π is Transcendental
A transcendental number is one that cannot be expressed as a solution of ax^n+bx^(n-1)+...+cx^0=0 where all coefficients are integers and n is finite. For example, x=sqrt(2), which is irrational, can be expressed as x^2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic.
It is very easy to prove that a number is not transcendental, but it is extremely difficult to prove that it is transcendental. This feat was finally accomplished for π by Ferdinand von Lindemann in 1882. He based his proof on the works of two other mathematicians: Charles Hermite and Euler.
In 1873, Hermite proved that the constant e was transcendental. Combining this with Euler's famous equation e^(i*π)+1=0, Lindemann proved that since e^x+1=0, x is required to be transcendental. Since it was accepted that i was algebraic, π had to be transcendental in order to make i*π transcendental.