Tom Rocks Maths intern Isaac Wood introduces the most amazing formula for pi – involving prime numbers and multiples of 4 – and shows you how to prove it.
The proof is broken down into several steps. We being by proving ‘Mini Result 1’ which gives the sum of an infinite geometric series. Next, we prove a result about the infinite limit of a sum of powers – this is ‘Mini Result 2′. We then set up the geometry of the problem and using results about similar triangles and Pythagoras’ Theorem, obtain a formula for the approximation of the arc length which is equal to pi divided by 4. This is known as the ‘Leibniz Formula for Pi’. Finally, using prime numbers and the fundamental theorem of arithmetic, we deduce the amazing result: Pi divided by four is equal to the product of each prime number divided by its closest multiple of 4.