Definition of Pi 📂Lemmas

Definition of Pi

Definitions

Geometric Definition

A circle is defined as the set of points in a plane that are at a given distance $r > 0$ from a given point.
The ratio of a circle’s circumference $l$ to its diameter $2r$ is defined as the Pi $\pi$. $$ \pi := {{ l } \over { 2r }} $$

Analytical Definition ¹

$$ E (z) := \sum_{k=0}^{\infty} {{ z^{k} } \over { k! }} $$ Let’s define the complex function $E : \mathbb{C} \to \mathbb{C}$ as the series expansion of an exponential function, and through it, define the following function similar to the cosine function $C$. $$ C(x) := {{ E (ix) + E(-ix) } \over { 2 }} $$ When $C(x)$, the smallest positive root that satisfies $C(x) = 0$ is called $x_{0}$, and twice this value is defined as Pi $\pi$. $$ \pi := 2 x_{0} $$

Description

In this post, both the geometric (simple) and the analytical (complicated) definitions are introduced. Mathematics students beyond their junior year of college might smile subtly at the analytical definitions.

Throughout human history, Pi has been an extremely important constant, being practically useful no later than the invention of the wheel. In particular, it is known that there were efficient and precise approximations like $$ {{ 22 } \over { 7 }} = 3.142857 \cdots \approx \pi $$ This value is far more accurate than what was taught during the so-called Yutori education era at the end of the 20th century in Japan, which significantly surpassed the educational standards of the time. (This was when they taught Pi as $3$ under the pretext of providing a relaxed education) ²