โœ–

Are there more even or odd numbers?

What do you think, which numbers are more? Dry numbers or odd numbers? Or are there the same number? How could we judge? What about the other numbers? Are there more even numbers or natural numbers? Natural numbers or rational numbers? It's not quite that simple, because all the numbers mentioned are infinitely many. Yet we are able to compare and see which numbers are more and which are less.

Let's try to demonstrate this on a small set of numbers. Imagine two sets of numbers: blue numbers and red numbers:

1
2
3
4
5
6
7
8
9

Is there any way to tell which numbers are more? Red or blue? I can clearly see that... What? That you're colorblind and can't see clearly? All right, from now on, we'll show the extra red numbers as squares:

1
2
3
4
5
6
7
8
9

Okay, now we can clearly see that there are more blue numbers than red. It was easy because we're working with finite sets of numbers, so we're able to count them. We can see that we have six blue numbers and only three red numbers. The infinite sets will not be as simple because we can't count them. So let's answer the question from the title: are there more dry numbers or more odd numbers? For simplicity, let's stick with positive numbers. Imagine a bag full of all odd numbers and a bag full of all even numbers. Which bag is bigger?

1
3
5
7
9
11
13
15
...
2
4
6
8
10
12
14
16
...

Well, imagining an infinite number of numbers isn't exactly easy, and counting them is even harder. Isn't there some other way we could compare the sizes of the sets?

Comparing two finite sets

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

๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ

๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต

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

๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ โŒ

๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต

Every wife has a husband, but one husband stays sad on vinegar without a wife. At this point, with one husband staying, we can say we have more husbands than wives. Similarly, when one bride will be staying

๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ ๐Ÿ‘ฐ

๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต ๐Ÿคต โŒ

we can say we have more brides than husbands.

Comparing two endless amounts of

Can we use this procedure to compare two infinite sets? Can we pair every odd number with some even number and every even number with some odd number, with neither number being able to be used more than once? We can! We can pair every odd number n with an even number n + 1. So we pair the number 5 with 5 + 1, the number 149 with 150, etc. The following animation shows the pairing:

1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50

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

Well, that wasn't entirely surprising, was it? Let's try comparing other sets of numbers: all natural numbers versus even numbers. Here the result will be more shocking.

Are there more natural numbers or more even numbers?

Which numbers are more? Natural numbers or even numbers?

1
2
3
4
5
6
7
8
...
2
4
6
8
...

We can obviously see that natural numbers contain all even numbers at the same time. And in addition, the natural numbers contain some extra numbers - odd numbers. So it is clear that there must be more natural numbers - they contain twice as many numbers!

Yeah, if only life were that simple!

We don't care if one set of numbers contains some extra numbers. We are interested in whether we can somehow pair numbers from two different sets. Are we able to pair every even number with every natural number? Even though there are more natural numbers? But yes, we can.

We can easily pair every natural number p with an even number 2 ยท p and vice versa - we can pair every even number n with a natural number n / 2. So we pair the natural number 5 with the even number 10, and we pair the even number 14 with the natural number 7, and so on. The pairing for the first few numbers is shown in the following animation:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50

Suddenly we have a unique even number for every natural number, and we have a unique natural number for every even number! So what if there are more natural numbers. There are infinitely many of both numbers and we will never run out. So from our point of view, the set of even numbers is as big as the set of integers! ๐Ÿ˜ฑ

It's a bit counter-intuitive, but it's true.

Are there more natural numbers or rational numbers?

Rational numbers are fractions. All numbers that can be written in the form p/q where p and q are integers are rational numbers. For example, ยฝ, ยพ, or 8 are rational numbers, but ฯ€ is no longer a rational number. There are many more rational numbers than natural numbers. For example, between the numbers 2 and 4 there is a single integer, namely the number 3, but infinitely many rational numbers. There are infinitely many rational numbers even between 0.001 and 0.002. There are so many rational numbers!

