have no other numbers left on the number line. What do the real numbers look like? The real numbers contain all the rational numbers and, on top of that, all the irrational numbers. If we write a number as a decimal, then a rational number:

• either has a terminating decimal expansion, e.g. 0.4841554,
• or has an infinite recurring expansion: 0.1313131313…

Irrational numbers, on the other hand, always have an infinite decimal expansion that is also not recurring. An example of an irrational number is π, which is equal to 3.141592653… (no recurring part). Let’s say right away that there are far more real numbers than rational numbers, and so more than positive integers. The set of real numbers is bigger than the set of positive integers. And how do we prove it?

Imagine we have found some pairing strategy that pairs all the positive integers with all the real numbers. Let’s say it’s this one:

1 → 0.3452807674449021…
2 → 0.8978319745321683…
3 → 0.8972169570849831…
4 → 0.0635759905832629…
5 → 0.0081863365749335…
6 → 0.4000559394278442…
7 → 0.8637537627594889…
…

Suppose we have all the positive integers on the left and all the real numbers on the right. If that were true, we would not be able to find any real number missing from the right-hand side. Since we have already given away that the set of real numbers is bigger, this means that some number is missing on the right. Yes, but which one? Well, for instance we can see that the number 0.4818063607183678 is missing. But what if it’s hiding under the number 8?

1 → 0.3452807674449021…
2 → 0.8978319745321683…
3 → 0.8972169570849831…
4 → 0.0635759905832629…
5 → 0.0081863365749335…
6 → 0.4000559394278442…
7 → 0.8637537627594889…
8 → 0.4818063607183678
…

And what about the number 0.4208424736008147? We don’t see that one either. But what if it’s hiding under the number 9?

1 → 0.3452807674449021…
2 → 0.8978319745321683…
3 → 0.8972169570849831…
4 → 0.0635759905832629…
5 → 0.0081863365749335…
6 → 0.4000559394278442…
7 → 0.8637537627594889…
8 → 0.4818063607183678
9 → 0.4208424736008147
…

This won’t work. Since we obviously can’t write out all the pairs, as there are infinitely many, we can’t just guess whether this number happens to be there or not. We would need to find a number that differs from all the numbers on the right of the table, even though we don’t know what all the numbers on the right are. How do we do that?

For two numbers to be different, it is enough for them to differ in one digit. For example, the numbers 0.123456 and 0.123457 are not the same – but they differ in only one digit. That is enough for us. So in our table we can always highlight the nth digit after the decimal point of the nth real number:

1 → 0.3452807674449021…
2 → 0.8978319745321683…
3 → 0.8972169570849831…
4 → 0.0635759905832629…
5 → 0.0081863365749335…
6 → 0.4000559394278442…
7 → 0.8637537627594889…
8 → 0.4818063607183678
9 → 0.4208424736008147
…

and build a new number so that its nth digit after the decimal point differs from the nth digit of the nth real number. So we construct the new number by starting with a zero: 0.??? and choosing the first digit after the decimal point so that it differs from the first digit of the first real number, that is, so that it differs from 3. Let’s write, say, 0.1???. The second digit must be different from the second digit of the second real number, that is, it must be different from 9. Let’s write a 2: 0.12???. And so on. The first nine digits could therefore look like this: 0.123123127 (see the last row of the table):

1 → 0.3452807674449021…
2 → 0.8978319745321683…
3 → 0.8972169570849831…
4 → 0.0635759905832629…
5 → 0.0081863365749335…
6 → 0.4000559394278442…
7 → 0.8637537627594889…
8 → 0.4818063607183678
9 → 0.4208424736008147
? → 0.123123127

We can see that this number is different from all the numbers, because it always differs from each of them in at least one digit. Even though our table contains an infinite number of numbers, our new number is made of infinitely many digits in such a way that it differs from every number in the table in at least one digit. And notice that we can construct infinitely many numbers that are not in the table! What follows from this? We are not able to pair the positive integers with the real numbers. We will always be left with infinitely many real numbers that are not paired with any positive integers. There are far more real numbers than positive integers.

Final notes

• For the sake of simplicity, we ignored negative numbers throughout the article. Of course, they make no difference, and for homework you can adapt the pairing strategies mentioned here so that they work with negative numbers too.
• The size of the set of positive integers even has its own name. We call it aleph-null.
• In maths we then often use the term cardinality instead of “size of a set”.