
Rational Math Catches Slippery Irrational Numbers - theafh
https://www.quantamagazine.org/how-rational-math-catches-slippery-irrational-numbers-20200310/
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svat
This is a nice article, and gently (with great explanation) introduces:

• The idea of approximating a real number by rational numbers (e.g. √2 ≈
99/70),

• Dirichlet's approximation theorem, that every real number x has a rational
approximation p/q such that |qx - p| < 1/(N+1) using a denominator q ≤ N, for
any integer N,

• The Duffin–Schaeffer conjecture, recently announced to be proved.

BTW, I got the above "√2 ≈ 99/70" (≈ 1.4142857…, better than 1414/1000 which
has a much larger denominator) from a webpage I made a while ago, to generate
the best rational approximations (with increasing denominators) for any
number:
[https://shreevatsa.net/fraction/best](https://shreevatsa.net/fraction/best)
(Of course the continued fraction for √2 has a very simple form so convergents
could be written down by hand.)

