I am confused about adding, subtracting, multiplying, and dividing negative numbers.

Dealing with negative numbers can be difficult for some people to grasp at first because it involves a level of abstract thought that isn't as necessary with positive numbers.

Basic math lessons are often linked to physical, real-world situations so that you can better visualize what is going on. You might be asked, "If Jordan has 9 cats and Aaron has 12 cats, how many cats do they have altogether?" as an illustration of addition, or "If you want to give an equal number of marbles to each of 8 people, and you have 24 marbles, how many marbles does each person get?" to illustrate division.

Negative numbers cannot be so easily illustrated with real-world examples. For example, you can imagine that Jordan doesn't have any cats, but what does it mean for Jordan to have –4 cats? How can he have fewer than zero cats?

Obviously, he can't. Negative numbers require you to think abstractly. Using a number line can help you visualize the math more easily, but over time, as your mathematical reasoning abilities develop, your ability to understand abstract negative numbers will improve, and you won't give negative numbers a second thought.

But until then, it might be a good idea just to memorize a few rules about negative numbers and how to use them.

Multiplying and dividing with negative numbers

Multiplication and division are two sides of the same coin, and when it comes to negative numbers, they both follow the same rules, which can be illustrated in a simple table:

 

The second number is positive (+)

The second number is negative (–)

The first number is positive (+)

The answer is positive (+)

The answer is negative (–)

The first number is negative (–)

The answer is negative (–)

The answer is positive (+)

 

Put another way, if both numbers have the same sign, the answer is positive; if the numbers have different signs, the answer is negative.

Adding and subtracting with negative numbers

First, recognize that adding and subtracting are, from one viewpoint, the same thing. Subtracting a number is the same thing as adding the negative of that number. For example, 4 – 12 is the same as 4 + –12 (which, because the order of terms doesn't matter with addition, is the same as –12 + 4). With that in mind, here are the rules for adding with negative and positive numbers:

  • If both numbers are positive, then the answer is positive.
  • If both numbers are negative, then the answer is negative.
  • If the numbers have different signs, the answer takes the sign of the higher number.
  • Subtracting a negative number is the same as adding the positive of that number. For example, 5 – –4 is the same as 5 + 4.