No, sqrt(2) is exact and no approximation is needed -- it's the diagnoal of a 1x1 square. Goats and grass don't care about decimal number systems.

As the article says, the solution to the goat problem isn't practically constructible:

"It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"

>No, sqrt(2) is exact and no approximation is needed -- it's the diagonal of a 1x1 square.

That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.

Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.

But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge.

Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.

But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.

You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.

A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)

> one that happens to be able to construct the goat-grass area exactly.

How? How did you get a string of the correct length?

Note that the tool you described is a compass.

> compass and straightedge

But those are not symbols. We're talking symbols (digits).

Who are we? The problem is to create a leash of a certain length. If you can make a leash exactly one unit long, then you can also make a leash exactly √2 units long; the fact that the ratio of the two numbers is irrational doesn't impede this in any way.

To remind, the context is:

> you can't write it down as a decimal

No, the context is whatshisface claiming that there is a (resolvable) tension between the concept of an exact representation and an infinite decimal expansion, and gowld responding that decimal expansions are irrelevant. That doesn't support the idea that we're talking about digital representation; it undermines it.