Multiplicative Property of -1: Definition & Examples

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Multiplying by -1 can be really confusing, especially when you have lots of them happening! In this lesson, we'll figure out what to do with those -1 problems and look through a few examples to help us get it down.

Moving in the Opposite Direction

Have you ever been headed out on your bike and then suddenly turned around and went the opposite way? Well, that is how you can think of the multiplicative property of -1, it's a change in direction. Multiplying anything by -1 makes it change the direction it was headed.

The multiplicative property of -1 says that any time you multiply something by -1, you change it into its opposite. The opposite of a number is that same number on the opposite side of 0 on a number line. For example, if you multiply 5 by -1 you'll get -5. If you multiply -5 by -1 you get 5. Take a look at Figure 1.


Figure 1
number line graphic


It's kind of like a bouncing ball. Every time you multiply by -1, the ball bounces back the other way.

Examples of Multiplying by -1

Okay, what do you think happens if you multiply a -1 by each of the following?

  • 10
  • .001
  • -3
  • 0

The opposite of 10 is -10, so if you multiply 10 by -1 you get -10. Same with .001. Just tag a '-' on the front of it, and you're there. Okay, what about the -3? Well, the ball is already on the negative side of the number line, so what happens when you bounce it back over? It becomes positive again! -3 times -1 makes a positive 3. In fact, any time you multiply your -1 by a number that's already negative, you're just going to blow away that '-' sign and make the thing positive again.

Now, what about the 0? Well, 0 has its own property that says anything you multiply by 0 becomes 0. That's true for -1, as well. Multiply 0 by -1 and you still have 0.

What if it's a variable instead of a number? You know how they sometimes use letters to represent quantities that you don't know yet? Well, that part's easy. If you have an x, for example, then multiplying it by -1 puts you in the same place on the other side of the 0 on the number line, which gives you -x. What if you start with -x? Well, as we said above, multiplying -x by -1 is going to knock off the '-' and make it into x again.

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