Introduction

In this episode we will give an overview about an interesting and enigmatic topic in mathematics. We are referring to transcendental numbers. Let’s start by just talking about this name “transcendental numbers”.  If you look for the meaning of the word “transcendental” in the dictionary,y you might find things like this:

1. “supernatural, beyond human experience”
2. “describes anything that has to do with the spiritual, religious or non-physical world”
3. “an experience that transcends – goes beyond – the regular physical realm”

But why the mathematicians, a lot already notable for not having their feet on the ground, were compelled to use such an extraordinary adjective for these numbers? Maybe the properties of these numbers were so unique that made them appear sort of “supernatural” even to first class mathematicians. But why? What are these transcendental numbers? How many of them exist? These are the questions that we will begin to tackle in this episode.

So, what are transcendental numbers? I think I good way to explain this is by defining what is not a transcendental number is the first place:

“A non-transcendental number also called an algebraic number is simply a number that is a solution of an algebraic equation.”

That’s it! For instance, if you have the algebraic equation below:

Well, it takes just a couple of steps to find the solution for this equation. First we pass the 6 to the other side, and then we pass the 2 dividing the 6, which give us that the solution is x equal to 3.  Therefore the solution 3 is an algebraic number or a non-transcendental number.

Let’s pause for a moment to see this also from another perspective. What is it that we just did? Well, essentially we applied several operation on the number x = 3 to reduce it to zero. First we multiply 3 by 2, the result is 6, and then we subtract 6 which gives us zero. Of course we could always avoid all of this and multiply 3 by zero, but this is what is called a trivial solution and we are not interested in those. We are interested in non-trivial solutions that can be express as algebraic equations.

This lead us to play a slightly different game and the question is this one now. Can we take a number and reduce it to zero by applying a series of algebraic operations like addition, subtraction, multiplication, division and potentiation (raising a number to some power).

We obviously can do this with natural numbers, but can we do this to other classes of numbers too? Long story short: yes. We can find algebraic or non-transcendental numbers also in the rational, irrational and complex numbers for instance. But, are there exceptions to this? That is, can we find examples of numbers that we cannot reduce to zero applying algebraic operations? As it happens yes, we have examples, maybe the most famous one is pi which is an irrational number, but is also not algebraic. There is no algebraic equation that has pi as solution. No doub pi is a very memorable example of a transcendental number but it was not the first one to be discovered.

The Liouville constant

The Liouville constant was one of the first numbers that was proven to be transcendental. The number was proposed in 1844 by French mathematician Joseph Liouville. Liouville was prolific in several areas of maths and physics but he had also great organizational skills as he founded the Journal de Mathématiques Pures et Appliquées that is still a prestigious publication today. Liouville also read and recognize the importance of the last manuscripts from Evariste Galois and published these manuscripts posthumously in his journal, but lets go back to the topic responsible for the presence of Liouville in this topic, the Liouville constant, a number proposed Liouville. This number can be constructed by assigning a “1” to each decimal place corresponding to a factorial and zeros everywhere for instance:

Now the Liouville constant is not a unique number. In fact, the Liouville constant is just the most famous example of the so called Liouville numbers. There is procedure to generate those numbers with some formulas that when given specific parameters produce the Liouville constant but the procedure can produce an infinite number of Liouville numbers. All of them are irrational numbers and all of them are also transcendental. That is, as we already explain, that no Liouville number is a solution of any algebraic equation.

There is more to go on Liouville numbers but in this episode we just wanted to mention them because of their historical importance as the first numbers to be proven transcendental.

Other famous transcendental numbers

But after the Liouville constant and more generally the Liouville numbers there were other mathematicians that found other examples. Most notably Charles Hermite proved in 1873 that the Euler number “e” was a transcendental number. Then Ferdinand von Lindemann proved that Pi was also a transcendental number in 1882. After that, as result and special cases of a couple of theorems there was also proved that “e to the power of Pi” is transcendental. Interestingly enough we have not been able to prove that “pi to the power of e” is also a transcendental number. Finally we also have a number like the Gelfond-Sneider constant “2 to the power of the square root of 2” is also a transcendental number.

On the difficulty to find transcendental numbers

Ultimately Georg Cantor amazed the mathematical community by showing that almost all real numbers are transcendental. That is a topic and proof that we might like to try in a separate episode but for now I just want to call attention for something interesting. Besides the effort done by Liouville, who discover a whole group of transcendental numbers, most of the efforts of other mathematicians have manage to find just a handful of other transcendental numbers in spite of the result from Cantor pointing to the abundance of these numbers.

Then why haven’t we be able to find more transcendental numbers? Why it seems so difficult to prove that a number is transcendental? To illustrate this, let’s imagine that you can have a list of all the possible algebraic equations. Then now imagine that you have a number “n” and you want to determine if this number is algebraic or transcendental. In a naïve way, what you can do is to check is “n” is a solution of the first equation, then check if “n” is a solution of the second equation, etc. However if you find an equation, let’s say equation 10, that has “n” as solution, then your search stops and you immediately know that “n” is algebraic.

On the other hand, imagine that you have check 1 billion equations and still you haven’t found an equation that has “n” as solution. At this point you still don’t know if “n” is algebraic or transcendental, but you know that to prove that “n” is transcendental you will have to test all the infinite possible algebraic equations and find that n is not a solution of anyone. As you see the burden of proof is much higher. From this perspective it kind of makes sense that proving that a number “n” is transcendental is more difficult and therefore it also makes sense that we have discovered a relatively smaller quantity of these very special numbers.

And with this last intuitions we arrive to the end of this episode on transcendental numbers. I hope that you now understand why these numbers have puzzled mathematicians since the time they were discovered and also that you had a glimpse of the complexities you could find working in this area of number theory.