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)

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.

A rational number refers to a number that can be expressed by dividing two whole numbers. Say, 219/53 for example. There are no two whole numbers which can be used to represent both the diameter and the circumference of a circle.

In fact, if the diameter is a whole number or even a rational number, then the circumference can only be an irrational number with infinite digits. Also vice versa.

(Although 355/133 is a pretty close approximation of pi for most practical uses. This is the best approximation with fewer than 5 digits in the denominator and numerator.)

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