
Rational Math Catches Slippery Irrational Numbers - rfreytag
https://www.quantamagazine.org/how-rational-math-catches-slippery-irrational-numbers-20200310/
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msla
I'm surprised this doesn't mention Cauchy sequences. It seems to get close,
but never quite hit the mark.

Anyway, a Cauchy sequence is one where the terms of the sequence eventually
become arbitrarily close to one another; that is, you can pick an epsilon, an
arbitrary small number, and after a certain point, the difference between
elements of the sequence is less than epsilon.

The intuition is an infinite list of rational numbers which, when plotted,
look like a ping-pong ball endlessly bouncing around a given point, getting
closer and closer... to what? Well, it depends on the precise sequence, but
that end-point might not exist in the rationals. If it's a_n/2 + 1/a_n, it
converges to the square root of 2, but we've known for a long time that _that
's_ irrational, so the ping-pong ball bounces forever, from one rational to
another, getting closer and closer to a spot that doesn't exist...

What if it did exist? What if _all_ those spots existed? Well, if you have
that, the set is now Cauchy-complete, also known as a complete metric space;
if you have that, plus the field axioms (so addition, subtraction,
multiplication, and division all work as you expect), plus ordering (so
greater-than and less-than always work), plus the Archimedean axiom (no
infinitesimal or infinite numbers), you have the real numbers. The reals are
the unique Cauchy-complete Archimedian ordered field; every other set with
those properties is isomorphic to the reals.

~~~
addcninblue
I'd also like to add one more clarification: Cauchy sequences require _any
two_ elements after N (the "certain point" mentioned above) to be within
epsilon of each other.

Why? I won't explain it fully, but hopefully a counterexample can convince you
so: Take the common harmonic series (a_n = 1+1/2+1/3+...+1/n) Each successive
element differs from the previous by 1/n. Therefore, informally we can see
that their pairwise differences gets arbitrarily small. However, this sequence
is _unbounded_: The infinite sum of the harmonic series tends to infinity. It
therefore cannot be Cauchy. (Cauchy "bounces around a fixed point")

~~~
msla
That is a very good point.

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btilly
I find that quanta entertains but does not inform.

See
[https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conje...](https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture)
for the conjecture in question.

Basically it is about under what measures of "a surprisingly good
approximation" we would expect to find infinitely many surprisingly good
approximations for any random real we pick. Where the definition of a
surprisingly good approximation is "within a bound dependent on the
denominator".

Obviously you can get a better approximation by simply taking a larger
denominator. The question is whether you get a good approximation relative to
the size of the denominator.

We probably aren't surprised to find that with a denominator of size q we can
get within 1/q of an arbitrary rational. But we likely are surprised to find
that we can get within 1/(100 q^2). In fact there are no approximations that
good of the golden ratio (1 + sqrt(5))/2\. But it is unusual, almost all real
numbers can be approximated that well. But 355/113 is such a good
approximation of pi.

The conjecture is about the existence, not how to find them. (In practice
continued fractions are usually the right way to do it.)

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pierrebai
There must be a subtlety I don't understand. To cover the number line at
precision of at least 10^n, just cover it with intervals of 1/10^n, surely?

IOW, to approximate an irrational number to 10 decimal places... use 10
decimal places? Geen, use 11 decimal places just to be double-sure and not
have to worry.

What am I missing?

~~~
btilly
The question isn't how to approximate a given number to desired accuracy. It
is whether we should be able to approximate it to desired accuracy relative to
the denominator.

Getting an approximation p/q of x within 1/q is easy. Getting it within 1/q^2
takes work. Within 1/(10 q^2) is only sometimes possible.

~~~
AnimalMuppet
> Within 1/(10 q^2) is only sometimes possible.

For _any_ q?

~~~
btilly
_For any q?_

Yes.

The hardest real number to approximate with rationals is (1+sqrt(5))/2\. Its
approximations that come closest are of the form fib(n+1)/fib(n) where fib(n)
is the n'th fibonacci number. The error converges to 1/(sqrt(5) fib(n)^2).
None are within 1/(10 q^2).

To learn more about this, read up on continued fractions.

