Why is $22/7$ a better approximation for $\pi$ than $3.14$? This seems counterintuitive, but $22/7$ is closer to $\pi$ than $3.14=314/100$ which has a significantly greater denominator.

Why is $22/7$ a better approximation for $\pi$ than $3.14$?

This has important implications:  e.g. Should "$\pi$-day" be the $14^{th}$ of March or $22^{nd}$ of July?
 A: It only seems odd to you because you are used to representing numbers in base 10. What if you used base 7?
A: Well, just measure $|\pi - 22/7|$ and $|\pi-3.14|$ ...
A: It has to do with the continued fraction expansion of $\pi$.  Suppose $[a_1, a_2, \ldots]$ is the continued fraction of an irrational number $\alpha$ -- that is, if $a_n$ is the (essentially unique) sequence of natural numbers such that if we define partial convergents by $x_1 = a_1$, $x_2 = a_1 + 1/a_2$, $x_3 = a_1 + 1/(a_2 + 1/a_3)$, $x_4 = a_1 + 1/(a_2 + 1/(a_3 + 1/a_4))$, and so on, then $\alpha = \lim_{n\to\infty} x_n$.  Then the partial convergents $x_n$ are rational numbers that approximate $\alpha$ better than anything that is not a partial convergent, in the following sense:  a rational number $\frac pq$ satisfies the inequality $|\alpha - \frac pq| < \frac 1{2q^2}$ if and only if $\frac pq$ is one of the convergents $x_n$.  (One could, of course, come up with different notions of what constitutes a "good" approximation.)
The continued fraction expansion of $\pi$ is $[3,7,15,1,292,1,1,\dots]$, so the first few convergents are $3$, $\frac{22}{7}$, $\frac{333}{106}$, $\frac{355}{113}$, etc.  Thus $\frac{22}{7}$ is a better approximation than $\frac{314}{100}$ (in the above sense) because it appears in the list of partial convergents, while $\frac{314}{100}$ does not.
Incidentally, the approximation $x_n$ is best when the coefficient $a_{n+1}$ is quite large, so the size of $a_5 = 292$ means that $x_4 = \frac{355}{113}$ is a particularly good approximation.
At the risk of self-promotion, I wrote a brief exposition of all this in a bit more detail -- you can find it on my website if you're interested, at http://www.math.psu.edu/climenha/contfrac.html.
