If Pi is continual and random-digited as scientists say, then how could it be the quoteint of the circumference and the diameter, because any thing that includes a fraction cannot be continual and random digited. The only explaination that keeps Pi continual and random digited is that the diameter and radius cannot be perfectly measured or that a perfect circle cannot be made. However pretty much any thing can be perfectly measured down to the very last atom. so circles must be unmakeable. But if circles are unmakeable, then there would be no way to caculate Pi. Would someone clear this up for me?
Anything of the form a/b where a and b are both integers (or by extension rational numbers) cannot be an irratinoal number like pi, howvere this doesn't apply to a/b when a and b are both real numbers. So what this means is that pi can be expressed by two numbers in the form a/b, but both those two numbers cannot be integers (or rationals).
No, you can't! A circle is not comprised with the atoms making up the circumference but with the points making up the circumference. Points have no dimension. A small circle would still have a slightly different ratio of circuference to diameter than a larger circle if you count the atoms. If you count by points you'll get an infinite length so therefore pi is gotten by approximation of series or other similar methods.
1-You can never measure to exactness, there will always be some uncertainty in the measurement you make even with a supermicroscope that could measure to the atom. 2 -a perfect cirlce is a CONCEPT, not a physical object, just like numbers.
Microscopes and physical measurements have NOTHING to do with a circle or the value of pi. A circle is, like all things in mathematics, a mathematical concept. It doesn't exist in real life. Circles, lines, planes, numbers or whatever are not physical entities. Quantization of the universe or whatnot has no influence whatsoever on mathematics (and the value of pi in particular).
It is misleading to say that the digits of the decimal representaion of [itex]\pi[/itex] are random since they can, in principle, be determined. Nevertheless, they are not cyclic.
Isn't this sort of like the uncertainty principle in quantum mechanics? And, to answer your question: what about a line with measure {sqrt(2)}? This must exist, right?
No, it's not. Define your terms. If you are talking about a "mathematical line" with measure √(2), then yes, such a thing exists. If you are talking about actually measuring a physical line, then, no, there is no physical line whose length is exactly √(2) (or exactly 1, for that matter).
jcsd is strictly right - there's nothing to stop one of the two (diameter or circumference) being any particular value, real or rational, but if one is rational, the other isn't.
There's a misuse of terms here. The measurement of PI has nothing to do with uncertainty. To an arbitrary level of measurement, you can be certain you have measured it accurately. The trouble is, that no matter how finely tuned you measure it, you are always stopping at some level. You will have only measured it to that level of accuracy. But you can always get a smaller ruler... It has nothing to do with atoms. A circle - or any measurement - has an infinitely divisible value. This is in contrast to definitions. 1 foot is exactly equal to 12 inches. Not 12.0 inches, and not 12.00000000000 inches, but *exactly* 12 inches - by definition.