History of Pi

Definition of Pi

For every circle, the ratio of its circumference to its diameter is the same. We call that constant $\pi$ (pi), and its value begins $3.14159$.

Mathematically, $\pi$ can be defined as:

The ratio does not depend on the circle's size. If every length doubles, the circumference and diameter double together, so their ratio stays unchanged.

Early Approximations

Surviving ancient sources record several approximations. A Mesopotamian approximation was $\pi \approx \frac{25}{8}=3.125$.

The Egyptian Rhind Mathematical Papyrus, dated to about $1650 \text{ BC}$, gives a rule equivalent to $\pi \approx \frac{256}{81}$. This is about $3.16049$. Neither fraction equals $\pi$ exactly, and the Mesopotamian value $3.125$ is actually the closer of these two approximations.

Method of Archimedes and Polygons

Archimedes of Syracuse, a Greek mathematician who lived around $287\text{-}212 \text{ BC}$, developed a systematic method for bounding $\pi$ more precisely. He compared regular polygons inscribed in and circumscribed around a circle.

The perimeter of an inscribed regular polygon is less than the circle's circumference, while the perimeter of a circumscribed regular polygon is greater. Increasing the number of sides brings those lower and upper bounds closer together.

Using $96$-sided polygons, Archimedes proved that $\pi$ lies between $\frac{223}{71}$ and $\frac{22}{7}$. In particular, $\frac{22}{7}$ is the upper end of this bound. Its value is slightly larger than $\pi$:

Contributions of Chinese Mathematicians

In the $5$th century AD, the Chinese mathematician and astronomer Zu Chongzhi refined the polygon method and established the bound $3.1415926 < \pi < 3.1415927$, an interval only $10^{-7}$ wide.

Zu Chongzhi also gave $\pi \approx \frac{355}{113}$, which equals $3.1415929$. The fraction agrees with $\pi$ through the first six digits after the decimal point. His seven-decimal bound was not surpassed until al-Kashi's work in the $15$th century, almost a millennium later.

Modern Era and Pi Symbol

The Welsh mathematician William Jones used $\pi$ with its present meaning in his $1706$ work "Synopsis Palmariorum Matheseos". The notation was not immediately universal.

Leonhard Euler later adopted the symbol in his influential work, helping $\pi$ become the standard notation still used today.

Special Properties of Pi

The number $\pi$ is irrational, so it cannot be written as a ratio of two integers. Johann Heinrich Lambert supplied the first rigorous proof of its irrationality in the $18$th century. Ferdinand von Lindemann proved in $1882$ that $\pi$ is also transcendental, meaning it is not a root of any nonzero polynomial with rational coefficients.

For many basic calculations, $\pi \approx 3.14$ or $\pi \approx \frac{22}{7}$ is sufficient. Scientific and engineering work uses as many digits as the required precision demands.

The ratio that defines $\pi$ in circle geometry is:

In these equations, $C$ is circumference, $d$ is diameter, $A$ is area, and $r$ is the radius of the circle.