A simple proof that π is rational∗ Peter Rowlett 1st April 2013 Abstract The number pi, written using the symbol π, is a mathematical con- stant that is the ratio of a circle’s circumference to its diameter, and has been claimed since antiquity to be an irrational number, mean- ing that it cannot be expressed exactly as a ratio of two integers, and that therefore its decimal expansion never ends or settles into a per- manent repeating pattern. Here a proof is given that π can indeed be 4 expressed as a ratio of two integers, 17 , a fact that has unbelievably been overlooked until now. Moreover, this proof is understandable to anyone with a basic knowledge of algebra and calculus and arises from 1 simply considering a standard integral at two values of x, x = 4 and x = 1. Of course I doubted the result at first, given that it has been overlooked for so many years, but I have checked the proof and verified it to be correct. This is a crucial and important revelation that will significantly alter all of mathematics. 1 A formula for π First we note that Z 1 1 x dx = tan−1 x2 + a2 a a and π tan−1(1) = . 4 1 Since x can take any value, consider what happens when x = 4 . Then: Z 1 −1 2 dx = 4 tan (1) = π. (1) 2 1 x + 4 ∗Cite this as: Rowlett, P., 2013. A simple proof that π is rational. Travels in a Mathematical World, 1st April. Available via: http://travels.aperiodical.com/2013/04/a- simple-proof-that-pi-is-irrational.html 1 1 2 Consider 2 2 1 x +( 4 ) 1 If x = 4 , then 1 1 2 = 2 2 2 1 1 1 x + 4 4 + 4 1 = 1 1 16 + 16 1 = 2 16 16 = 2 = 8. Then ! 12 1 = 8 x2 + 4 1 1 = 8x2 + 2 1 8x2 = . 2 Now, note that x2 is equal to x × x, that is x added to itself x times: x2 = x + x + ... + x (x times). Then 8x2 = 8(x + x + ... + x). Multiply each side by x: 8x2 = 8x(x + x + ... + x). Now, take the derivative of each side. The left hand side is trivial, since d 8x2 = 16x. (2) dx The right hand side requires the product rule. First let u = 8x & v = x + x + ... + x. Then u0 = 8 & v0 = 1 + 1 + ... + 1. 2 Note that v0 is 1 added to itself x times, i.e. v0 = x, and v is x added to itself x times, i.e. v = x2. Then the product rule gives: uv0 + u0v = 8x × x + 8 × x2 = 8x2 + 8x2 = 16x2 (3) Since we have, from (2) and (3), that 16x = 16x2 16 it follows that x = 16 = 1. Putting x = 1 into 1 , we have 2 1 2 x +( 4 ) 1 1 = 2 1 2 1 1 + 16 1 + 4 1 = 16 1 16 + 16 1 = 17 16 16 = . (4) 17 3 The final step It follows from (4) that Z 1 Z 16 2 dx = dx 2 1 17 1 + 4 16 = x (5) 17 1 From (1), we know that this equation equals π when x = 4 , i.e. 16 1 4 π = × = . (6) 17 4 17 4 Clearly, 17 is rational, and therefore π is rational. 3.