Length in Number Line Geometry

Remember, a number line has its origin (0), its positive direction and its unit length, these characteristics distinguish it from normal lines.

So let's get started.

Hmmm... Wait, $0-2=-2$, but the distance can't be negative...

So here comes the absolute value. It basically makes everything positive (or 0)!

We use two vertical bars around the number for its absolute value. For example, $|2|=2,|-2|=2,|0|=0$.

Try this out.

Nice! When we apply it to the distance on number line, absolute value can become very handy!

Now you get the hang of it. Absolute values are pretty useful in geometry, since all lengths and distances are positive.

Given a line, we can represent all the numbers in the set $\mathbb{R}$ (the real numbers).

This line used to represent all real numbers is more commonly known as the number line.

We can define something called the distance between two points on the number line:

Given two points $a$ and $b$, we define the distance between these two points as $|a-b|$ where $|\ldots |$ is the absolute value function. As we can see, the distance between two points is always non-negative.

For example, the distance between the points $4$ and $7$ is $|4-7|=|-3|=\boxed{3}$.

As you may see later, we can generalize this number line into a two-dimensional number plane, which represents all ordered pairs in $\mathbb{R}^2$.

On a number line in geometry, every point corresponds with a value on the number line.

Consider the number line below.

Point $\color{red}{L}$ is paired with $-3$

Point $\color{blue}{M}$ is paired with $-2$

Point $\color{green}{N}$ is paired with $-1$

Point $\color{orange}{O}$ is paired with $0$

Point $\color{pink}{P}$ is paired with $1$

Point $\color{limegreen}{Q}$ is paired with $2$

Point $\color{purple}{R}$ is paired with $3$

The distance between two points on a number line is the absolute value of the difference of their coordinates. Because it is the absolute value, the distance will always be positive.

Consider the number line below.

$\overline{BE}=|-4-(-1)|=|-3|=3$

This says the length of line $BE$ is $3$.

When solving for the length between two points on a number line, the order in which we subtract does not matter because we are using absolute value.

$\overline{QV}=|1-(-4)|=5$

$\overline{QV}=|-4-1|=5$

In both instances we are left with a length of $5$.