Is Pi a Rational or Irrational Number?

Pi (π) is an irrational number because its decimal expansion is non-terminating and non-repeating. It starts 3.14159265358979… and never ends or forms a pattern, which means π cannot be written as a simple fraction p/q where p and q are integers and q ≠ 0; even though we use 22/7 or 3.14 as convenient approximations in school, they are not equal to π. Pi is a magical number that has puzzled mathematicians for centuries, and the answer to whether it can be expressed as a simple fraction is a definitive no, which is exactly what makes π irrational. In this guide you will learn what it means for a number to be rational or irrational, why pi (π) is irrational with clear reasoning about its decimal expansion, why 22/7 and 3.14 are only approximations and not equal to π, how the circle ratio (circumference ÷ diameter) leads to π, and common exam questions and how to answer them confidently.

 

Table of Contents

• What is Pi (π)?
• Why Pi is Irrational?
• Proof: π is Irrational
• Why is 22/7 Not Equal to π?

What is Pi (π)?

Pi (π) is the ratio of a circle's circumference to its diameter. 

Definition of Pi:

π = Circumference ÷ Diameter

π = C / d     (where d = 2r)

π ≈ 3.14159265358979323846…

This ratio is identical for every circle, regardless of size.

The Greek letter π (pronounced "pie") was first used to represent this ratio by Welsh mathematician William Jones in 1706. The famous Leonhard Euler then popularised the notation in the mid-1700s, and it has been universally used ever since.

 

Why Pi is Irrational?

A rational number, when written as a decimal, must either stop at some point or eventually start repeating a pattern.

Pi does neither. Let's look at its digits.

First 100 decimal digits of π:

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510

     5820974944 5923078164 0628620899 8628034825 3421170679…

There is no pattern, no repetition and no end. The digits of π have been computed to over 100 trillion decimal places by computers and not a single repeating cycle has appeared.

The three reasons π is irrational:

1. Its decimal expansion never terminates (it goes on forever)

2. Its decimal expansion never repeats (no recurring block exists)

3. It cannot be expressed as a fraction p/q of two integers

Proof: π is Irrational

The formal proof that π is irrational was first given by Swiss mathematician Johann Heinrich Lambert in 1768. While the complete proof involves calculus, here is a clear, student-friendly version of the argument.

 

Proof by Contradiction (Conceptual)

Assumption: Suppose π is rational.

Then π = p/q, where p and q are integers, q ≠ 0, and the fraction p/q is in its lowest terms (no common factors).

Every rational number p/q, when written as a decimal, must either terminate or eventually repeat.

But we know that π = 3.14159265358979… has decimal digits that never terminate and never repeat.

Which is a contradiction. Therefore, π cannot equal p/q for any integers p and q.

∴ Our assumption was wrong.

∴ π is NOT rational. π is irrational.

 

Why is 22/7 Not Equal to π? 

22/7 is only an approximation of π, accurate to 2 decimal places.

22 ÷ 7 = 3.142857142857142857… (the block '142857' repeats forever)

π = 3.14159265358979323846… (no block ever repeats)

The two decimals differ from the third decimal place: 22/7 gives …42857…, while π gives …15926…

Since 22/7 is rational (it has a repeating decimal) and π is irrational (it does not repeat), they cannot be equal.

∴ 22/7 ≠ π. 22/7 is only an approximation.