If Pi is an IRRATIONAL number, then why is it commonly defined as...?

Question

..... the RATIO of a circle's circumference to its diameter?

Answer 1

Because it IS. A rational number is not just any ratio, but the ratio of two INTEGERS. You will find that it is quite impossible to construct a circle in euclidean space where both the circumference and the diameter have integer length (in geometric terms, we say that the circumference and diameter are incommensurable).

Answer 2

If π is an IRRATIONAL number, then why is it commonly defined as the RATIO of a circle's circumference to its diameter?

Because that is exactly what π is. But this ratio cannot be expressed as the ratio of two integers.

It is not the fact that it can be expressed as a ratio that makes it irrational, but the fact that it cannot be expressed as the ratio of two integers.

After all √2 is irrational and yet √2 = √12 / √6 = √30 / √15
= 6/ √18 = √50/5..... etc

I think our friend below does not understand what irrational numbers are:

An irrational number can not be expressed as either a repeating or a terminating decimal so 1/3 = 0.3333333333333.... is definitely NOT irrational

Here is a proof of the irrationality of π

Theorem: pi is irrational

Proof: Suppose pi = p / q, where p and q are integers. Consider the functions f_n(x) defined on [0, pi] by

f_n(x) = q^n x^n (pi - x)^n / n! = x^n (p - q x)^n / n!

Clearly f_n(0) = f_n(pi) = 0 for all n. Let f_n[m](x) denote the m-th
derivative of f_n(x). Note that

f_n[m](0) = - f_n[m](pi) = 0 for m <= n or for m > 2n; otherwise some integer

max f_n(x) = f_n(pi/2) = q^n (pi/2)^(2n) / n!

By repeatedly applying integration by parts, the definite integrals of
the functions f_n(x) sin x can be seen to have integer values. But
f_n(x) sin x are strictly positive, except for the two points 0 and
pi, and these functions are bounded above by 1 / pi for all
sufficiently large n. Thus for a large value of n, the definite
integral of f_n sin x is some value strictly between 0 and 1, a
contradiction.

ref: http://www.math.clemson.edu/~simms/neat/math/pi/piproof.html

Answer 3

Because a rational number is not just any ratio, it's a ratio of integers. And, no matter how you construct a circle or what units you use, you cannot make a circle whose circumference and whose diameter can both be measured in the same whole number units.

If you wanted to, you could make up your own system of measurements, where a "circle inch" is the unit you measure the diameter in, and the "circle foot" is the unit you measure the circumference in. But there would be no integer ratio for converting "circle inches" to "circle feet".

The ancient Greeks tried all sorts of experiements along these lines, but got nowhere. Eventually it was proven impossible, by a similar arguement that square root of 2 is irrational, only involving calculus.

Answer 4

It is defined as the ratio of the circumference to the diameter. That number is irrational, ie, not the ratio of two integers. This means you will never have the circumference and the diameter as rational numbers.

It might be eaier to see that the diagonal of a square is an irrational number. Let the square have sides 1. What is the diagonal? Pythagorus tells us it is √2. It is not to hard to show that this is irrational, but our "nice" sides are integers.

Don't confuse measuring versus assigning real numbers to quantities.

Answer 5

Irrational means that the value of Pi cannot be represented as a finite sequence of digits. And it is defined as the ratio of a circle's circumference to its diameter.

Answer 6

You're forgetting one minor fact: an irrational number multiplied by any number will yield an irrational number. Therefore, if a number is represented by a ratio of two unknowns, it does not violate the definition of irrational; it can be safely assumed that one of the unknowns is irrational. Remember that the definition of irrational is "a number that cannot be expressed as the quotient of two rational numbers." If we cannot prove that both the circumference and the diameter are rational, then we cannot prove that pi is rational.

Answer 7

A rational number is defined as the ratio of two integers, or a/b, with the restriction that the denominator b not equal 0. The ratio given is not of this form, for any circle.

Answer 8

That's an excellent question: it drove the ancient Greeks nuts!

They had a real geometric sense of numbers. You intuition is saying that the circum. is a number, the diameter is a number, so the ration must be a rational number. But it ain't.

Look at the early calculations of pi. I like the one where you say that it must be between an inscribed polygon and a circumscribed polygon. Then let the number of sides go to infinity and you get convergence on something that we call pi!

Answer 9

The question is not logical. Pi is that ratio. However, in applications, people need less than perfectly accurate dimensions. So pi is defined as 3.14, or for more accurate situations, 3.1416, etc. People have tried to find out if pi is truly irrational, and it seem to be so.

Answer 10

That's what it is. That means that either the circumference, the diameter or both are irrational themselves.