Finding the exact area of a circle?

Question

Background:

I recently began taking calculus and it has come to alter the way I look at circles, and curves. The equation of a circle is $\pi r^2$, traditionally in school we have always left the answer in terms of pi (i.e.) if the radius $r=2$ then the area $A = 4\pi$.

Question:

If one were to attempt to write the area of a circle in decimal form (i.e.) if the radius=2 then the area $A = 4\pi$, but $\pi$ doesn't have an end, it has (per what I have learned in school) an infinite number of decimal places so it is $3.14159\ldots$ therefore if one multiplied $4\cdot 3.14159\ldots$ one would have to approximate ones answer.

Does that mean that it is impossible to calculate the exact (without approximation) area of a circle?

Thanks for any responses,

Joel

Answer 1

$\pi$ by itself is an exact value as is $10$. But as it happens to be, $10$ and $\pi$ don't math together nicely.

We could've have lived in a base-$\pi$ world where $\pi=10_\pi$. But if you are looking for this value in base-$10$, we get an infinite decimal expansion with no apparent pattern. Similarly, if you wanted to write $10_{10}$ in its base-$\pi$ representation, you would get a weird inconvenient

$$100.010221222211211220011112102..._{\pi}$$

which is just as bad as what $\pi$ looks in base-$10$. But both $\pi$ and $10$ have exact values associated with them. Attempting to write a number in base-$10$ just give one particular representation of that value. Particularly a representation we are more comfortable with, but is not ideal for every value. For a value of $1/3$, the decimal expansion is also unfortunate, and so we keep it in its fraction representation.

So something like $4\pi$ is the exact value for your circle, just not in a base-$10$ format.

Answer 2

Yes, you can't express the exact area of a circle with radius $r=2$ (exact). However, this doesn't seem a problem in real life, because in real life you never know the exact value of any quantity (except natural ones...). That applies to the radius, too, so given a radius known with a limited precision you can determine the circle's area only within a limited precision, thus considering a never ending chain of digits is useless. Anyway, we have no tool and no time to write an infinite decimal expansion.