What is Pi ($\pi$)?

1. Introduction: 3.14 is a "Lie"

In elementary school, we learn it as "3.14". In higher mathematics, we use the Greek letter $\pi$ (Pi). However, this number $3.14$ is merely an "approximation."

Why don't textbooks teach us the "exact value"? The answer is simple: "It is impossible to write the exact value using numbers."

Let's unravel the true identity of this mysterious number $\pi$ and the history of how humanity has chased its digits.

2. Definition: The Constant Hidden in Circles

What is $\pi$ exactly? Its definition is very simple.

Whether it's a tiny circle or a massive one, if the diameter is "1", the circumference will always be "3.141592...". It is one of the fundamental constants that govern our universe, unchanging no matter where you go.

Why can't it be a fraction? (Irrational Numbers)

Numbers that can be expressed as a ratio of integers (fraction $\frac{a}{b}$) are called "Rational Numbers." However, $\pi$ cannot be written as a fraction. Such numbers are called "Irrational Numbers."

What if $\pi$ were rational?

Suppose $\pi$ ended exactly at $3.14$. This would mean the circumference is exactly $\frac{314}{100}$ times the diameter.

This leads to a strange conclusion. If we can count the circumference as an exact number of "straight line segments" (based on the diameter), it implies that the circle is made up of "tiny straight lines (a polygon)."

A true circle is never "jagged," no matter how much you zoom in. It curves smoothly forever. Because there is always a "gap that can never be measured by straight lines," the ratio $\pi$ must also never end, continuing infinitely.

3. How to Calculate? (Archimedes' Method)

Over 2000 years ago, without supercomputers, how did Archimedes reach the precision of "3.14"? He used a method that can be considered the origin of numerical analysis: "Squeezing the circle with polygons."

He manually increased the number of sides: 6 → 12 → 24 ... up to a 96-gon, narrowing the range to: $$ 3.1408 < \pi < 3.1428 $$ This is the origin of "Pi $\approx$ 3.14".

4. How do Computers Calculate?

Modern computers don't draw polygons. They use more efficient mathematical formulas.

1. Leibniz Formula (Infinite Series)

A beautiful but famously slow-converging formula. By infinitely adding and subtracting odd fractions, you eventually get $\pi$.

2. Monte Carlo Method (Probabilistic Simulation)

We can calculate $\pi$ by "randomly dropping rain."

1. Draw a circle inside a square.
2. Randomly throw points (darts) at it.
3. The ratio of "points inside the circle" to "total points" approaches $\frac{\pi}{4}$.

This is a type of "numerical integration" and is a vital technique frequently used in modern physical simulations.

5. Dr.WataWata's Insight