# Chaitanya's Random Pages

## March 26, 2011

### Some Mathematical Coincidences

Filed under: mathematics — ckrao @ 3:42 am

Here are some mathematical coincidences I quite like. There are more at the references below.

• $\displaystyle 2^{10} = 1024 \approx 10^3$ – this leads to such approximations as a doubling corresponding to a 3dB increase, or a kilobit of data being represented by 10 bits.
• $\displaystyle 3^2 + 4^2 = 5^2$ and $\displaystyle 3^3 + 4^3 + 5^3 = 6^3.$
• $\displaystyle 10^2 + 11^2 + 12^2 = 13^2 + 14^2$ (= 365, the number of days in a year)
• $\displaystyle 1.08^9 = 1.999004...$, $\displaystyle 1.02^{35} = 1.999889...$ so now you know the doubling time for investments at 8% or 2% return.
• $\pi \approx 22/7$ to 0.04% or $\pi \approx 355/113$ to six decimal places
• $\displaystyle \left(\frac{2143}{22} \right)^{1/4}= 3.14159265258...$, so $\displaystyle \pi - \left(\frac{2143}{22} \right)^{1/4} < 10^{-9}$
This was found by Ramanujan, who knows how he came up with it!
• $\displaystyle e^{\pi}-\pi = 19.9990999792$
• 1 year is approximately $\pi \times 10^7$ seconds to 0.45% (1 year = 31,556,926 seconds)
• Some involving 666: $\phi(666) = 6.6.6$, 666 is the sum of the squares of the first 7 primes, $\sum_{i=1}^{6\times 6}i = 666$
• $\displaystyle \int_0^{\infty}\prod_{i=1}^{\infty}\cos\left(\frac{x}{n}\right)\ dx \approx \frac{\pi}{8}$, where the difference only occurs after the 42nd decimal place!

#### References

http://en.wikipedia.org/wiki/Mathematical_coincidences

http://en.wikipedia.org/wiki/Almost_integer

Weisstein, Eric W. “Almost Integer.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/AlmostInteger.html

## 1 Comment »

1. This one made me laugh, frankly. It’s as if there is some cosmic sense of humour at work here.

Comment by Radhika — April 14, 2011 @ 10:42 pm

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