Pi

Pi is one of the most fundamental constants in all of mathematics. It is normally first encountered in geometry where it is defined as the ratio of the circumference of a circle to the diameter: π = C/d where C is the circumference and d is the diameter. This fact was known to the ancient Egyptians who used for π the number 22/7 which is accurate enough for many applications. A closer approximation in fractions is 355/113. Students often use a decimal approximation for π, such as 3.14 or 3.14159.

Actually, the number π is not even a rational number. That is, it is not exactly equal to a fraction, m/n where m and n are whole numbers or to any finite or repeating decimal. This fact was first established in the middle of the eighteenth century by the German mathematician, Johann Lambert. Even further, it is a transcendental number. That is, it is not the root of any polynomial equation with rational coefficients. This was first proved by another German mathematician, Ferdinand Lindeman, in the latter half of the nineteenth century.

There are many infinite series that can be used to calculate approximations to π. One of these is where the denominators are the consecutive odd numbers.