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I also love his explanation of "How Regexes Work":
https://perl.plover.com/Regex/article.html
Before reading it, I could write regexes with decent success, but I was never 100% comfortable that they'd do quite what I expected. After, the fog cleared, and I felt like I finally understood what I was asking the computer to do when I wrote a regex.
Also, whenever I think of state machines, in my mind I still see pennies moving around on circles.
(FYI, it was written for a Perl programmer audience, but it doesn't involve much Perl.)
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.
Part of the confusion may come from poor(?) naming. An irrational number is clearly understood as one that's not representable as a ratio of integers. Non-algebraic works for me, but is a bit of a mouthful.
This is much the same as we call non-composite numbers prime, i.e. numbers missing the property of being composable as a product. Explaining what a prime number is harder than explaining what a composite number is, because it's the negative space. Similarly a transcendental number is the negative space when you remove rational and algebraic numbers from the set of real (or complex) numbers. I don't know of a way to describe such a negative space in a direct/'positive' way.
composite numbers are integers that are evenly divisible by more than two numbers (including 1 and itself)
How about this.
If we take a number X, and there are no series of multiplications and additions by numbers that can be found without knowing X beforehand that would end up with zero for X but not for every other number, then X is 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".
And you still haven't captured all the algabraic numbers. Once a polynomial reaches degree 5, then even having an operation for taking the nth root of a number does not nessasarily allow you to find a root of the polynomial.
> 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".
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.
All numbers are transcendental, with a negligible list of exceptions. The exceptions are just the numbers you could write down if you spent infinitely long at it.
I'm not sure I agree with that characterization of the exceptions. What exactly did you mean by spending infinitely long writing the number down?
If "infinitely long" means some unbounded, finite time then that gives a subset of the rationals, and the other rationals would be exceptions that we missed. If it means that there exists a program capable of spitting out successive digits then those are the computable numbers, but some of those (like pi) are transcendental. If it means that they're represented in at least one way by an infinite string of digits then our list of exceptions is now all the reals.
Infinitely long means, literally, _exactly the opposite_ of "finite time".
In infinite time you can write, exactly, countably many digits, countably many fractions, _and_ countably many roots. I.e., the number of exceptions is countably-infinite. But "countable infinity" is a more sophisticated concept than the problem statement called for.