Do transcendental numbers outnumber real numbers?

Question

I am not a mathematics student, but just out of curiosity I was checking out a website which explains the basics of 'Chaos Theory' to the layman. In this site was the sentence :

transcendental numbers outnumber real numbers with an infinite factor.

So I checked wikipedia for the definition of transcendental numbers. I think I understand what transcendental numbers are now but I still don't comprehend this statement. Could someone please explain it in layman terms.

The website is: http://www.abarim-publications.com/SelfSimilarity.html#.U4alxHIluXs.

P.S: I also have no idea what tags to put for this question. Feel free to add any as you deem appropriate.

Answer 1

Real numbers comprise rational numbers and irrational numbers (these don't overlap, i.e. they are disjoint subsets).

Irrational numbers comprise algebraic irrational numbers and transcendental numbers (again, disjoint subsets).

Hence transcendental numbers are a proper subset of the real numbers. They cannot "outnumber" the real numbers by any measure.

However, the cardinality of the set of transcendental numbers is equal to the cardinality of the set of real numbers (known as the cardinality of the continuum).

You can also say that the "vast majority" of real numbers are transcendental, but this is an imprecise statement.

Answer 2

Part I

What are real numbers? For the mathematician they are the unique Dedekind-complete ordered field; or the metric completion of the rational numbers (which themselves are an extension of the integers, which are an extensions of the natural numbers; which themselves are an extension of $1$ and $0$).

For the layman? Well, it's hard to explain to the layman without lying to them or completely leaving them in the dark. Feynman was once asked to describe the electromagnetic force, but he refused to do so because any analogy can be eventually reduced to the electromagnetic force, so the analogy would be circular.

One of the common ways to describe the real numbers to the layman would be to tell them that these are the numbers which can be written with finitely many terms on the left of the decimal points, and infinitely many terms on to its right (possibly most of them are $0$).

The rational numbers are easier to explain, and we don't need much for that. These are numbers which are the ratio of two integers. Simple, isn't it?

Real numbers, it turns out, can be divided into two parts, one part which is the rational numbers and the other part which is the irrational numbers, meaning the numbers which are not rational. And while it is true that every irrational number has an infinite decimal expansion, it is not true that every number with an infinite decimal expansion is irrational. $0.9999\ldots$ is really just a fancy way of writing $1$, and $\frac13$ cannot be written in any form other than $0.333\ldots$ in decimal form.

But we can consider another division of the real numbers into two parts. We can look at numbers which are "simple" enough, for example $\sqrt{2}$ or $\frac13(\sqrt[5]{42}+\sqrt{12})$. These numbers are called algebraic numbers, and these are exactly the roots of polynomials whose coefficients are integers.

Numbers which are not algebraic numbers are called transcendental, and these two can be divided into many families. But I'm not going to get into that now.

Part II

If we want to talk about "outnumber" we need to have a solid notion for what it would mean that one collection outnumber another collection. Clearly, $\{1,2,3\}$ outnumbers the collection $\{4,5\}$. One has three elements, and the other has only two.

Formally, we call this notion cardinality. It is one of many ways to measure the size of a collection. How did we say that $\{4,5\}$ has a smaller cardinality than $\{1,2,3\}$? We tried to match up the elements of the two sets, and saw that no matter what we did we could never get all the elements of $\{1,2,3\}$. We can match $4$ with $1$ and $5$ with $2$, but then $3$ doesn't have a partner; or we can match $4$ with $3$ and $5$ with $1$, but then $2$ doesn't have a partner.

We can't match both $2$ and $3$ with $4$, because the matching has to give a unique value for each point.

Mathematically, we say that there is an injection from the set $\{4,5\}$ into the set $\{1,2,3\}$; but there is no surjection from $\{4,5\}$ to $\{1,2,3\}$. An injection means that we match $4$ and $5$ into distinct elements, and a surjecion would be the case that we managed to cover the entire set we tried to. For example, we could match from the other direction, $1$ and $2$ are matched with $4$ and $3$ is matched with $5$.

Note that this matching still gave a unique value for each element in $\{1,2,3\}$, but it gave two elements the same match. That's fine. And you can work the options and see that any such matching between these two sets must have similar properties.

We say that one set $A$ has the same cardinality (or size) as the set $B$ if there is a matching which is both injective and surjective; if there is only an injective matching, but there are no surjective matchings, then $B$ is strictly larger than $A$.

It follows that a set cannot have a strictly smaller cardinality than any of its subsets. So in particular it is impossible that there are more transcendental numbers than real numbers, since every transcendental number is a real number.

One fact which may surprise you, is that the rational numbers, the fractions, have the same cardinality as the set of integers, and as the set of natural numbers. But the real numbers have a strictly larger cardinality than these sets.

Since we mentioned the algebraic numbers, these also have the same size as the integers. And it follows from all these facts that the transcendental numbers are strictly larger. In fact the cardinality of the real numbers (transcendental and algebraic together) is the same as the cardinality of the transcendental numbers alone.

Part III

Let's see what the site you linked to has to offer

And besides that, the decimals of the number pi go on forever. That means that if we try to express the relationship between diameter and circumference of a circle in numbers we need infinite detail to stay truthful. But infinite detail does not exist, and so we must yield to the rather shocking conclusion that the before mentioned relationship can not be expressed in numbers, and that pi is not a number at all!

Well, since there is no formal definition of "number" to begin with, I can't fully argue with that. But certainly if we want that the real numbers be a model of the notion of length, being able to say that the length of the circumference of a circle of radius $\frac12$ is a real number would be expected. And what would this length be? That's right, $\pi$.

But it is true that the ratio cannot be expressed as the ratio of two integers, but that's fine. The ratio between the side of a square and its diagonal cannot be expressed as the ratio of two integers either.

The same goes for that other famous 'number' e, and all so-called transcendental numbers (numbers that go on forever after the dot; in other words, numbers that represent infinite detail). And to make matters worse, transcendental numbers out-number real numbers with an infinite factor.

As I wrote above, this is not a definition of what are transcendental numbers. But if we do consider, for a moment, that the author was really just talking about rational numbers (which would match some of the other mistakes), and in that case it is true. There are way more transcendental numbers than rational numbers.

But what does it mean "with an infinite factor"? I have no idea. Because you can try and prove that between two integers there are infinitely many rational numbers. And yet, the two sets have the same cardinality. The same size.

Epilogue

And finally,

It may seem a bit paradoxical but since mathematics can not release its detailed accuracy, it loses connection with the real world around quantum level.

What??