# Why isn’t pi a rational number when it’s circumference/diameter?

155 views

I remember asking my math teacher this when we first learned irrational number but they said they didn’t know. Not sure if it’s just a stupid question or not.

In: 10

A rational number is a number that can be written as a ratio specifically of two whole numbers. There’s no circle whose circumference and diameter are both whole numbers.

(edit: unless you count a single point as “a circle with radius 0,” since the diameter and circumference would both be 0, but in that case the ratio between them would be undefined)

A rational number must be able to be expressed as the fraction of other two rational number. but it’s impossible to get both circumference and diameter rational at the same time so you will get rational/irrational or irrational/rational, the result of both case will be irrational.

A rational number is one that can be expressed as the ratio of two integers. Pi cannot be expressed as the ratio of two integers. For any circle, either the diameter or circumference must be irrational, or both.

Oh I forgot to also include that I’ve heard that irrational numbers don’t exist in real life is that just like saying there’s no perfect circle?

I was surprised that proof that π isn’t a rational number is so recent:

> In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction *a*/*b*, where *a* and *b* are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus.

https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

Unlike, say, that √2 is irrational, which has been known for ages.

Is there any reason why it *should* be a rational number?

Is there any reason why the length across a circle should be an exact fraction or multiple of the length around the outside of a circle?

Similarly, if you draw a square, and the diagonal across the middle, is there any reason why that diagonal should be a multiple of the length of the side of the square?

To respond to another question you’ve asked, irrational numbers do exist in the real world to the extent that any other number (maybe other than 1 and 0) exists. You cannot have exactly 1/2 of something, and without getting into too much philosophy you cannot have 3 of something (either you have one thing, and a different thing, and another thing, or you have one three-thing). Numbers are mathematical constructs that are useful for understanding the world.

Definitely not a stupid question- in fact, others have thought the same way and tried to make pi rational. Back in the day, lots of places around the world used the fraction 22/7 for pi. Since math class doesn’t really require 100% accuracy with very intricate circles, this was considered “close enough” for students to use.