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
You seem to be having a bit of confusion about “an infinite number of decimal places” versus “infinity, the concept”.
I’ll start with “infinity, the concept”, then I’ll try to explain “infinite number of decimal places”, and how the two are different, and why an infinite number of decimal places will not make your number “reach” infinity.
Firstly, infinity, the concept. It’s an idea mathematicians most often use to describe a number so big, that it’s bigger than all other numbers.
An infinite number of decimal places – such as what pi has – does not make the number the same “size” as infinity. Let me explain this with a hopefully simple example.
Take the numbers 2.5 and 2.6. You’ll understand that 2.5 is less than 2.6. So, let’s take 2.5 and add another decimal place to it. 2.59. Still less than 2.6. Ok, let’s add another decimal place to it. 2.597. Still less than 2.6.
If you keep adding decimal places to that original 2.5, you’ll never get a number bigger than 2.6.
This same logic is what keeps pi’s infinite decimal places from ever being able to make pi equal infinity, even though pi has infinitely many decimal places. Pi won’t even get bigger than 3.15, because its decimal places start with 3.14.
Now, because we know pi has a size bigger than 3.14 and smaller than 3.15, we can know the radius of the circle in your example is somewhere very close to (or precisely) 25.00cm².
Pi’s infinite amount of decimal places will not make the circle’s area infinite, but it can make the circle’s radius and / or area have an infinite number of decimal places as well.
Even if we were to construct a universe-sized circle out of subatomic particles, there would still be inconceivably small imperfections. This is why pi is irrational, because we’re trying to use math to describe a perfect circle, but there’s no such thing.
As we continue to calculate pi into infinitesimal amounts of digits, all we’re really doing is creating units of measure far beyond the scope of known existence. We can’t even observe anything small enough at this time to make use of the most accurate calculations of pi that we have. The endless nature of pi doesn’t mean a circle has no limit, but that our ability to calculate that limit is flawed.
Your question is like asking how can an ocean dry up if there is still a single water molecule lodged in a microfissure in a grain of sand. At a certain point, it just isn’t practical to care about it anymore.
My isn’t unlimited/infinite. It is an exact value between 3 and 4. And we *can* get exact value for areas and circumferences if we state them in terms of *pi*. However, if we want to represent them in other ways, then we will likely have to round.
But, there are trivial ways of getting exact values for areas and circumferences. For example, the circumference of a circle whose radius is 1/pi is exactly 2.
25.00cm² doesn’t mean *exactly* 25cm². It means somewhere between 24.995cm² and 25.005cm². If we used more digits of pi (and knew the radius/diameter precisely), then we could get better precision. But in practice we only use as many as we need. Since every digit gives us 10x as much precision, we only ever need the first 30 or so digits. That’s enough to calculate the circumference of the entire observable universe to be accurate to the size of a hydrogen atom.
> how can we say that a circle has an area of 25.00cm2 if pi is irrational??
3 < pi < 4. Therefore, 3 r^(2) < pi r^2 < 4 r^(2). The size of the circle is bounded and therefore finite. The sharper you bound pi, the better the estimate of the area will be. With a sharp enough bound, the difference between the estimated area and the actual area will be too small for you to measure. At that point, does it really matter?
> How can a circle, a closed shape, have a limited area if pi is unlimited??
pi is not infinite. It has an infinite decimal expansion. These are two completely different concepts.
An infinite decimal expansion just means that you cannot write the number exactly as digits. The number has an exact, finite value, it just takes infinitely many digits to write it down on paper. No matter how many digits you write down, the number on the paper will always be just a bit higher/lower than the actual value. Adding more digits gets you closer to the exact value, but you’ll always be just a little bit off.
Pi isn’t special here. All irrational numbers have this property. If you only use base-10, even some rational numbers have it. 1/7 has an infinite decimal expansion in base-10, for instance.
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).
It’s unlimited precision, more precise than could ever be useful in the physical world. It’s actually the fraction 22/7, so do your math in fractions instead of decimals and you get the exact result.
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The exact level of pi continues however pi has been calculated so accurately so far that a circle extending out to the entire limit of the universe would be out by less than an atom.
It all depends on what precision you need. Of course, theorically, you’ll never be able to get an totally exact value for an area from the radius or diameter.
But you can get values exact enough for your use.
For a circle with a radius of 5cm, you have an area of
78.5cm2 if Pi = 3.14 or
78.5398163397 cm2 with Pi = 3.141592
Thats 7853.98163397 mm2. And probably 0.0000000097 mm2 could be negligible.
Not precise enough ? Take pi = 3.141592653589793238462643383279502884197169399375105820974944592307816406286. The imprecision here should be around an helium atom diameter…
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