What is a Negative Number? - Definition & Rules

Instructor: Jason Furney

Jason has taught both College and High School Mathematics and holds a Master's Degree in Math Education.

This lesson will walk you through what negative numbers represent and how they interact with our equation operations like addition, subtraction, multiplication, and division. A quiz will be included at the end to test your understanding.

What is a Negative Number?

The first thing you should do is take a look at a number line.

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Notice the two different colors. Red is on the left of the zero and blue is on the right of the zero. Also, take note that 0 is neither blue nor red. Which numbers are considered the positive ones? Which are the negative ones?

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Negative numbers are any number to the left of zero on the number line. They are represented by the - sign attached on the left. You can have -1, -2, -10, -1000000000000, - ½, -3, etc. etc. Negative numbers can be any number you can imagine between 0 and negative infinity. That means that over a million people can make a list of 100 negative numbers and no two lists ever have to be the same! That is a lot of negatives in our lives! But are these negatives a bad thing?

I don't know about all of you, but when I open my mail at the end of the month I get my cable bill, my credit card bill, and my bank statements. I often get loan bills too, but I like to pretend those aren't there for as long as I can. Anyway, the point is that these bills tend to show negative numbers often. The first question you should ask yourself is whether or not these negative numbers are good for you or bad for you. I know when we think about the word negative, negative thoughts tend to arise, but that doesn't always have to be true. Negative numbers are used in our everyday lives to show decreases. These decreases can be both good and bad! Let's see how:

Ex 1. Your bank statement comes in the mail, and you open it to find the following numbers.

Deposit 300

Gas -50

Department Store -45

Total Balance 205

In this case, the 50 and the 45 are negative numbers and are represented with the - sign. Our initial deposit into our imaginary bank account was 300 dollars. That is a positive number and is very good! Everyone loves to add money to their account. Under that, however, we have some monthly expenses. We had to buy gas to get to school and work. That cost us 50 dollars, and it shows up as a negative in our account. We also decided to get a new outfit at the department store, which cost us another 45 dollars. So how do we get our remaining balance at the end of the month? We take our initial deposit of 300 and subtract 50 and subtract 45 because they are negative. In this case, these negative numbers were a bad thing. They took money away from us.

Ex 2. You now get to open your credit card statement. How many negative numbers will we see here?

Last Month's Balance 432

Last Month's Payment -432

Dinny's Diner 28

Gas 52

Wally's Mart 200

Wally's Mart Credit -200

Balance Due 80

Looking at the pretend statement above, we see some more negative signs. In this case however, they are beneficial for us. The -432 dollars paid off our previous balance of 432 dollars. They canceled each other out. Later on, we bought food and gas which added 80 dollars to our credit card statement. We then went to Wally's Mart and bought something expensive, which added another 200 dollars to our bill! But wait, that item was then brought back and our card was credited 200 dollars as shown by the -200.

For example number two, we see that the negative numbers are, again, subtracting from our total, but are they good or bad in this case?

In the first example the subtractions were bad because they decreased our funds but in the second example the subtractions were good because they lowered our bill. So you see, negative numbers have good and bad effects depending on the context they are used in. The one thing that stayed constant throughout both examples was that they subtracted from something.

Negative Number Rules!

Negative numbers follow some very precise rules when used in computations. As you saw from the examples above, sometimes they are used to subtract things.

Ex 3.

4 + -3 = 1

4 - 3 = 1

-3 + 4 = 1

All three equations say the exact same thing. Whenever you have a positive 4 and you are combining (adding) it with a negative 3, you are going to get 1. It doesn't matter which number is written first, it matters where the negative signs are.

Now things start to get tricky, so I am going to give you an example of a few different equations and then we will create some rules for them.

Adding/Subtracting

Ex 4.

a) 5 + -3 = 2

b) -2 + 6 = 4

c) 4 - 2 = 2

d) -7 + 3 = -4

e) 2 - 5 = -3

Every example above is considered an addition/subtraction property.

Rule 1:

In fact adding negative numbers is the same thing as subtracting the positive. Let's re-write each equation above just for understanding.

a) 5 - 3 = 2

b) 6 - 2 = 4

c) 4 - 2 = 2

d) 3 - 7 = -4

e) 2 - 5 = -3

Rule 2:

The number directly to the right of the - sign is the negative number.

The negative numbers are, respectively:

a) -3

b) -2

c) -2

d) -7

e) -5

Rule 3:

Any time you are adding a negative and a positive, subtract the smaller number from the bigger and then carry the sign that is attached to the 'bigger' number.

a) Original: 5 + -3 Solve it: 5 - 3 = 2 and the bigger number is positive so our answer is positive

b) Original: -2 + 6 Solve it: 6 - 2 = 4 and again the bigger number is positive so our answer is positive

c) Original: 4 - 2 Solve it: 4 - 2 =2 and the bigger number is positive so our answer is positive.

d) Original: -7 + 3 Solve it: 7 - 3 = 4 but in this case the 7 was a negative so our answer is negative and we get -4

e) Original: 2 - 5 Solve it: 5 - 2 = 3 but the bigger number was negative so our answer is -3.

If you have time you can also try using the number line at the top. Take -7 + 3 for example. Start your pencil on -7 on the number line and then 'add' or move forward 3 spaces. What do you get?

Rule 4:

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