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 highlyupvoted 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 quotingthedefinition 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 gradeschool 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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