Multiplication, Division, and Integers

Background

We've covered how to add and subtract negative numbers. How do we multiply and divide them?

Question

Evaluate the following:

• 8 x 4; (-8) x 4; 8 x (-4); (-8) x (-4) 
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Answer

• 8 x 4 = 32; (-8) x 4 = -32; 8 x (-4) = -32; (-8) x (-4) = 32
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Analysis

This topic is all about the number -1. While we worked with what happens when we subtract something bigger than what we started with (such as 3 - 5 = -2), multiplying and dividing by negative numbers boils down to dealing with -1.

Ok - so let's talk about -1. The multiplication rules that we're introducing in this entry are the following:

(-1) x 1 = -1
(-1) x (-1) = 1

Or in other words, if we multiply a negative number with a positive number, we'll get a negative number. If we multiply a negative number with another negative number, we'll get a positive number.

Let's now work a few multiplication problems.

We've already seen from the multiplication table that 8 x 4 = 32. So what happens when we start adding negative signs to the numbers?

(-8) x 4

Keep in mind that -8 = -1 x 8. So let's rewrite the question:

(-8) x 4 = (-1) x 8 x 4

We already know that 8 x 4 = 32. So let's rewrite again:

(-1) x 8 x 4 = (-1) x 32

And now following our rule in green that any positive number multiplied by a negative number results in a negative number, we can finish up:

(-1) x 32 = -32

So (-8) x 4 = -32

We can use the exact same process to find 8 x (-4) = -32

Now let's do the last one:

(-8) x (-4)

And rewrite the numbers:

[(-1) x 8] x [(-1) x 4]

We can rearrange the numbers using the commutative property:

(-1) x (-1) x 8 x 4

Again, we know what 8 x 4 is:

(-1) x (-1) x 32

So now we have (-1) x (-1). From our rules above, we know that (-1) x (-1) = 1. So let's rewrite:

1 x 32

And lastly we know that this is simply 32.

With division, it follows the same pattern. If one of the numbers we're using in our division is negative, the quotient will be negative. If neither are negative or both are negative, we get a positive.

So with , we use regular division and get 2.

With , one of the numbers is negative, so we get -2.

And with , both are negative and so we get 2.

And now we can do one last set involving both sets of rules at once.

What if we have a negative sitting in front of the fraction? Like this:



This is the same thing as having -1 multiplying the fraction:



We know what 8 divided by 4 is and we also know what happens when we multiply something by -1:



In like fashion, we can work out the other questions:







A trick people use to help keep track of the negative sign is if you end up with one in the top or bottom numbers of the division, so instead stick it out front - it doesn't change the value of the number at all. So we can do this:



and this



Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Numbers:

• Integers

Operations:

• Multiplication
• Division

Operations with different kinds of numbers:

• Multiplication, Division, and Whole Numbers
• Addition, Subtraction, and Integers

Associated Operations:

• Factors and Factoring

Where might we go?

Operations with different kinds of numbers:

• Multiplication, Division, and Integers - Practice Problems
• Multiplication, Division, and Rational Numbers