In simple English, what does it mean for a number to be “transcendental”?
blog.plover.com
blog.plover.com
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.)
Some people, when confronted with a problem, think "I know, I'll use regular expressions." Now they have two problems.
Jamie ZawinskiOr in other words... https://owasp.org/www-community/attacks/Regular_expression_D...
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 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.
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.
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.
I've restricted myself to "multiplications and additions by numbers". The solution doesn't require knowing the value of X beforehand. It works for all real numbers - algebraic or transcendental.
Wanna try again?
I completely agree but I think it captures the spirit of what a transcendental is succinctly.
> 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 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".
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.
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.
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.
You might choose to write some random set other than the rationals and roots, but what you could write (absent infinite repetition) would have exactly as many members as those sets, and thus map to them one-to-one; so, just identify them possibly more abstractly. But the set will in any case remain negligible vs. the uncountably-infinite real numbers.