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All The Numbers In The World

We know that the natural numbers 1,2,3,… go off to infinity. But what happens when we consider negative numbers -1,-2,-3,… as well? How many numbers do we get? And what about fractions? Do we have multiple infinities?

Welcome to another part of the “Maths For Normal People” series. You will find all the previous parts in this article.

So far, we have been talking only about <$\mathbb{N}$>, the set of natural numbers. As the name implies, these are the numbers that come to mind most naturally and so they were invented first. As mathematical thought developed, new numbers had to be invented.

To facilitate subtracting larger numbers from smaller ones, humans developed the notion of whole numbers, whose set we denote <$\mathbb{Z} = \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$>. The dots “in both ways” mean that there is an infinity of negative numbers and an infinity of positive numbers.

To facilitate division of numbers that are not commensurable (i.e. can’t be divided without a remainder), we devised the idea of rational numbers, or fractions. Their set is denoted <$\mathbb{Q}$> and can no longer be easily described by listing a few elements and adding dots. Instead, we say that

<$$ \mathbb{Q} = \{\text{all numbers }\frac{p}{q}\text{ such that }p\text{ is whole and }q\text{ is natural}\}. $$>

And finally, to make it possible to take square roots of all positive numbers, we invented real numbers, whose set is denoted <$\mathbb{R}$> and contains, roughly speaking, all numbers with all possible combinations of decimal digits. The genesis of number sets is a deep and interesting topic that I am going to cover in an article of its own. For now, let’s settle for this short introduction.

Now, the obvious question is: how many numbers are there in the respective sets <$\mathbb{Z},\mathbb{Q},\mathbb{R}$>? We have already established that there are <$\aleph_0$> numbers in <$\mathbb{N}$>, and that this <$\aleph_0,$> or “Aleph-0” is a name we give to infinity.

Negativity Brings Nothing New

Perhaps this won’t come off as a surprise: there are just as many whole numbers as there are natural numbers. If you take the set of natural numbers and give a minus sign to each of them, you get all the negative whole numbers. Merging this set with the original natural numbers and adding zero, you arrive at <$\mathbb{Z}$>. But concerning set cardinalities, all you’ve done is you multiplied <$\aleph_0$> by two and added one. But according to the rules we have derived, <$2\cdot\aleph_0 + 1 = \aleph_0$>. Therefore,

<$$ |\mathbb{N}| = |\mathbb{Z}|. $$>

If you don’t believe me, I’ve got a better proof. We can easily find a bijection between <$\mathbb{N}$> and <$\mathbb{Z}$>. One way to do it is to assign odd natural numbers to positive whole numbers and even natural numbers to negative whole numbers. (You might want to refresh your infinity counting skills in the “Infinity Plus Three” article I wrote back in March.)

Rationally More

The question of rational numbers is not all that obvious. Fractions are created by making all possible combinations of whole and natural numbers, and so there are many, many more fractions than whole numbers.

Right?

No.

There are no more fractions than natural numbers, and we can prove it by numbering all the fractions by natural numbers, which really is nothing else than creating a bijection between fractions and natural numbers. (Whenever you are counting things, you are establishing a bijection between those things and natural numbers.)

Let’s write all the positive fractions out in a table and number them diagonally in the direction of the arrows. Well, you’ll forgive me if I don’t write out all the positive fractions, right? :-)

Numbering of positive fractions.

We see that it is indeed possible to number all positive fractions in this manner.

Having established that there are as many positive fractions as there are natural numbers, we can infer that this is valid for all fractions. The negative ones can be obtained simply by putting a minus sign in front of the positive ones, so their number is obviously the same. And putting the positive and negative ones together creates a set with cardinality <$2\cdot\aleph_0$>, which obviously equals <$\aleph_0$>. So we have

<$$ |\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}|. $$>

What About The Reals?

We’ve discovered that that there are as many natural numbers as there are fractions and all whole numbers. You can be either surprised or not, depending on your idea of infinity…

It remains to be answered how many real numbers there are. The answer is not easy and has far-reaching consequences not only for mathematics, but also for philosophy.

Is <$|\mathbb{R}| = |\mathbb{N}|$> or not? And why? We’ll see next week!

Want to read more? You will find the outline of all parts of the series in this article.

September 22, MMXI — Mathematics, Maths For Normal People, Infinity.