You can't even tell me what value pi has unless you TRUNCATE pi...
3.1 > 3.0
3.14 > 3.1
3.141 > 3.14
3.1415 > 3.141
3.14159 > 3.1415
See? For every decimal place you add another number to pi, it gets larger in value.
The truth is that pi is none of the values you have written. pi is the number you get if you keep adding decimals
forever. This does not actually require that a person sit there adding decimals forever, by the way. We smart humans
understand that, as we keep adding decimals, the value of pi
converges to a particular value, even if we can't write that value out as a decimal, because it would need an infinite number of decimal places to do that.
All of the values given above are
approximations to pi. What you get when you truncate the
actual pi to a certain number of decimal places is no longer pi, but a number a little less than pi. pi is the 'limit' you get when you keep all the decimal places (i.e. an infinite number of them).
It is a failure of imagination to assume that pi must have only a finite number of decimal places. There is no reason to make such an assumption. Besides, it doesn't work.
You can not tell me a FINITE pi value, so you don't even have a value for pi, which means you have no finite area for the circle...
Be careful that you use the words "finite" and "infinite" correctly.
pi is a finite number. It is certainly a number greater than 3.140 and less than 3.142, for instance. An actual infinite number would be greater than any number you cared to write down.
It is true that it would require an infinite number of decimal places to write out pi exactly. That means that, in practice, we can't write out the exact value of pi as a decimal. But it doesn't matter. There's no reason we'd ever need to do that. What we can do is to write it out to as many decimal places as we're willing to wait for a computer to calculate. You can decide how many decimal places you want and how long you're willing to wait to get them, and you can get a VERY good approximation to pi.
The circle's area is FINITE! If the circle's area is finite that means pi MUST be a finite number!
Right on both counts. pi is a finite number, as previously explained. Areas of circles are finite too. No problem.
What is that finite number that you claim pi to be in order for the circle to have a finite area????
It's that number between 3.14 and 3.142 that I just talked about.
So according to you the circle is too small too, but it is growing with pi with every decimal place for pi! (rolls eyes)
No! The area of the circle is pi times its radius squared. It is what it is. It's not
our fault if
you can't calculate it to the degree of accuracy you think you need.
In practical terms, if you're drawing a circle on a piece of paper with a pencil, how accurate do you want the area, anyway? How accurately are you going to measure the width of the pencil marks, for example? If you get that wrong, it doesn't matter how accurate the pi value is you use: if your radius value is incorrect, you'll find the "wrong" area (though it might be "close enough", depending on your aims).