# Why does this expression equal pi?

I was fiddling with numbers when I noticed that $$50 \times 1.05^{168} \times \frac{12600}{727767941} \approx \pi$$ I understand it's an approximation. Does anyone know why?

• That's probably just a coincidence. – Arthur Nov 10 '15 at 15:52
• Ho many digits accuracy does it give you? That's a fairly random expression - given any huge numerator, we can likely find some denominator that gets somewhat close. – Thomas Andrews Nov 10 '15 at 15:53
• If you gave us a reason or something interesting about that denominator, then it might be worth exploring, or if these values came out of another question, but just blindly, a number close to $\pi$ is a dime a dozen - there are lots of them. There isn't a "reason" for it - there are lots of expressions equally close to every real number. – Thomas Andrews Nov 10 '15 at 16:01
• There are occasionally really incredible approximations of $\pi$. For example: $$\frac{\ln 262 537 412 640 768 744}{\sqrt{163}}$$ is unreasonably close, because there is a deep reason for $e^{\pi\sqrt{163}}$ to be remarkably close to an integer. – Thomas Andrews Nov 10 '15 at 16:24
• $262537412640768744=640320^3+744.$ – Lucian Nov 10 '15 at 16:44

By my calculator this expression only agrees with $\pi$ to about 13 decimal digits. Since there are 22 separate digits in the expression, we should expect that very many different expressions of that shape approximate $\pi$ with a similar precision -- there's nothing particular remarkable about one of them, nor any "explanation" other than it just happens to be close to $\pi$.
• In fact, $1.05^{168}\times\frac{80034}{92454253}$ is a better approximation with fewer digits, but $\frac{80143857}{25510582}$ without the power of $1.05$ is even better. – Henning Makholm Nov 10 '15 at 16:41