# Show that $\pi ≈ 355/113$ is the best rational approximation of $\pi$ with a three-digit denominator

Given that the continued fraction expansion of $$\pi$$ is $$[3, 7, 15, 1, 292, ...]$$. Prove that $$\pi ≈ [3, 7, 15, 1]= 355/113$$ is the best approximation of $$\pi$$ with a 3-digit denominator.

From the following theorem:

1. Any convergent $$x/y$$ of an irrational number $$\alpha$$ satisfies $$\lvert\frac{x}{y} - \alpha\rvert < \frac{1}{y^2}$$. $$x$$, $$y$$ are positive integers and gcd(x, y) = 1.

2. If $$\alpha$$ is an irrational number and x/y satisfies $$\lvert \frac{x}{y} - \alpha\rvert < \frac{1}{2y^2}$$, then $$\frac{x}{y}$$ is a convergent of $$\alpha$$.

Have $$\lvert\frac{355}{113} - \pi\rvert < \frac{1}{113^2}$$.

I tried to consider $$n = \frac{a}{b}$$ where $$\lvert \frac{a}{b} - \pi \rvert < \lvert \frac{355}{113} - \pi \rvert$$ and hope to raise a contradiction.

I considered the cases when $$\frac{a}{b} - \pi$$ is positive and negative but wasn't able to conclude anything from the results.

Could someone please point me in the right direction? Thank you!

• How do you define best approximation? I guess through $|q\alpha-p|$ or $|q^2\alpha-pq|$, but it is not stated anywhere, so the initial question is not really meaningful at the moment. – Jack D'Aurizio Dec 10 '18 at 22:39
• It's not stated in the original problem but I'm assuming it means an approximation with the smallest error? – Lin Dec 10 '18 at 22:52
• Hint: Contrapositive to 2 is: If $\frac{x}{y}$ is not a convergent of $\alpha$ then $|\frac{x}{y} - \alpha| \ge \frac{1}{2y^2}$. – Daniel Schepler Dec 10 '18 at 23:27

The approximation $$\pi\approx\frac{355}{113}$$ is not only good, it is exceptionally good, since $$\left|\pi - \frac{355}{113}\right|<\frac{1}{\color{red}{291}\cdot 113^2}<\frac{1}{2\cdot 1000^2}.$$ In particular if some approximation $$\frac{p}{q}$$ with $$q\leq 1000$$ achieves an absolute error which is less than the absolute error achieved by $$\frac{355}{113}$$, such approximation is a convergent of the continued fraction of $$\pi$$.
$$\frac{355}{113}$$ is a convergent, the next convergent is $$\frac{103993}{33102}$$, with a denominator much larger than $$1000$$.
It follows that the Chinese approximation $$\pi\approx\frac{355}{113}$$ is the best rational approximation (in the absolute error sense) of $$\pi$$ with a denominator less than $$1000$$.
• Thanks! So I tried to compare $\lvert \frac{355}{113} - \pi \rvert$ and the difference between the 3rd and 4th convergents and got $\lvert \frac{355}{113} - \pi \rvert<\frac{1}{113*33102}$, which eventually leads to $\lvert \frac{355}{113} - \pi \rvert<\frac{1}{2*1000^2}$. But not sure if that's how you got 291? – Lin Dec 11 '18 at 14:17
• @Lin: $$\pi=[3;7,15,1,\color{red}{292},1,\ldots]$$ – Jack D'Aurizio Dec 11 '18 at 18:07