Let’s imagine that you have a universe with only two points in it. You now draw a line segment between the two points and choose one of them to be the reference point. The location of any point along the line segment is now determined by the distance it is from the reference point. Now make a third point on the line segment that is exactly half-way between the two points. This new point is located at exactly half of the distance between the two end points.

Can you see the symmetry that is formed by these three points? The distance from the center point to either of the two end points is exactly the same. If you now were to choose the center point to be the reference point, then both end points would be the same distance away, but in opposite directions.

You would now need more than just the distance measurement alone to determine the location of points along the line. With three points, now direction becomes important. Again, using the axiom of choice and can choose which direction from the center point is positive, and which direction is negative.

We can then, finally, determine the location of every point on the line by specifying both a distance and a direction (positive or negative) from the center point. This arrangement of determining the locations of points on a line is called a “number line”. We call the distance and direction of each point on the line the Point’s coordinates. A Point’s coordinates uniquely determine its location on the line. Finally, we have a way of determining the location of a Point.

The center point is a special point of symmetry between the two endpoints. We can “rotate” the direction (exchange negative for positive directions) and the location (distance and direction) of the two identical endpoints would be exactly the same. This is called “rotational symmetry”. It means you can switch reference directions and the list of all the point coordinates will be the same. Remember that since you can’t tell the difference between points, the order of the list of points is not important.

Please believe me, taking the time to understand these concepts makes math so much more fun and easier later on. Learning these concepts is like learning the basic vocabulary of the math language.

How can the exact location of the Point be described? This is a conceptual problem. It is also where some very advanced concepts of math (found in calculus and number theory) come to play. After some thought, it becomes obvious that if there was just one point in the universe, all by itself, it does not have an absolute location until you look at where it is relative to another point in the universe.

If there were only two points in the universe you could imagine a line segment drawn between the two points, and then there is some distance between the two points, you could pick one of the points as the reference point. Then you could describe the location of the second point as being a certain distance from the reference point. You only need one (1) distance measurement from the reference point to describe the location of the Point. This is why such a line is called 1-dimensional space.

You have now discovered quite a few interesting concepts, including the concept of the number “2”. Two identical objects in space that are distinguished by the distance between them. You have also discovered the concept of the number “1”, the concept of distance, and you are also starting to realize why Einstein’s theory is called the theory of relativity. Distances and times are only measurable relative to a reference point, and you can pick any of the points to be the reference point.

The idea that you can pick any of the points to be the reference point is called the axiom of choice. An axiom is something that is assumed to be true even though it has no definite proof. The axiom of choice forms the conceptual foundation for all mathematics.

Another concept that comes from using the distance from a reference point to describe location is the idea of symmetry. You can exchange the locations of two points, and you have not changed anything. It is impossible to tell the two points apart, and both points still have the same locations relative to every other point. This is called exchange symmetry. Symmetries makes math and physics much easier.