Time for the Number Line

In the world of numbers, you will usually work with whole numbers. Those are the numbers that are zero and greater on a number line. There are no parts of numbers to worry about with whole or natural numbers. When you start to work with integers, you will start using numbers that are less than zero. For example, you might have -3, -25, or -147. Once you start working with integers, you'll also be adding negative and positive numbers. The easiest way to look at addition with negative numbers is to use a number line.

integers below and above zero

You've seen these lines before, but now that we have negative numbers, we can count values below zero. They are still numbers that are "whole" (not parts of numbers), but they have negative symbols in front of them. When you count, add, or subtract, you will bounce around the line as you always have. Positive numbers bounce to the right and negative numbers will bounce to the left.

positive values to the right and negative values to the left

What Is a Negative Number?

A negative number is the opposite number of a positive number. That seems easy. The opposite of 2 is -2. The opposite of 18 is -18. The same is true for negative numbers. The opposite of -3 is 3. The opposite of -24 is 24. Let's do a subtraction problem...

5 - 8 = ?

When we were using whole numbers, we couldn't do this problem. Now that we are using integers, we can do this problem, because we can use negative numbers. Looking at the number line, you can see how 5 - 8 = -3.

number line showing 5 - 8

But what is 5-8? Take a step back and look at the way the problem is written. You wrote down a positive five and a negative eight. In this example, you were actually adding five and negative eight. Let's rewrite the problem...

5 - 8 = 5 + (-8) = -3

You added a negative number to a positive one. That's a big idea in math and integers. But what if you want to subtract a value from a negative number? Let's look at -5 – 8 = ?.

-5 - 8 = (-5) - 8 = (-5) + (-8) = -13

number line showing -5-8

See how we switched it over to an addition problem? You can see how we did it on the number line too. The negative values had us moving to the left. Work these examples out on your own to see if you get the correct answers.

8 - 6 = 2
6 - 8 = -2
-6 - 8 = -14

Subtracting Negative Numbers

We have one more type of number to subtract. You can already subtract a positive from a positive (4 - 2) and a positive from a negative (-2 – 4). What about subtracting negative numbers? Your next big step is to understand the idea of combining signs before numbers. When you subtract a negative number you are actually doing this… 5 - (-2). Do you see how there is a minus sign and a negative sign next to each other? In math, when you get two negative signs in a row, the value becomes positive, or, in our case, a plus sign.

For example:
8 - 5 = 8 + (-5) = 3
-8 - 5 = -8 + (-5) = -13
-8 - (-5) = -8 + 5 = -3

Look at the way two dashes became a plus symbol. Just remember that two dashes make a plus sign. The overall pattern is about odds and evens. If you have an even number of dashes together, you change to a plus sign. If you have an odd number of dashes, you keep a minus/negative sign. An extra credit problem on a test might look like this...

8 - - - - - - - - - - 5 = ?

Since there are ten minus signs in a row, you can switch it to a plus sign. 8 + 5 = 13.

8 - - - - - 5 = ?

With five minus signs in a row, you have a negative sign. 8 - 5 = 3.

No real math problem would ever be written like that, but it might be a final test question. Mathematicians like to keep things organized. If you want it written correctly, it will look like this...

8 - (- (- (- (-5)))) = ?

Do you get the same answer if we write it this way?
8 - (0 - (0 - (0 - (0-5)))) = ?

Can you see how -5 is the same value as 0-5? We just placed some zeros in the empty spaces of the problem.


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