A circle circumference is composed of a finite number of line segments. An arc continuum is undefined by logic. Is this true? Refutable?
You want to measure the circumference of a circle with a subatomic ruler. Pick a point to start (S) on the circumference. To measure any length, you must move from S to some first point. You can only move in one direction at a time. This is your first segment. Two endpoints in one direction is a line segment. It is not and cannot be curving. If you claim that these little segments are infinite in number, the reverse proof of arithmetic disproves this. An infinite number of any extension would equal infinity, and to say an infinite number of miles and inches is the same is nonsense. This fundamental proof destroys the infinite math conjecture. If the segments are small enough, the eye sees a continuum. OK, and when we speak of a thing as a whole, it is a continuum by that point of perspective. Yet analytically, mathematically, there is no continuum of an arc. What do you think?
- skeptikLv 73 weeks ago
No, because mathematically, what you just described is not a circle.
A figure composed of a finite number "N" of straight segments is a polygon of N sides - an "N-gon." And attempting to measure the circumference of a circle as if it's actually such a polygon will *ALWAYS* result in a total that's shorter than the circle it's attempting to approximate.
Claiming that - at the quantum level - that's how you would measure it, just means you've left the realm of mathematics and entered the world of physical measurement. Which are not the same world.
- JonLv 63 weeks ago
The existence of a continuous arc and the possibility of measuring the length of a continuous arc are two different questions. Disproving the second doesn't disprove the first. And in fact it is possible to define the length of an arc. It's the supremum -- that is, the least upper bound -- of all possible measurements made in the manner you describe.
For more information, consult a text on "measure theory". A good one is Probability and Measure Theory, by Ash and Doleans.
- Mr. SmartypantsLv 73 weeks ago
OMG, that's the dumbest thing I ever heard. Not easy to follow either (but that's not your fault).
You know the old riddle that you can never get from this side of the room to the other side because you have to go halfway first, and half of that first, and half of that first, etc. ad infinitum. So there is an INFINITY of positions between you and the other side of the room, so you can't ever get there. While it's intellectually amusing, you CAN get to the other side of the room. Easily! Just try it! It's an infinity of -infinitesimal- points.
In the real world, an arc is sometimes hard to do. A computer printer or pen plotter breaks a circle up into line segments. A CNC machine does also. If you have to break up a circle into cartesian coordinates, you have to break it up into line segments--how many depends on your desired precision and accuracy. In any of these cases, you're not dealing with an arc, you're -simulating- an arc, it's an approximation of an arc.
That doesn't mean arcs don't exist. A compass makes a perfect true arc. Your car makes an arc when you turn a corner. If you swing a weight on a string, that's an arc. When you open a door, the doorknob swings in a perfect arc.
- ted sLv 73 weeks ago
you have a logic error...an infinite number of positive numbers does not always become infinite....imagine you are in a room and each second you travel half the distance from one side of the room to the other side... after each movement you still have half the remaining distance to cover...so you NEVER reach the other ' wall ' even though you are doing an infinite number of steps