To prove that 4π is irrational, assume the product is rational and set it equal to ab, where b is not equal to 0: 4π=ab. Isolating π gives π=a4b. The right side of the equation is irrational or rational
. Because the left side of the equation is
rational or irrational
, this is a contradiction. Therefore, the assumption is wrong, and the number 4πis
rational or irrational
.