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This is a request for help, not an attempt to challenge anything.

Since $\pi$ is irrational, this tells me that it's impossible to express the distance around a circle in terms of the distance accross.

That boggles my mind, but maybe it should not.

I think crazy thoughts like: "this means that a the path of a circle around a unit lenth line segment has a non existant length".

Is there a way to accept that the number is irrational and not break from reason?

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Knowing what rational and irrational actually mean would be a good start. The irrationality of $\pi$ certainly does not mean that it's impossible to express the circumference of a circle in terms of the diameter. – Chris Eagle Feb 22 '12 at 17:15
apologies.. maybe i have to delete this – Aaron Anodide Feb 22 '12 at 17:16
Why it tells you that it is impossible to express it? You can express it using $\pi$. You can't say that is is a nonexistant length it is just a length that is not a fraction. Like the length of the diagonal of the unit square. Nothing special. – savick01 Feb 22 '12 at 17:17
@PradipMishra We do not call numbers without decimal representation irrational. ($1/3=0.33333\ldots$ is rational but has no representation) We call numbers irrational if they are not rational, so they are not a ratio $a/b$ of some integers $a,b$. – savick01 Feb 22 '12 at 17:24
It's of interest to note that the modern connotation of "irrational" arose from the consternation expressed by those who first realized $\sqrt 2$ is not a ratio (of integers), at least according to some sources... – David Mitra Feb 22 '12 at 17:41

2 Answers 2

Some numbers are indeed impossible to express. Those are the numbers that are not definable. Easier to deal with, but still difficult, are numbers that are not computable.

However, it is not so difficult to provide a clear definition of the quotient of the circle's circumference by its diameter. We can let the matter stand there and simply name the quotient "pi", or we can do like Archimedes and define that quotient to be the limit of the regular polygon's perimeter to its diameter (using either choice of the inner or outer diameter) as the number of sides approaches infinity. We have the epsilon-and-delta definition of limits to help convince us that this is a valid definition.

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I disagree. Some non-computable numbers are expressible. – Michael Hardy Feb 22 '12 at 17:23
thank you for the help – Aaron Anodide Feb 22 '12 at 17:25
Michael Hardy, I'm sure your objection is correct. I have edited my answer in an attempt to be more accurate. – minopret Feb 22 '12 at 17:29
out of curiousity, what is an example of a non-computable yet expressible number? – Aaron Anodide Feb 22 '12 at 17:30
Chaitin's omega is expressible but not computable because it measures halting probability. The halting problem is famously not solvable by computation. – minopret Feb 22 '12 at 17:32

That $\pi$ is irrational does not mean that it's impossible to express the distance around a circle in terms of the distance across. It only means it can't be expressed as a ratio of integers.

BTW here you can read some proofs that $\pi$ is irrational.

It is quite easy to prove that $\sqrt{2}$ is irrational. It's even easier to prove that $\log_2 3$ is irrational. But it's hard to prove $\pi$ is irrational.

Here is a related question that may be of interest, where you will read about how to prove some numbers are irrational.

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