I find school somewhat slow and my constant questions used to annoy the teacher…i now have teachers that ignore me. I instead sleep. 
Cool…this is probably the best “class” i’ve had!!
I will look into them…
As long as you at least give the impression that you are reading the answers and trying to understand them, you will find that people on this board actively enjoy answering questions.
Acording to Bill Plummer III, Manager of the National Softball Hall of Fame in Oklahoma City, the name “softball” was coined in 1926 by Walter Hakanson. Details here.
Its difference is best described by Giuseppe Peano, who developed the system of arithmetic that is in popular use today. Peano’s Third Axiom states that zero is not the successor to any number. (He establishes that zero IS a number in his First Axiom.) Details here.
The reason why division by zero is undefined is because it violates the Law of Noncontradiction, a fundamental axiom of first order logic. Details here.
Actually, ball bearings were used in ancient times, but they were made of wood. The technology required for metal ball bearings was not possible until many technologies came together to allow for precision machining. Details here.
The reason why pi is irrational is that there is no integer relation between the diameter of a circle and its circumference. A circle is a planar figure, and plane coordinates are in integers. That’s why a circle can’t be squared in Euclidean space. But it can be squared in Gauss-Bolyai-Lobachevsky space. Details here.
Libertarian said:
Could you explain what you mean by this?
It just means that all the points of a circle are on the same plane.
It was more the second part I was thinking of, “plane coordinates are in integers”.
In any given plane, an arc between two points can not be measured exactly.
Yep. That’s it. When the plane is Euclidean.
I’m afraid I’m still not quite seeing it. Could you explain it in a little more detail?
I thank Libertarian for giving me some reading material for me to ponder over.
Pi is an irrational (and transcendental) number. That’s not the same as being “inexact.” Pi has an exact value, but that exact value cannot be expressed in a finite number of terms in decimal notation. See Dex’s Straight Dope Staff Report Is pi an inexact number?
It’s not possible to zoom in on the curve and focus, you hit that whole irrational number thing. The closer you look, the more you run into that damn lazy “8”…
Consider that no transcendental numbers are constructible, and that constructible numbers correspond to line segments. Line segments are defined as having endpoints on the plane.
As far as I can tell, Libertarian and Darth Nader, your statements refer to the limits of accurate measurement. This is a perfectly worthwile subject, but it has nothing to do with the mathematical question of whether or not we can calculate the length of the arc of a circle, still less what it means to say that “plane coordinates are in integers”. Do the coordinates (0.3,0.5) not define a point in the Euclidean plain?
Well, yes. But the question by the OP was in the context of accurate measurement, specifically areas and volume.
OK, I understand the point you were making now. Thank you for your time.
I know, I messed up on the stupid [symbol]Ö[/symbol] character. Forgot to use Symbol font, and it didn’t go through. :o
Points in the plane have integer coordinates? What?
That’s not what I meant. Given that you could take that interpretation, I worded it very badly.
I was trying to comment succinctly on the transcendental nature of pi when circles are rendered on planes of rectangular coordinate systems defined by integer units of distance.
If, on such a plane, a circle is drawn and the unit of measure is set to the radius of the circle, then the circle will have an area of pi. Constructing a square of equal area is impossible because the square’s side would have to have a length of [symbol]Ö[/symbol]pi.
The reason I used squaring circles on integer coordinate systems as illustration is because the OP was drawing a comparison between circles and other figures whose areas can be expressed as ratios of integers.
I thought the fact that a circle cannot have all rational coordinates on its circumference would help to illuminate the problem.
I regret the poor and misleading wording. My apologies to everyone in General Questions and to its moderators.
No problem. I didn’t read the last few replies, and now that I have, I understand what you meant.