So one would expect that the set of rational numbers must be larger than the set of natural numbers. But at the same time, we all sort of suspect there's a catch. Okay, let's try pairing the rational numbers with the natural numbers. Let's start by writing down the rational numbers in a table like this:

1โ„1
1โ„2
1โ„3
1โ„4
1โ„5
1โ„6
2โ„1
2โ„2
2โ„3
2โ„4
2โ„5
2โ„6
3โ„1
3โ„2
3โ„3
3โ„4
3โ„5
3โ„6
4โ„1
4โ„2
4โ„3
4โ„4
4โ„5
4โ„6
5โ„1
5โ„2
5โ„3
5โ„4
5โ„5
5โ„6
6โ„1
6โ„2
6โ„3
6โ„4
6โ„5
6โ„6

The pattern should be clear: all fractions in the first row have a one in the denominator (above the fraction line) and successively increasing numbers 1, 2, 3, etc. in the numerator. In the column it is the other way around. If we continued indefinitely, we would have all the existing fractions in this table. For example, the fraction 1478/517 is in row 1478 and column 517. Now let's try to create our matching function. We could create it like this:

  • We pair the natural number 1 with the first fraction in the first row, i.e. 1/1
  • We pair the natural number 2 with the second fraction in the first row, i.e. with 1/2
  • Pair natural number 3 with the third fraction in the first line, i.e. 1/3
  • Pair natural number 4 with the fourth fraction in the first row, i.e. 1/4
  • ...

Did we manage to pair all the natural numbers in this way? Well, yes. Every natural number has a counterpart. The natural number 174 has as its counterpart the fraction 1/174. But does the reverse also apply? Does every fraction have a counterpart in the natural numbers? What counterpart does the fraction 2/3 have ? None, because we never got to the second line with our matching function. We were only able to pair the first row of fractions.

Does that mean there are more rational numbers than natural numbers? Unfortunately no - just because one matching strategy doesn't work doesn't mean another won't. Let's try taking it diagonally this time:

1โ„1
1โ„2
1โ„3
1โ„4
1โ„5
1โ„6
2โ„1
2โ„2
2โ„3
2โ„4
2โ„5
2โ„6
3โ„1
3โ„2
3โ„3
3โ„4
3โ„5
3โ„6
4โ„1
4โ„2
4โ„3
4โ„4
4โ„5
4โ„6
5โ„1
5โ„2
5โ„3
5โ„4
5โ„5
5โ„6
6โ„1
6โ„2
6โ„3
6โ„4
6โ„5
6โ„6
  • Pair the natural number 1 with the fraction 1/1 (first diagonal),
  • pair natural number 2 with the fraction 1/2 (second diagonal),
  • pair natural number 3 with the fraction 2/1 (second diagonal),
  • pair natural number 4 with the fraction 1/3 (third diagonal),
  • pair natural number 5 with the fraction 2/2 (third diagonal),
  • pair natural number 6 with the fraction 3/1 (third diagonal),
  • etc.

You can see the pairing strategies in the animation:

1โ„1
1
1โ„2
2
1โ„3
4
1โ„4
7
1โ„5
11
1โ„6
16
2โ„1
3
2โ„2
5
2โ„3
8
2โ„4
12
2โ„5
17
2โ„6
3โ„1
6
3โ„2
9
3โ„3
13
3โ„4
18
3โ„5
3โ„6
4โ„1
10
4โ„2
14
4โ„3
19
4โ„4
4โ„5
4โ„6
5โ„1
15
5โ„2
20
5โ„3
5โ„4
5โ„5
5โ„6
6โ„1
21
6โ„2
6โ„3
6โ„4
6โ„5
6โ„6

In this way we can cover all fractions. No diagonal is infinite, so we don't hit "forever" on any diagonal. Eventually every fraction will have some natural number associated with it, and of course every natural number will have some unique fraction associated with it.

Well... not really. If we look at our fraction table again, we'll see that there are a lot of repeating numbers. For example, the fractions 1/1, 2/2, 3/3 etc. represent the same number - the number 1. Similarly, the fractions 1/2, 4/2 and 6/3 represent one half. So in effect we are mapping the natural numbers 1, 5, 13 etc. to the same rational number 1! We have several grooms marrying the same bride. That wouldn't work!

Fortunately, we can easily fix this. In short, we cross out all fractions from our table that are not in base form, i.e. we cross out all fractions that can be truncated. After this deletion, we will be left with only unique rational numbers in the table and our diagonal trick will work. Each natural number will be paired with a unique natural number and vice versa.

But that means that the set of natural numbers is as big as the set of rational numbers!

So far it looks like all infinite sets are the same size. Is there any infinite set that is larger than the set of natural numbers?

Are there more natural numbers or more real numbers?

Our last resort comes into play: the real numbers. We don't have any more numbers on the real axis. What do the real numbers look like? The real numbers contain all the rational numbers plus all the irrational numbers. If we write a number in decimal notation, then it is a rational number:

  • has either a finite decimal expansion, e.g. 0.4841554,
  • or it has an infinite periodic expansion: 0.1313131313...

