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# Integer Multiplication

## Understand the rules for multiplying positive and negative numbers.

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Practice Integer Multiplication
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Integer Multiplication

### Let’s Think About It

Credit: Tatiana Vdb
Source: https://www.flickr.com/photos/kit4na/9965831784
License: CC BY-NC 3.0

Marybeth just helped give her baby brother a bath. As the water is draining out of the tub, Marybeth notices that the height of the water in the tub is decreasing by about 2 inches per minute. How could Marybeth describe the change in the height of the water in the tub after 3 minutes?

In this concept, you will learn how to multiply integers.

### Guidance

Integers are the set of whole numbers and their opposites.

When you multiply integers, the numbers that you are multiplying are called factors. Your answer after multiplying is called a product.

Here is an example.

\begin{align*}3 \times 7=21\end{align*}

In this example, 3 and 7 are the factors. 21 is the product.

Multiplying integers is very similar to multiplying whole numbers. The only difference is you will have to decide if your product is negative or positive.

To multiply two integers:

1. First, multiply the absolute value of the factors.
2. Next, determine the sign of your product according to the following rules:
• A positive number times a positive number equals a positive number.
• A positive number times a negative number equals a negative number.
• A negative number times a negative number equals a positive number.
• A negative number times a positive number equals a negative number.
• Any integer times zero equals zero.
3. Your product will be your result from step 1 with the sign from step 2.

It might help to remember that if your original two integers have the same sign, then their product will be positive. If your original two integers have different signs, then their product will be negative.

Let's look at an example.

Multiply \begin{align*}(-4)(-3)\end{align*}.

The first step is to multiply the absolute value of the factors. This means you are basically ignoring the negative signs for now.

\begin{align*}(4)(3)=12\end{align*}

The next step is to decide on the sign of your answer. Both factors were negative and a negative times a negative equals a positive. This means your product is positive.

\begin{align*}(-4)(-3)=12\end{align*}

The answer is 12.

Here's another example.

Multiply \begin{align*}-5 \cdot 8\end{align*}.

Again, the first step is to multiply the absolute value of the factors. Don't worry about the negative sign for now.

\begin{align*}5 \cdot 8=40\end{align*}

The next step is to decide on the sign of your answer. The two factors had different signs. A negative times a positive is a negative. This means your product is negative.

\begin{align*}-5 \cdot 8=-40\end{align*}

The answer is -40.

You can use the same process to multiply more than two integers together. Just work from left to right and multiply two factors at a time.

Here is an example.

Multiply \begin{align*}(-8)(-3)(-2)\end{align*}.

First you will multiply \begin{align*}(-8)(-3)\end{align*}. You know that \begin{align*}(8)(3) = 24\end{align*}. You also know that a negative times a negative equals a positive. So you have

\begin{align*}(-8)(-3) = 24\end{align*}

Next, take your product of 24 and multiply by the third factor, -2.

\begin{align*}(24)(-2)\end{align*}

You know that \begin{align*}(24)(2) = 48\end{align*}. You also know that a positive times a negative equals a negative.

\begin{align*}(24)(-2) = -48\end{align*}

The final answer is \begin{align*}(-8)(-3)(-2) = -48\end{align*}.

### Guided Practice

The number of students voting in a school election has been decreasing at a rate of 15 students per year. Represent the change in the number of students voting over the last 3 years as an integer.

First, represent the decrease in the number of students voting as an integer.

The problem states that the number of students voting has been decreasing by 15 students each year. To show a decrease, use a negative integer -15.

To represent the decrease in the number of students voting over the last 3 years, multiply the number of years by the integer representing the decrease.

\begin{align*}3 \times (-15) =?\end{align*}

Find the product to solve the problem.

Multiply the absolute value of the factors.

\begin{align*}3 \times 15=45\end{align*}

Next, decide on the sign for your product. A positive times a negative equals a negative, so your product is a negative.

\begin{align*}3 \times -15=-45\end{align*}

The answer is -45.

