why is pi called an irrational number

Pi is called an irrational number because it cannot be written as a fraction of two whole numbers, and its decimal expansion goes on forever without ever falling into a repeating pattern.

What “irrational number” means

• A rational number is any number that can be written as a fraction a/ba/ba/b, where aaa and bbb are integers and b≠0b\neq 0b=0.

• In decimal form, rational numbers either terminate (like 0.5) or eventually repeat (like 0.333… or 2.145145145…).

• An irrational number cannot be written as such a fraction, and its decimal digits continue forever with no repeating block or pattern.

Pi falls into this second category.

Why pi is irrational

Pi is defined as the ratio of a circle’s circumference to its diameter: π=C/d\pi =C/dπ=C/d.

Mathematicians proved that no matter how precisely you measure, this ratio is never exactly equal to any fraction of integers like 22/7 or 355/113; those are only approximations.

Key facts about pi’s decimal form:

• It never ends: computers have calculated over 100 trillion digits and still found no end.

• It never repeats: there is no recurring block of digits, which rules out the possibility that it represents a rational fraction.

Historically, Johann Lambert gave the first rigorous proof that pi is irrational in 1768 using properties of the tangent function, showing that if tan⁡(x)\tan(x)tan(x) behaves a certain way for rational xxx, then π/4\pi/4π/4 cannot be rational, and hence π\pi π is irrational.

“Irrational” and “transcendental”

Pi is not just irrational; it is also transcendental , meaning it is not the root of any polynomial equation with integer coefficients.

All transcendental numbers are irrational, so proving pi is transcendental automatically confirms that it must be irrational as well.

Quick intuitive picture

• Imagine walking around a perfect circle and measuring its circumference and diameter more and more precisely.
• You keep trying to express C/dC/dC/d as a fraction, refining your guess: 3, 22/7, 355/113, and so on.
• Each time, the fraction gets close, but never exactly matches the true ratio; there is always some tiny mismatch left over.

Because no fraction ever hits that value exactly, and because the decimals of pi go on forever without repeating, we call pi an irrational number.

Bottom note: Information gathered from public forums or data available on the internet and portrayed here.