The continued fraction of the square root of 2

$\sqrt 2$ too can be written as a periodic continued fraction, and its truncations give the best rational approximations of the number.

Example — Continued fraction of $\sqrt 2$

We look for $x>0$ with $x^2=2$, that is $x = 1 + \dfrac{x^2-1}{x+1} = 1 + (x-1)$. We rewrite $x-1 = \dfrac{1}{x+1}$ obtaining $x = 1 + \frac{1}{1+x}.$ Substituting $x$ on the right at each step: $\sqrt 2 = 1 + \cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cdots}}}.$ All $2$s after the first digit: $1,2,2,2,\ldots$. The successive convergents are $1,\ 3/2,\ 7/5,\ 17/12,\ 41/29,\ 99/70,\ldots$ (ratios of the Pell numbers); each is the best rational approximation of $\sqrt 2$ with denominator less than or equal to the given one. Check: $99/70 = 1{,}41428\ldots$ against $\sqrt 2 = 1{,}41421\ldots$ (error $<10^{-4}$).

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Topics: Second-degree equations
Concepts: Convergents · Continued fraction · Pell numbers
Skills: Solving equations · Estimating