# Definition:Pi

## Definition

The real number $\pi$ (pronounced pie) is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$

### Geometric Definition

Take a circle in a plane whose circumference is $C$ and whose radius is $r$.

Then $\pi$ can be defined as $\pi = \dfrac C {2r}$.

### Algebraic Definition

The real functions sine and cosine can be shown to be periodic.

The number $\pi$ is defined as the real number such that:

the period of both sine and cosine is $2 \pi$.

## Decimal Expansion

The decimal expansion of $\pi$ starts:

$\pi \approx 3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$

In binary:

$\pi \approx 11 \cdotp 00100 \, 10000 \, 11111 \, 1011 \ldots$

## Also see

It is a common fallacy that the value of $\pi$ is dependent upon the geometry in which it is defined.

• Results about pi can be found here.

## Historical Note

Every ancient society that considered circles was aware of $\pi$, although in general only as a rough approximation.

In the Old Testament, the implication is that $\pi = 3$:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
-- $\text I$ Kings $7 : 23$

The Egyptian scribe Ahmes in the Rhind Papyrus used the approximation that the area of a circle equals the area of a square whose side is $\dfrac 8 9$ that of the diameter of the circle, leading to a value of $\pi$ of $\left({\dfrac {16} 9}\right)^2 = 3 \cdotp 16049 \ldots$

By calculating the areas of regular polygons of $96$ sides, Archimedes of Syracuse determined that $3 \dfrac {10} {71} < \pi < 3 \dfrac {10} {70}$, that is:

$3 \cdotp 14085 \ldots < \pi < 3 \cdotp 142857 \ldots$

That last value:

$3 \cdotp 142857 \ldots$

more often given as $\dfrac {22} 7$, is commonly used in schools as a good working approximation to $\pi$. This sequence is A068028 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

In binary notation it has the repeating pattern:

$\pi \approx 11 \cdotp 00100 \, 1001 \ldots$

Archimedes also found more accurate approximations still.

Ptolemy used $\dfrac {377} {120}$, which is approximately $3 \cdotp 14166 \, 66 \ldots$

The Zu Chongzhi fraction is an exceptionally accurate approximation to $\pi$ (pi):

$\pi \approx \dfrac {355} {113}$

whose decimal expansion is:

$\pi \approx 3 \cdotp 14159 \, 292$

Zu Chongzhi and his son Zu Geng determined that:

$3 \cdotp 14159 \, 26 < \pi < 3 \cdotp 14159 \, 27$

Jamshīd al-Kāshī calculated $\pi$ to $16$ decimal places.

The Ludolphine number is the expression of the value of $\pi$ to $35$ decimal places:

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \ldots$

It was calculated by Ludolph van Ceulen between $1596$ and $1610$.

Improvements in trigonometric techniques allowed for better methods for calculating the digits of $\pi$.

Willebrord van Royen Snell calculated $34$ places using the same techniques that Ludolph van Ceulen used to calculate $14$.

Christiaan Huygens achieved $9$ places just by considering the geometry of the regular hexagon.

François Viète was the first to devise a formula for $\pi$, which he did in $1592$.

John Wallis was next, with Wallis's Product.

Isaac Newton devised a formula in $1666$, and Gottfried Wilhelm von Leibniz devised one in $1673$.

The latter is unfortunately too inefficient to be useful.

## Linguistic Note

While the conventional contemporary prounciation of $\pi$ in Western English is pie, it is worth noting that the "correct" Greek pronunciation of the name of the letter $\pi$ is in fact the same as the letter p is pronounced in English.

It is just as well that $\pi$ is pronounced pie, otherwise the opportunity for confusion between $\pi$ and $p$ in spoken language would be too great.