# Am I wrong? Is this correct thinking? Am I weird? Am I crazy?

I was counting sheep last night and I change my vote. I can give you 1 sheep, you cannot give me pi sheep, eve if you cut a 4th sheep in half, there are a finite number of atoms in a sheep, so they cannot divide in (pi - 3)|(4-pi) You could then split the atoms but there is still a finite number of particles that make them up. You could argue this comes down to measuring, or you could argue this comes down to "Is matter infinitely divisibe?" Which I would think it is not.
That way, I also can't give you 1/3 sheep...

So I would say repeating numbers are equally as irrational as pi, but less "real" than 1, or other finite decimal numbers.

"Irrational" is a bad word. It has a definite meaning in mathematics.

But finite decimal numbers have the same problem. I can't give you 1/2 or 2/10 sheep as well...

Ah I see, unless it is an even-number of atoms in the sheep..which is not guaranteed.

I disagree on that as well. Pi still behaves as any other number in your example, and here's why I think that:

If we define the radius to be one, I agree that we cannot accurately measure the circumference, if we are talking about a mundane form of measurement. I also agree that if we define the circumference to be 1, we cannot mundanely measure the radius of that circle accurately. By mundanely, I simply mean with an apparatus alone.

Do you agree that if I define the edge of a square to be 3, we cannot accurately, mundanely measure its area? Apparatus alone, could we get its exact area to an infinite number of decimal places? No.

It is only by a mathematical relationship that we know the area of that square is exactly 9.
And I also know by a mathematical relationship that the circumference of your circle is exactly 2pi when the radius is our unit standard.
I think I see it simulary to you. Mathamaticaly the length units "work well" with length units.

I think curves complicate this, perfect circles infinitely so. (i.e. circumference / diameter) Lengths don't divide well into curves. Circle gets the square!

I certainly don't think Pi can be thought of as a physical unit.

Consider the way we measure the length of string of length 10 cm:

You can start by placing your ruler down, and starting at 0 cm, count by 1 cm until you get to the end of the string, and at at 10 cm - which is equivalent to adding 1 + 1 + 1... + 1 until you get 10.

Suppose it was now 10 and 1/3 cm:

You add 10 cm together, and then add an extra 1/3 cm. It doesn't matter if you can't write down what 1/3cm is in terms of mm or any other decimal, physically you can divide 1 cm into exactly three pieces - there's nothing stopping you from doing that.

Now try to get to 10 + pi cm - you start with adding 10 cm up, then you can add three more, then you have to add 1 mm, and then a little more, and a little more, and just a little bit more... and so on forever. You have to add an infinite number of smaller and smaller measurements to ever achieve that extra pi cm. You can never actually measure out an irrational physical quantity - and hence I don't think it's possible for physical quantities to obtain irrational values.

The fact that the billiard ball doesn't have an irrational surface area is because it's not a perfect sphere, and never will be. Just like you can't ever draw a perfect triangle with a hypotenuse of Sqrt(2).

I certainly don't think Pi can be thought of as a physical unit.

Consider the way we measure the length of string of length 10 cm:

You can start by placing your ruler down, and starting at 0 cm, count by 1 cm until you get to the end of the string, and at at 10 cm - which is equivalent to adding 1 + 1 + 1... + 1 until you get 10.

Suppose it was now 10 and 1/3 cm:

You add 10 cm together, and then add an extra 1/3 cm. It doesn't matter if you can't write down what 1/3cm is in terms of mm or any other decimal, physically you can divide 1 cm into exactly three pieces - there's nothing stopping you from doing that.

Now try to get to 10 + pi cm - you start with adding 10 cm up, then you can add three more, then you have to add 1 mm, and then a little more, and a little more, and just a little bit more... and so on forever. You have to add an infinite number of smaller and smaller measurements to ever achieve that extra pi cm. You can never actually measure out an irrational physical quantity - and hence I don't think it's possible for physical quantities to obtain irrational values.

The fact that the billiard ball doesn't have an irrational surface area is because it's not a perfect sphere, and never will be. Just like you can't ever draw a perfect triangle with a hypotenuse of Sqrt(2).
Again, counting exactly 1 cm and counting exactly pi cm are equally daunting tasks. You can measure 1.00000000000 cm perhaps, but that's no easier than measuring 3.141592653589 cm. Both are approximations to 1 and pi respectively when we are talking about measurement. 1.0 cm is not equal to exactly 1 cm, it means we stopped measuring after millimeters. You can't just ignore that in the rational number's case.

The issue here that computations with pi (and irrational numbers) can only work with rational approximations. This is a technological limitation, it has nothing to do with "not knowing pi exactly"; we do know pi exactly, we just can't work with it exactly. It's just a number.

Deveno
assuming we know how to define it, any real number is "knowable" in the sense that we can compute it to any desired accuracy.

i mean what is a mathematical relationship?

when we say a = f(b), for some formula involving b, there is a tacit appeal to a and b being "the same kind of something". if our formula is:

A = $\pi$r2, than A is not a rational number. so we have to have some kind of notion that allows us to say what equality of two real numbers IS.

analytically, we say that A converges to $\pi$. some people have trouble resolving this notion of convergence to our everyday notion of "is-ness". probably because it relies on the notion of a "completed infinity" ($\pi$ is not finitely expressible as a rational, it is an infinite sequence of such rationals).

my point being, saying a = b, when a and b are two real numbers, sweeps a lot of complicated machinery "under the rug", the truth of such a statement (and how complicated it actually is) is hidden behind a formalism which makes it appear as if it is just as simple as a statement like 2+2 = 4, or 1/3 + 2/3 = 1.

to go a bit further, how does one actually prove that the formula for an area of a circle actually holds? if you can show me a derivation that does not, in fact, rely on some limiting process (and thus a "completed infinity"), i will withdraw my statements.

pwsnafu
to go a bit further, how does one actually prove that the formula for an area of a circle actually holds?
Before one can do that you need to define the problem of finding "area of a circle" without analysis.

Deveno