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The description omits the part why this seemingly convoluted game is needed in the first place.
When we, for example, learn about fractional numbers, we immediately visualize them as something between integers. We talk about the need to have something more than 2 but less than 3.
However here we look at a seemingly arbitrary game which somehow - after a convoluted process (intentional emphasis everywhere) divides numbers into having a property and not having it, without explanation why the property is there or not, or why it's important, and there is no "feeling" of the property.
I don't think this explanation is complete, even though it has good parts.
> _"The description omits the part why this seemingly convoluted game is needed in the first place."_
The game isn't needed regardless. We don't _need_ this classification. The original poster was asking how to explain what a transcendental is, and the answer does exactly that.
As a different question, asking _why_ we care about transcendentals is reasonable, but for you to complain that this post is not answering a question that wasn't asked seems a little unfair.
But even so, the submitted question says this:
_"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."_
So it looks like the post really does answer the question you raise, and I don't know why you're saying the answer is incomplete.
Not a mathematician so someone please correct me, but isn't `Pi` derived from `circumference/diameter`? Is this itself not a polynomial?
If you had a case of the circumference and the diameter both being integers, then pi would be a rational number.
There is no polynomial f(x) with integer coefficients such that f(pi) = 0.
In the language of the post, if you have pi, and you're allowed to add, subtract, multiply, and divide by integers, and you're allowed to multiply by pi, you can never and up with 0.
So pi can be expressed as the ratio of the diameter to the circumference, but when you do so, at least one of the diameter and circumference will themselves be transcendental.
"You can't write it down in terms of basic arithmetic"
I don't think that captures the full detail of exactly what a transcendental is. When you say "basic arithmetic" do you include square roots? Or cube roots? And if you do (which you must) then you are already outside of "laymen's terms".
> I don't think that captures the full detail of exactly what a transcendental is
I completely agree but I think it captures _the spirit_ of what a transcendental is succinctly.
That's true, but it's not answering the original question as asked. The original question says:
> _So in layman's terms, what exactly does it mean to be transcendental? How would a transcendental number be different from an ordinary number, say 5._
Saying that you "can't write it down in terms of basic arithmetic" doesn't make clear what you mean by that, and I feel it would be unsatisfying for the original poster. To say you CWIDITOBA is not saying _exactly_ what it means, and I can imagine the original poster coming back and asking ... "But what does that mean? What is basic arithmetic?"
The answer submitted here explains exactly that.
The answer covers the original question exactly.
The original question conveniently gives an example of an "ordinary number" as 5. Which indicates what they mean by ordinary numbers is indeed integers, and the answer completely explains how transcedental numbers differ from "ordinary numbers".