An irrational number, on the other hand, always has infinite decimal expansion while not being periodic. An irrational number is, for example, the number ฯ€, which is equal to 3.141592653... (no period). Let's just say right up front that there are many more real numbers than rational numbers and therefore than natural numbers. The set of real numbers is larger than the set of natural numbers. And how do we prove this?

Imagine that we found some pairing strategy to pair all natural numbers and all real numbers. Let's say it's this one:

1 โ†’ 0.3452807674449021... 2 โ†’ 0.8978319745321683... 3 โ†’ 0.8972169570849831... 4 โ†’ 0.0635759905832629... 5 โ†’ 0.0081863365749335... 6 โ†’ 0.4000559394278442... 7 โ†’ 0.8637537627594889... ...

Suppose we have all natural numbers on the left and all real numbers on the right. If this were true, we would not be able to find any real number that is missing on the right. Since we've already spoilered that the set of real numbers is larger, that means we're missing a number on the right side. Yeah, but what number? So, for example, we see that there's a missing number 0.4818063607183678. But what if it's hiding under the number 8?

1 โ†’ 0.3452807674449021... 2 โ†’ 0.8978319745321683... 3 โ†’ 0.8972169570849831... 4 โ†’ 0.0635759905832629... 5 โ†’ 0.0081863365749335... 6 โ†’ 0.4000559394278442... 7 โ†’ 0.8637537627594889... 8 โ†’ 0.4818063607183678 ...

What about the number 0.4208424736008147? We don't see that there either. But what if it's hiding under number nine?

1 โ†’ 0.3452807674449021... 2 โ†’ 0.8978319745321683... 3 โ†’ 0.8972169570849831... 4 โ†’ 0.0635759905832629... 5 โ†’ 0.0081863365749335... 6 โ†’ 0.4000559394278442... 7 โ†’ 0.8637537627594889... 8 โ†’ 0.4818063607183678 9 โ†’ 0.4208424736008147 ...

It won't work that way. Of course, since we can't list all the pairs, since there are infinitely many, we can't just guess whether or not this number happens to be there. We would need to find a number that is different from all the numbers on the right in 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, they only need to be different in one digit. For example, the numbers 0.123456 and 0.123457 are not the same - but they only differ in one digit. That's enough. So we can always highlight n-th real number in our table with n-th digit after the decimal point:

1 โ†’ 0.3452807674449021... 2 โ†’ 0.8978319745321683... 3 โ†’ 0.8972169570849831... 4 โ†’ 0.0635759905832629... 5 โ†’ 0.0081863365749335... 6 โ†’ 0.4000559394278442... 7 โ†’ 0.8637537627594889... 8 โ†’ 0.48180636071836789 โ†’ 0.4208424736008147 ...

and compose the new number so that the n-th digit after the decimal point differs from the n-th digit of the n-th real number. So we construct the new number by starting with zero: 0,??? and choosing the first digit after the decimal point to be different from the first digit of the first real number, i.e. different from three. For example, we write 0,1????. The second digit must be different from the second digit of the second real number, so it must be different from 9. For example, write a two: 0.12????. And so on. So the first nine digits might look like 0.123123127 (see the last row of the table):

1 โ†’ 0.3452807674449021... 2 โ†’ 0.8978319745321683... 3 โ†’ 0.8972169570849831... 4 โ†’ 0.0635759905832629... 5 โ†’ 0.0081863365749335... 6 โ†’ 0.4000559394278442... 7 โ†’ 0.8637537627594889... 8 โ†’ 0.48180636071836789 โ†’ 0.4208424736008147? โ†’ 0.123123127

We can see that this number is different from all numbers because it always differs from all numbers in at least one digit. Although our table contains an infinite number of numbers, our new number is made up of infinitely many digits such that it differs from all the numbers in the table by at least one digit. And notice that we can construct infinitely many numbers that are not in the table! What does this imply? We are unable to pair natural numbers with real numbers. We are always left with infinitely many real numbers that will not be paired with any natural numbers. There are many more real numbers than natural numbers.

Notes at the end

  • We have ignored negative numbers throughout this article for simplicity. But of course, they don't change anything, and for homework you can modify the pairing strategies mentioned here to work with negative numbers.
  • The size of the set of natural numbers even has a name. We call it Aleph-zero.
  • In mathematics, then, we often use the term cardinality or cardinality of a set instead of set size.