What is a Number Line?

Right now I hope we have a good understanding of real numbers as all the rational and irrational numbers. There is a nice quality to real numbers that no two real numbers are equal and we can always tell whether a real number is greater or smaller than the other. Thus we can simply arrange all the infinite numbers in a single line called the number line. Let me be further clear, we can make a number line for any number set. Here are the lines for each of the number lines that we have discussed.

Natural Number Line

Note how there is no solid line connecting the numbers, meaning this is a discrete line. Which means that between 1 and 2, there is no other real number. The arrow on the right side means that the line continues infinitely long to the right. But the lack of an arrow on the left side means it starts with 1. There is no natural number before 1.

Whole Number Line

Note the similarities between the two number lines. Both are discrete, both are infinitely long. The only difference being whole number line starts from 0.

Integer Number Line

Progression, now the number line stetches infinitely on both the left and right side. There are infinitely many integers behind 0 as there are after it which is denoted by the arrowheads on either side. However, since this is still a discrete line, there is no line connecting these integers.

Rational Number Line

The difference between the Integer Number Line and the Rational Number Line can be seen in the dots placed between the numbers. These dots represent that there are infinitely many rational numbers between 1 and 2 and each other rational number. However, even rational numbers are not continuous. There are numbers between 1 and 2 that are not rational. Hence we still have the faint opaque line between the numbers.

Real Number Line

Finally the real number line is ready. The solid line represents that the real number line is continuous. There are no gaps. All numbers can be found on this line, irrational, rational, positive, negative, and transcendental.

From One-Dimensional to Two-Dimensional Thinking

So far, we’ve represented numbers along a one-dimensional line — the number line — where each value has a single position along a horizontal axis. But what if we want to represent more than one number at a time? What if we want to show a relationship between two different values, like time and distance, or x and y?

To do that, we need to move beyond just one dimension. We need to consider two dimensions. This brings us to a powerful mathematical tool: the Cartesian Plane.

The Cartesian Plane

The Cartesian Plane is made up of two number lines:

• The x-axis — a horizontal number line
• The y-axis — a vertical number line

These two lines intersect at a point called the origin, which is at $(0, 0)$.

Coordinates: Plotting Points

Every point on the Cartesian Plane is described using a pair of numbers in the form$(x, y)$.
The first number tells us how far left or right the point is (along the x-axis), and the second number tells us how far up or down it is (along the y-axis).

For example:

• $(3, 2)$ means move 3 units to the right, and 2 units up from the origin.
• $(-2, -4)$ means move 2 units to the left, and 4 units down.

The Four Quadrants

The two axes divide the Cartesian Plane into four regions, called quadrants. Here is how they are arranged:

It is important to note that points lying directly on the axes like $(0, 4)$ or $(3, 0)$ are not considered to be in any quadrant.

Activity: Match The Columns!

Reorder the Condition to match them with the correct Quadrant.

Other multdimensional numbers include Complex Numbers, Matrices, Vectors, Tensors, and more. They will be discussed in their due time.