The change in the number of students voting over the last 3 years can be represented as -45.

### Examples

#### Example 1

Multiply \begin{align*}-9(-3)\end{align*}.

First, multiply the absolute value of the factors.

\begin{align*}9(3) = 27\end{align*}

Next, decide on the sign for your product. A negative times a negative equals a positive, so your product is a positive.

\begin{align*}-9(-3) = 27\end{align*}

The answer is 27.

#### Example 2

Multiply \begin{align*}(-3)(12)\end{align*}.

First, multiply the absolute value of the factors.

\begin{align*}(3)(12) = 36\end{align*}.

Next, decide on the sign for your product. A negative times a positive equals a negative, so your product is a negative.

\begin{align*}(-3)(12) = -36\end{align*}

The answer is -36.

#### Example 3

Multiply \begin{align*}(-4)(3)(-2)\end{align*}

First you will multiply \begin{align*}(-4)(3)\end{align*}. You know that \begin{align*}(4)(3) = 12\end{align*}. You also know that a negative times a positive equals a negative. So you have

\begin{align*}(-4)(3) = -12\end{align*}

Next, take your product of -12 and multiply by the third factor, -2.

\begin{align*}(-12)(-2)\end{align*}

You know that \begin{align*}(12)(2) = 24\end{align*}. You also know that a negative times a negative equals a positive.

\begin{align*}(-12)(-2) = 24\end{align*}

The final answer is \begin{align*}(-4)(3)(-2) = 24\end{align*}.

### Follow Up

Credit: Kristian Thøgersen
Source: https://www.flickr.com/photos/flottenheimer/8560910193/
License: CC BY-NC 3.0

Remember Marybeth and the bathtub? She noticed that the height of the water in her baby brother's bathtub was decreasing by 2 inches every minute. She wondered how she could represent the change in the height of the water after 3 minutes.

First, represent the decrease in the height as an integer.

The height of the water is decreasing by 2 inches per minute. Use a negative integer, -2, to show the decrease.

To represent the decrease in the height of the water after 3 minutes, multiply the number of minutes by the integer representing the decrease.

\begin{align*}3 \times (−2) =?\end{align*}

Find the product to solve the problem.

Multiply the absolute value of the factors.

\begin{align*}3 \times 2=6\end{align*}.

Next, decide on the sign for your product. A positive times a negative equals a negative, so your product is a negative.

\begin{align*}3 \times -2=-6\end{align*}

The answer is -6.

The change in the height of the water after 3 minutes can be represented by -6 inches.

### Video Review

The video below reviews multiplying integers.

### Explore More

Use integer rules to find each product.

1. \begin{align*}(-3)(9)\end{align*}
2. \begin{align*}8(-9)\end{align*}
3. \begin{align*}12 \times 8\end{align*}
4. \begin{align*}(-4)(7)\end{align*}
5. \begin{align*}(-2) \cdot (-11)\end{align*}
6. \begin{align*}3 \cdot (-25)\end{align*}
7. \begin{align*}5 \times (-6) \times (-1)\end{align*}
8. \begin{align*}(-8)(-7)\end{align*}
9. \begin{align*}(-2)(3)(-4)\end{align*}
10. \begin{align*}(9)(1)(-1)\end{align*}
11. \begin{align*}(-9)(2)(-1) \end{align*}

### Vocabulary Language: English

Factors

Factors

Factors are numbers or values multiplied to equal a product.
Integer

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Product

Product

The product is the result after two amounts have been multiplied.

### Image Attributions

1. [1]^ Credit: Tatiana Vdb; Source: https://www.flickr.com/photos/kit4na/9965831784; License: CC BY-NC 3.0
2. [2]^ Credit: Kristian Thøgersen; Source: https://www.flickr.com/photos/flottenheimer/8560910193/; License: CC BY-NC 3.0

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