COMS Tutorial Corner

A surprising expression for π

Math π : the ratio of circumference to diameter of a circle, is an important constant. A variety of π approximations have been made, but the following proposal by a genius Indian mathematician, is both surprising and captivating now even as the time it was proposed:

π = 3 ⁄ √̅5̅ (1+ (3 ⁄ √̅5̅ ) ) — (**)

By standard notation, √̅5̅  is the square root of 5. Grab a calculator to verify how close it is to the genuine value π = 3.1415 .. And in neat form too!

It is still puzzling today how ** came to this value. We have guessed that he came to this particular form of π from one of the following equations:

(1/x) (1+ (1/x)) = π — Eq.(1)
or
x = π x2 – 1 — Eq. (2)

Both share a positive root that is pretty close to √̅5̅ /3.
From the second equation, if we set out to solve π in terms of x, then we will find

π = 1/x (1 + 1/x) = (1+x) ⁄ (x2)

that’s exactly the first equation!

We can get the π -value  expression (which appears at the very beginning of the article), by simple replacing x with the root √̅5̅ /3.

Now take another look at the expression π , and rewrite it in the form below:

π = (√̅5̅ /3 + 1) (32 ⁄ 5)

If picking out all digits at the right hand side! but ignore operations and square /square root, then What do we get? We get 53135: a palindrome. That’s double neat!!

Comments are closed.