Do We Have Any Mathematical Proof That Pi Is Infinite?

The proof for the infinite length of pi comes from mathematician Johann Lambert, who proved that pi is irrational, and therefore must be infinite.

The sixteenth letter of the Greek alphabet in mathematics holds as much significance in this universe as pepperoni in pizza holds for the average reader. From determining the size of your frisbee to calculating the expanse of our universe, this symbol has changed the world.

Any guesses on what it is?

π.

Two vertical lines flagged by one horizontal line: π (pi). You’ve probably heard or even used this symbol in your math class. The circumference of a circle is 2πr, where r is the radius of the circle.

Have you ever wondered about the origin of pi? And do we have any proof that pi is infinite? Moreover, is pi really what we think it is?

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The Origin Of Pi

Our life has been dominated by circular objects from the very beginning. Wooden wheels back then, Hot Wheels today.

The same discovery cropped up independently in ancient civilizations all over the world — from India and Greece to Egypt and China: the circumference of a circle is always proportional to its diameter, no matter how large or small the circle is.

That is, the ratio of the circumference of a circle to its diameter always yielded a constant that was independent of the dimensions of the circle. The Welsh mathematician William Jones was the first to denote this proportionality constant by the symbol π in 1706, picking it as the first letter of the Greek word for perimeter (περίμετρος, perimetros) and for periphery (περιφέρεια, periphereia). The notation was popularised by Leonhard Euler in the 1730s and 1740s, and it has stuck ever since.

No, Pi Isn’t 22/7 – Here’s Why

Many of us were taught in school that pi is twenty-two divided by seven. However, that's only an approximation — pi is something more subtle than that, and it definitely isn't equal to 22/7.

Pi is an irrational number.

So in essence, it cannot be expressed as the ratio of two integers that have no other common factor other than one. Then, why 22/7 you ask?

Well, this is actually just an approximation.

22/7 ≈ 3.142857, whereas pi ≈ 3.141592 — the two agree to the second decimal place and diverge at the third. For its highest-precision interplanetary navigation, NASA uses only the first 15 or 16 digits of pi (3.141592653589793); even at the scale of the solar system, more digits than that don't appreciably change the answer.

Imagine if they had done the Apollo calculations with 22/7: that ~0.04% error, over a quarter of a million miles to the Moon, would have been enough to make Neil Armstrong and Buzz Aldrin miss the lunar surface by more than 150 km.

Is Pi Infinite? Why?

When Swiss mathematician Johann Heinrich Lambert proved, in 1761, that pi is irrational, the fact that its decimal representation is infinitely long came along at the same time. The reason is that all irrational numbers, by definition, have infinite, non-repeating decimal expansions. Roughly a century later, in 1882, Ferdinand von Lindemann went further still and proved pi is transcendental — meaning it isn't the root of any polynomial equation with rational coefficients.

Pi belongs to the group of transcendental numbers — numbers that are not the root of any non-zero polynomial equation with rational (or equivalently integer) coefficients. Every transcendental number is automatically irrational, because any rational number p/q is the root of the simple polynomial qx − p = 0 and is therefore algebraic of degree one.

We just noted that irrationals can't be expressed as a ratio of two integers — and that's exactly why their decimal expansions are both non-terminating and non-recurring (the digits never end and never settle into a repeating pattern). The argument runs the other way too: any finite decimal, say 0.2378, can be written as 2378/10000 or 1189/5000.

So a finite decimal can always be expressed as a fraction, which makes it rational. A repeating decimal (e.g. 0.333… = 1/3) is also rational. For a number to be irrational, then, its decimal expansion has to be both infinite and non-repeating.

Don't confuse pi's infinite decimal expansion with an infinite value. Pi itself is a finite number — its value is bounded between 3 and 4 (in fact between 3.14 and 3.15) — but the decimal string we write down to describe it never terminates.

3 < π < 4

Hence, pi is a real number, but since it is irrational, its decimal representation is endless, so we call it infinite.

How Do We Calculate Pi?

There are numerous ways of calculating pi. You can draw a circle, then measure its diameter using a ruler and its circumference using a piece of string. Now, divide the circumference of the circle by its diameter and there you have it—the value of pi! Voila!

The above method only gives us an approximate value of pi that is somewhere near 3. When I said that NASA only uses about 40 digits of pi, the simplest question that must have popped up in your head must be how did we come up with those 40 digits of pi?

Mathematicians all over the world have worked over the centuries to develop theorems and formulae that help us in calculating pi. The most respected methods for finding pi are:

1. Gregory- Leibniz
2. Newton
3. John Machin
4. Wallis
5. Ramanujan

The Gregory- Leibniz, Newton and Machin series are simpler to grasp and easier to comprehend, whereas the other means of calculating involve higher level of mathematics. A fun fact about pi

How Pi Is Related To ‘G’ – The Gravitational Constant

Take out your calculator and compute the square root of the acceleration due to gravity.

== ~

The square root of g is almost equal to the value of pi! Is that a coincidence?

There is an equation that typically depicts the relation between the time period and the length of a pendulum.

For an ideal pendulum of length one meter, the time period is 2 seconds. Boom! Using T=2 and L=1, we get,

π² = g

Pi is thus related to gravity!

The Life Of Pi

As you can see, pi is indeed an influential part of our lives! So influential, in fact, that it has a day of its own—pi day.

Celebrated on the 14th of March every year since its date stamp is 3.14, pi also features alongside 0, 1, e, and i in Euler's identity (e^iπ + 1 = 0), often called the most beautiful equation in mathematics for tying five of its most important constants into a single line. Even the speed of computers is decided by how fast they can calculate the value of pi. The current world record for the most computed digits of pi stands at about 202 trillion digits — achieved in June 2024 by Jordan Ranous and the StorageReview team using a single high-end server running the Y-cruncher algorithm.

The life of pi is as endless as its decimal expansion. The journey of pi has just begun and there are plenty of mysteries still to be unravelled!