Rational Math Catches Slippery Irrational Numbers
quantamagazine.org
quantamagazine.org
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.
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")
One can also arrive at the reals by constructing from the rationals through completion.
https://en.wikipedia.org/wiki/Complete_metric_space#Completi...
Edit: There is a nice wiki page about the different approaches
https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
See https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conje... 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.)
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?
A different measure of goodness takes into consideration not only the distance between the real number and its rational approximation, but how big the denominator of the approximation is. Note that, in the scheme you describe, a denominator of q guarantees an error of 1/q. But if you are allowed to choose both the numerator and the denominator of the rational approximation, then perhaps you can make the denominator smaller?
It turns out you can! Dirichlet showed that for every irrational alpha there are infinitely many "good approximations" of the form p/q where |alpha - p/q| < 1/q^2. It turns out the golden ratio shows that you can't do better than 1/(sqrt(5) * q^2) if you want the result to hold for every irrational number. This leads to the problem of identifying "interesting" sets of irrational numbers that allow approximations substantially better than Dirichlet's bound (this is the problem of Diophantine approximation [0]). Klaus Roth got his Fields Medal primarily for a result in this area [1].
[0] https://en.wikipedia.org/wiki/Diophantine_approximation [1] https://en.wikipedia.org/wiki/Roth%27s_theorem
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.
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.