What makes a number transcendental?

Question

I read wikipedia about transcendental numbers and I honestly didn’t understand most of what I read, nor why it should be important that e and pi (or any numbers) are transcendental.

In: Mathematics

Answer 1

Most numbers are transcendental – it’s not a special property. A better question is: What *prevents* a number from being transcendental?

A number is *not* transcendental if it can be totally described using a polynomial made from nothing but integers. So, for instance, the Golden Ratio is NOT transcendental because it solves the equation x^(2)-x-1=0 which is nothing but some simple combinations using the golden ratio which all eventually cancel out. That is, the golden ratio is “not far” from the integers, even if it is irrational. Pi *is* transcendental. No matter how long you take or what combinations you use, you can never simply relate pi to the integers in this way.

We expect *most* numbers to be transcendental, so if we think a number is *not* transcendental then we usually have a reason for it. An example that is kinda surprising is the Look-and-Say Constant. The Look-and-Say Sequence is the sequence of numbers starting at 1 where the next number is what you get by reading off the last entry. The first entry has one 1, so the second entry is 11. This entry has two 1s, and so the third entry is 21. This entry has one 2 and one 1, so the fourth entry is 1211. It then goes on like that, 111221, 312211, 13112221, etc. This seems like a totally arbitrary sequence, dependent on human language and quirks, so we shouldn’t really expect it to have much mathematical interest.

However, if you look at the ratio of consecutive values, like 11/1 then 21/11 then 1211/21, then 111221/1211 etc, then as this ratio goes on forever it becomes a not-transcendental number! In fact, it solves a degree 71 polynomial that mathematician John Conway figured out ([see here for the polynomial](https://wikimedia.org/api/rest_v1/media/math/render/svg/430b78be283355e717f8463ac68258d193d55bd9)). It was a bit of a surprise, not only that it wasn’t transcendental but additionally that we could actually write down the polynomial it solves! What this means is that there is actually some meaningful mathematical – specifically algebraic – structure to this sequence that we neglected to think about before.