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
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.
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.
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