Premise: In mathematics , the irrational numbers are all the real numbers which are not rational numbers , the latter being the numbers constructed from ratios ( or fractions ) of integers . The number is a mathematical constant , the ratio of a circle 's circumference to its diameter , commonly approximated as 3.14159 . Being an irrational number , can not be expressed exactly as a fraction ( equivalently , its decimal representation never ends and never settles into a permanent repeating pattern ) .
Hypothesis: Pi is an imaginary number.
Label: CONTRADICTION