Can anyone Prove a line has 2 sides mathmatically?

Answer 1

This depends on what space the line is embedded in. For example, a line in three dimensions only has one side. So I'm going to assume that you are tlaking about a straight line embedded in the plane.

Now, you have to precisely define what you mean by the phrase 'has two sides'. This is actually non-trivial. We say that a set A embedded in a topological space X has 2 sides if for every open set U containing A, the set U-A is disconnected and there are arbitrarily small such open sets where U-A has exactly two components.

In the case of a stright line in the plane, we might as well assume that the line, A, is the x-axis. If U is any open set containing A, then U-A={(x,y) in U-A:y>0} union {(x,y) in U-A: y<0}. Both of the sets in this union are open and non-empty, so U-A is disconnected. Furthermore, it is easy to find U so that U-A has exactly these two components. Hence, A is two sided.

Fun, but easy, project: Find a space where some embedded line is three sided.

Answer 2

This question only makes sense if you are talking about a line drawn in the plane. So we will suppose the line that we are considering is the X-axis. We can just rotate the plane to get to this if we need to.

If we take a point on the top half of the plane and one on the bottom half,, then they are on the same side of the line if we can draw a continuous curve from one to the other that does not cross the line.

If we consider the intermediate value theorem:
http://mathworld.wolfram.com/IntermediateValueTheorem.html
and take some small liberty with it, we get that the only way to get a curve between the two points is to cross the line.

Thus, in the plane, a line has 2 sides.

Answer 3

Alright, let's take a rectangle. Now suppose dat rectangle got two sides wit zero length. Let da other two sides have length x. Then, mathematically there's no difference between that and a line of length x. So mathematically, a line has 2 sides, or else, it would not be possible for da rectangle wit infinitesimal sides ta exist, which is mathematically not true. If there can be a guy wit no eyes and he's still called a guy, then there can be a rectangle with two zero-length sides and still be called a rectangle, or a line.

Answer 4

A point has 360 degrees.A line bisects the angle of 360 degrees into 2 angles of 180 degrees. which leaves you with 180 degrees on the upper side of the line and 180 degrees on the lower side of the line.
So there the line now has two sides.

Answer 5

all line should ,must have 2 ends without bend then it can be consider as a line otherwise it is something which can be called as a picture

Answer 6

line does not have width and therefore have not two sides.