E.g. how can we say that a circle has an area of 25.00cm^2 if pi is irrational?? How can a circle, a closed shape, have a limited area if pi is unlimited??
In: Mathematics
The decimals of pi go on forever. The more decimals of pi you know, the more precisely you can do your calculations.
If I give you the radius of a circle, say 2 cm, and I ask you to tell me its area, you’ll have to use an estimate of pi to some level of precision. A very rough approximation of pi would be pi=3, and using that estimate you get an estimate of the area of the circle that is 3*2^(2) = 12.
A better approximation is pi=3.1. This leads to an area of 3.1*2^(2)=12.4. We can keep going:
pi=3.14, area=12.56
pi=3.141, area=12.564
pi=3.1415, area=12.5660
pi=3.14159, area=12.56636
The more decimals of pi that we know, the more precisely we can also calculate the area of the circle. You can also see that the area of the circle isn’t *infinite*. It’s around 12.5, which is decidedly *finite*.
Is there some infinity somewhere? Yes, in a way. Because the decimals of pi go on forever, so do the decimals of the area. That is, you can never say “the area of the circle is exactly X”, no matter how many decimals you give for X. There’s always another decimal, and another, and another.
In practice this doesn’t really matter. We rarely know numbers with perfect precision anyway, unless we’re talking about counts. E.g. I can know that there are exactly two coffee mugs on the table in front of me. But can I tell you *exactly* how much water each of them can hold? No. Any measurement method I might use always has some limited precision to it (e.g. maybe I can tell you to the nearest ml).
So in practice, you don’t actually know the radius of the circle perfectly either. You don’t know that it’s exactly 2 cm, because the calipers you used to measure this radius have a precision of, say, 0.1 mm. So in practice you know that the radius is somewhere between 1.99 and 2.01 cm. Even if you knew pi exactly, you couldn’t calculate the area more precisely than your original measurement allows. For instance, imagine a universe in which pi=3, then your calculations in this case would tell you that the area of this circle is between 12.44 and 12.69 cm^(2).
For practical purposes, therefore, it is enough to just know the first N decimals of pi, and you can pick N in such a way that the result you get is precise enough for your purposes.
(By the way, there are some theoretical examples where you *can* know the area of circle directly. For instance, suppose we have a circle with radius r = 1/sqrt(pi). Then r^(2) = 1/pi, and therefore the area of the circle equals pi * r^(2) = pi * 1/pi = 1.)
As a final note, this isn’t just true for pi, or for circles. For instance, suppose we have a square with sides of length pi. Its area would be equal to pi^(2). Since we don’t know pi with perfect precision, we can’t know the area of this square perfectly either. Or, let’s say we have a rectangle with one side of length sqrt(2) and one side of length 3. sqrt(2) is an irrational number as well, so again we cannot know the area of this rectangle perfectly, because it would be 3*sqrt(2).
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