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In simple English, what does it mean to be transcendental?

I've been meaning to write this up for a while, but somehow never got around to it. In my opinion, it's the best Math Stack Exchange post I've ever written. And also remarkable: its excellence was widely recognized. Often I work hard and write posts that I think are really good, and they get one or two upvotes; that's okay, because the work is its own reward. And sometimes I write posts that are nothing at all that get a lot of votes anyway, and that is okay because the Math SE gods are fickle. But this one was great and it got what it deserved.

I am really proud of it, and in this post I am going to boast as shamelessly as I can.

The question was:

In simple English, what does it mean to be transcendental?

There were several answers posted immediately that essentially recited the definition, some better than others. At the time I arrived, the most successful of these was by Akiva Weinberger, which already had around fifty upvotes.

… Numbers like this, that satisfy polynomial equations, are called algebraic numbers. … A real (or complex) number that's not algebraic is called transcendental.

If you're going to essentially quote the definition, I don't think you can do better than to explain it the way Akiva Weinberger did. It was a good answer!

Once one answer gets several upvotes, it moves to the top of the list, right under the question itself. People see it first, and they give it more votes. A new answer has zero votes, and is near the bottom of the page, so people tend it ignore it. It's really hard for new answers to surpass a highly-upvoted previous answer. And while fifty upvotes on some stack exchanges is not a large number, on Math SE fifty is a lot; less than 0.2% of answers score so high.

I was unhappy with the several quoting-the-definition answers. Because honestly "numbers… that satisfy polynomial equations" is not “simple English” or “layman's terms” as the OP requested. Okay, transcendental numbers have something to do with polynomial equations, but why do we care about polynomial equations? It's just explaining one obscure mathematical abstraction in terms of second one.

I tried to think a little deeper. Why do we care about polynomials? And I decided: it's because the integer polynomials are the free ring over the integers. That's not simple English either, but the idea is simple and I thought I could explain it simply. Here's what I wrote:

We will play a game. Suppose you have some number !!x!!. You start with !!x!! and then you can add, subtract, multiply, or divide by any integer, except zero. You can also multiply by !!x!!. You can do these things as many times as you want. If the total becomes zero, you win.

For example, suppose !!x!! is !!\frac23!!. Multiply by !!3!!, then subtract !!2!!. The result is zero. You win!

Suppose !!x!! is !!\sqrt[3] 7!!. Multiply by !!x!!, then by !!x!! again, then subtract !!7!!. You win!

Suppose !!x!! is !!\sqrt2 +\sqrt3!!. Here it's not easy to see how to win. But it turns out that if you multiply by !!x!!, subtract 10, multiply by !!x!! twice, and add !!1!!, then you win. (This is not supposed to be obvious; you can try it with your calculator.)

But if you start with !!x=\pi!!, you cannot win. There is no way to get from !!\pi!! to !!0!! if you add, subtract, multiply, or divide by integers, or multiply by !!\pi!!, no matter how many steps you take. (This is also not supposed to be obvious. It is a very tricky thing!)

Numbers like !!\sqrt 2+ \sqrt 3!! from which you can win are called algebraic. Numbers like !!\pi!! with which you can't win are called transcendental.

Why is this interesting? Each algebraic number is related arithmetically to the integers, and the winning moves in the game show you how so. The path to zero might be long and complicated, but each step is simple and there is a path. But transcendental numbers are fundamentally different: they are not arithmetically related to the integers via simple steps.

This answer was an immediate hit. It rocketed past the previous top answer into the stratosphere. Of 190,000 Math SE, answers, there are twenty with scores over 500; mine is 13th.

The original version left off the final paragraph (“Why is this interesting?”). Fortunately, someone posted a comment pointing out the lack. They were absolutely right, and I hastened to fix it.

I love this answer for several reasons:

• It's not as short as possible, but it's short enough.

• It's almost completely jargonless. It doesn't use the word “coefficient”. You don't have to know what a polynomial is. You only have to understand grade-school arithmetic. You don't even need to know what a square root is; you can still try the example if you have a calculator with a square root button.

• Sometimes to translate a technical concept into plain language, one must sacrifice perfect accuracy, or omit important details. This explanation is technically flawless.

• One often sees explanations of “irrational number” that refer to the fact such a number has a nonrepeating decimal expansion. While this is true, it's a not what irrationality is really about, but a secondary property. The true core of the matter is that an irrational number is not the ratio of any two integers.

  My post didn't use the word “polynomial” and took a somewhat different path than the typical explanation, but it nevertheless hit directly at the core of the topic, not at a side issue. The “path to zero” thing isn't some property that algebraic numbers happen to have, it's the crucial issue, only phrased a little differently.

• Also I had some unusually satisfying exchanges with critical commenters. There are a few I want to call out for triumphant mockery, but I have a policy of not mocking private persons on this blog, and this is just the kind of situation I intended to apply it to.

This is some good work. When I stand in judgment and God asks me if I did my work as well as I could, this is going to be one of the things I bring up.

[ Addendum 20211230: More about one of the finer points of this answer's pedagogical approach. ]

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