how do you prove that a circle has an inside and an outside?

Answer 1

Build a circular fence around your house. To get inside you can come through the gate, or climb over the fence. To get outside, vice versa.

Answer 2

The Jordan Curve Theorem is overkill if it's just a circle.

By definition, a circle is a set of points equidistant from a given point (the center). Just use polar coordinates and define the origin to be the center of the circle, then the circle is the set of points (r,t), such that r = R, where R is the radius of the circle.

Now, we know that for any continuous curve containing the points
(r0,t0) and (r1,t1) where r0 < R < r1, there must exist some t2,
such that the curve also contains (R,t2). Therefore the circle divides the plane into at least two regions.

To prove that it is exactly two regions, we can show that for any pair of points, (r0,t0) and (r1,t1) such that
r0 <= r1 < R ...or... R < r0 <= r1 ,
there exists a continuous curve which does not intersect the circle and which contains both points. Namely, the union of the curves
r=r0 as t varies from t0 to t1 and
t=t1 as r varies from r0 to r1

Thus there is an inside and outside.

Answer 3

In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside". It was proved by Oswald Veblen in 1905.

The formal proof is 200,000 lines (follow the link) so you will excuse me if I don't include it in the margin here.

You sure get tough homework.

Addendum: using polar coordinates is a great idea; i was trying to think how you would do it in cartesian, and my head got a clot in it or sthing

Answer 4

each and every enclosed curve has inside and outside. this topic will come in 10+1or10+2 standard. there is a topic about circles .

Answer 5

u attack this from your basics .a plane or a surface is given by boundaries .for what??????
to specify the region.here acircle is a sotof plane
which has boundaries in order to seperate it from its surroundings ie froom out an in
so it has an out &in