Pi is a Transcendental number, meaning it cannot be expressed finitely. Also, there is no patterns, or repetitions, meaning with Pi you can never be 100% accurate. Transcendental numbers cannot be expressed as the root of any algebraic equation with rational coefficients. Pi CAN be expressed as an endless continued fraction or as the limit of an infinite series.
pi transcends the power of algebra to display it in its totality. Using a fixed-size font, it can't be written on a piece of paper as big as the universe.
Programs that calculate Pi ask, "to what decimal place" and fill your CPU until it finds out that it equals 3.141592654...
if you feel like doing a home experiment, take a glass (round one), and a bendable measuring tape, measure the diameter (across the glass at the widest) and the circumference (around the glass where you measured the diameter) divide the circumference by the diameter... you should be close to Pi, otherwise your measuring skills suck...
Engineering is the science of making life simple, by making it more complicated.