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

Advertisements

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 | Reply


RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: