Learn how to perform multiplication and division with negative numbers.
There are many things in math that people memorize and use but never really understand. Among these, none are more famous than “a positive times a negative equals a negative” and “a negative times a negative equals a positive.” But why are these statements true? And how does multiplication and division really work when you throw negative numbers into the mix? The answers to these questions and more are the subject of today’s article.
Review: What is Multiplication?
As we talked about in the article called Is Multiplication Repeated Addition? in most circumstances it’s best to think of multiplication as something that scales a number to be some other number of times its original size. You can think of the problem 2 x 5 as stretching out a 5-unit-long stick laying along the number line until it’s 10 units long—twice its original length.
That’s easy enough, but how about something even easier: 1 x 5? What does that mean? This problem just says to stretch the number 5 out along the number line until it’s 1 times it’s original length. In other words, the magnitude of the original number stretched out along the number line does not change in this problem. (In case you need a refresher, we talked about magnitudes in the article on absolute values.) So that’s how multiplication with positive numbers works, but what happens when one or both numbers are negative?
How to Multiply Positive and Negative Numbers
To start with, what does it mean to multiply a positive number by a negative number? Well, according to our definition of multiplication, a problem like –1 x 5 tells us to stretch the number 5 out along the number line until it’s –1 times its original length. We’ve seen how to stretch a number out to be 1 times its original length, but –1…what does that mean? The quick and dirty tip is that multiplying any number by –1 simply flips it from one side of the number line to the other. If the number multiplied by –1 is positive, the result is negative. If the number multiplied by –1 is negative, the result is positive.
So we now know that –1 x 5 = –5. Notice that just like when multiplying 1 x 5, multiplying –1 x 5 doesn’t change the magnitude of the original number 5—it only changes its sign. How about the problem –2 x 5? Well, in that case two things happen: First, the magnitude of the number 5 is stretched out along the number line until it’s twice its original size. And second, the number that we get as a result of this stretching, 10, is flipped across the number line to give us the final answer: –2 x 5 = –10.
It’s no mystery that these two things happen since what we’ve learned so far tells us that the number –2 can also be written –1 x 2. Which means that the problem –2 x 5 = –1 x 2 x 5. So the first step, 2 x 5, does the stretching, and the second step flips the result across the number line. And thus: “A positive times a negative equals a negative.”
How to Multiply Negative Numbers
That’s how things work when multiplying a positive number by a negative number. But what about multiplying two negative numbers together? To see what happens, let’s think about the problem –1 x –1. This problem says to flip the number –1 to the other side of the number line, giving an answer of 1. How about the more complicated problem –2 x –5? Start by using the fact that –2 = –1 x 2 to instead think of this problem as –1 x 2 x –5. This just says to first stretch –5 until it has twice its original magnitude (which means that it’s equal to –10), and then to flip this result across the number line. So the answer is 10. And thus: “A negative times a negative equals a positive.”
How to Do Division with Negative Numbers
But how do you figure out the sign when doing division with negative numbers? The rule for figuring out the sign when you divide a positive number by a negative numbers (or vice versa) or a negative number by a negative number is exactly the same as when you multiply them. Just swap in the operation of division instead of multiplication and everything works the same. In particular, dividing a positive number by a negative number (or vice versa) results in a negative number. And dividing a negative number by another negative number results in a positive number. Why?
[[AdMiddle]Well, think back to the connection between fractions and division that we talked about in an earlier article. In that article, we learned that dividing some number by a number like 2 has the exact same effect as multiplying the original number by the fraction 1/2. Which means that dividing a positive number by a negative number like –2 is the same thing as multiplying it by the fraction –1/2. That means that any division problem can be turned into an equivalent multiplication problem, and that all of the logic for figuring out signs that we talked about for multiplication works for division too!
Number of the Week
Before we finish up, it’s time for this week’s featured number from my post on QDT’s blog The Quick and Dirty. This week’s number isn’t actually a number, but is instead an explanation of the hypothetical job interview puzzle I gave in last week’s article. The puzzling question is: “How many points are there on a sphere where after walking one mile south, then one mile east, and finally one mile north, you end up right back where you started?”
It’s pretty easy to come up with one point that works: the North Pole. When you’re standing at Earth’s North Pole, any direction you walk is south. So walking one mile south (which means in any direction), then one mile east, then one mile north, must put you right back at the North Pole.
But are there other locations that work too? Indeed, there are. In fact, there are an infinite number of them! If you want to find out where all those points are, check out my blog post on The Quick and Dirty.
And that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new number of the day and math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at firstname.lastname@example.org.
Are you tired of trying to figure out whether there are “healthier” food options at the supermarket? Check out Nutrition Diva’s Grocery Store Survival Guide.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Image courtesy of Shutterstock