Number Line — Definition & Uses

(Video) The Number Line

by Eddie Woo

Here's an excellent video by Eddie Woo where he explains the concept of number lines, how to construct them, and how to use them. As you'll see, one of the main uses of number lines is to compare numbers. Given two numbers, which is bigger, and which is smaller? Let's find out.

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How to Make a Number Line

Making a number line is very simple. First we draw a long horizontal line.

Next, we make a small dash to mark the middle.

This will be our zero mark. We can write the number $0$ above or below the line to indicate this.

Now let's add some more equally spaced ticks to the right of the $0$. These will increase in size from $1$ to $2$ to $3$ etc. We can remember this by thinking that every time we move to the Right, we Raise the unit.

Now how far do we continue incrementing by one? How long can you go to the right before the integers stop? Well, the answer is forever! They go to infinity. So instead of making ticks forever we put an arrow at the end to indicate that it continues in that direction.

Now what about the left hand side? Well that is the negative numbers. Number that have a negative symbol in front of them, i.e. $-1$. They work the same way but instead of getting bigger, as you go to the left you get smaller. We can remember this by thinking that when we move to the Left, we Lower the unit. Let's add in ticks and mark down some.

Just like with the right hand side, the numbers can continue getting smaller forever. There's an infinite number of negative numbers. So we again denote this with an arrow.

Positive and Negative

Number lines are broken up into two parts. To the right of the zero is the positive numbers.

Similarly, to the left of the zero is the negative numbers.

Sadly, zero is all alone by itself. It is neither positive nor negative. Bummer. Still include it though!

We can put points on number lines. These are filled in circles or dots ON the line to show information. If we wanted to convey the number $-2$, we just put a dot on the line at $-2$.

Number Line Extensions and Examples

A quick note before we begin, this explanation is made to go over different types of problems you could see that use number lines. If you haven't already, check out this explanation on number lines to get the basics before tackling these extensions!

Example 1: Gaps In The Number Line

Some problems will give you number lines that haven't been completed. The one shown below has ticks that are $3$ apart. What number is denoted by the dot on the example below?

It was stated that each tick is $3$ apart, and this is confirmed by $9$ being $2$ ticks away from $3$. All we need to do is count ticks to find what is covered by the dot. The dot is one tick to the right of $9$, so it is $12$.

Word Problem 1: Counting Spaces

Not every problem is going to give you a visual aid. Some will instead give you information about how something moves along the line, and you will need to visualize it on your own. I encourage you to draw the number lines yourself to help with this.

A problem like this requires us to assume that the number line has $1$ unit for each tick. Moving $3$ units to the right on a number line with $1$ unit between ticks will take us from $0$ to $3$. We then move $6$ units to the left of $3$. This is the same as subtracting $6$. $$3-6=-3$$ We are going to end our movement at $-3$. Let's look at this on a number line.

Word Problem 2: Total Steps

Some problems will ask us to find the total distance traveled along the number line. Let's use the same set of moves as the last problem. We are moving $3$ spaces to the right then $6$ spaces to the left. We will add these together to find the total number of spaces covered. $$3+6=9$$ The movement covers $9$ total spaces.

Word Problem 3:

Some word problems will add context to give a framework for the question.

The point that would be the farthest left would be the lowest number out of our options. Grace lost the most money at the casino, so her earnings are the farthest left on the number line. Jake's number is positive because he made money; Sarah lost money but not as much as Grace did.

Wrap-Up

Each of these types of problems require some manipulation because you aren't given a fully filled in number line, but we are always a few steps away from getting there. As long as we can remember the basics of number lines (distance between ticks, right vs. left, etc.), we can find our answer.

(Video) The Number Line

by Mahalo

This video by Mahalo which explains the number line.

$\large{\text{Summary}}$

A number line is a horizontal line with ticks at intervals to show integers. At the end are arrows to show that the numbers keep going even though we didn't write them.

One thing number lines are really good for is to determine if a number is bigger than another. This might seem obvious to some but it can get confusing when trying to figure out which negative number is bigger.

If we select a number, everything to the right of it will be bigger. Everything to the left of it will be smaller.

Let's pick the number $-2$ for example.

Everything to the right of $-2$ is bigger than it. This means that $-1$ is bigger than negative two. Similarly, $0$, $1$, $2$, etc. are also all bigger than $-2$.

Everything to the left of $-2$ is smaller than it. This means that $-3$ is smaller than negative two. Similarly, $-4$, etc. is also smaller than $-2$.

We use inequalities to indicate which number is smaller, and which one is bigger.