Explain why is the number Pi infinite?

Understanding the irrationality of the number Pi

An irrational number is simply a number that cannot be written in the form of a fraction like a/b with a and b being integers. And guess what: Pi is exactly that kind of number. Unlike ordinary fractions like 1/2 or 3/4, Pi never ends and never settles into a regular repetition. This means that there are always new digits after the decimal point without any specific pattern. This is not an isolated oddity: in fact, precise mathematical proofs have long established that Pi is not related to any fraction, which necessarily implies a chaotic infinity of digits.

Why Pi does not have a repetitive sequence

A decimal number like 1/3 has a clearly repetitive sequence (0.333... with the digit 3 repeating indefinitely). But Pi belongs to the so-called irrational numbers, meaning it cannot be expressed as a simple fraction, and it never repeats. Why? Because of what is called its transcendence. This simply means that no matter how long you search, no repetitive or logical pattern will emerge in its digits: the sequence of Pi's decimals always appears just as unpredictable and random, regardless of how far you push the calculations. This complete absence of a repetitive sequence has even been rigorously demonstrated by mathematicians, confirming the infinitely chaotic nature of Pi.

Mathematical formulas highlighting the infinity of Pi.

Since antiquity, mathematicians have been trying to pinpoint Pi with increasingly precise formulas. But as we advance, we notice that its decimal value continues indefinitely, never stabilizing. Some famous formulas, like those of Leibniz or the infinite series of Madhava, clearly highlight that to calculate Pi exactly, one would need to add an infinite number of terms without stopping. This simply means that Pi cannot be represented by a fraction or a finite decimal number: there will always be a way to add an additional digit after the decimal point, without ever seeing a repetition or an end. In other words, these formulas quietly reveal that absolute precision in knowing Pi is simply impossible.

Analytical and geometric approaches explaining its infinitude

When you look at Pi from a geometric perspective, you often start with a circle. If you try to draw a polygon inside this circle (like a hexagon, octagon...) and add more and more sides, you gradually approach the actual perimeter of the circle, but you never quite reach it exactly. It is precisely this impossibility of obtaining exactly Pi with a geometric shape having a finite number of sides that shows that there will always be more decimals, in short, that Pi is infinite.

From the standpoint of mathematical analysis, it comes down to the same idea, but with infinite sequences, that is to say, series of numbers that progressively get closer to Pi. These mathematical series (like Leibniz's or Euler's) allow for the calculation of more and more decimals. But these infinite sums never land precisely on Pi; they always yield a slightly different result, requiring even more calculations to gain precision. That’s why we will never reach an end: Pi is a number with infinite decimals.

Practical and theoretical consequences of the infinity of Pi.

The fact that Pi is infinite implies that we can never calculate the exact circumference of a circle precisely, but can only approach it with a high degree of accuracy. In practice, this forces us to use approximations constantly, both in geometry and in physics or engineering. As a result, these approximations can lead to calculation errors in distances or measurements when extreme precision is required, such as in space calculations or advanced scientific simulations. On a more theoretical level, the infinity of Pi leads to deep reflections in mathematics and philosophy: it highlights the difference between our perfect abstract representation of circles and the concrete reality where everything is approximate. This is also why Pi is so fascinating, because despite its apparent simplicity, we are still only scratching the surface of the mysteries of this strange number.