Integers: Learn It 6

Multiplying Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that [latex]a\cdot b[/latex] means add [latex]a,b[/latex] times. Here, we are using the model shown in the graphic below just to help us discover the pattern.

This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.

 

Now consider what it means to multiply [latex]5[/latex] by [latex]-3[/latex]. It means subtract [latex]5,3[/latex] times. Looking at subtraction as taking away, it means to take away [latex]5,3[/latex] times. But there is nothing to take away, so we start by adding neutral pairs as shown in the graphic below.

This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.

 

In both cases, we started with [latex]\mathbf{\text{15}}[/latex] neutral pairs. In the case on the left, we took away [latex]\mathbf{\text{5}},\mathbf{\text{3}}[/latex] times and the result was [latex]-\mathbf{\text{15}}[/latex]. To multiply [latex]\left(-5\right)\left(-3\right)[/latex], we took away [latex]-\mathbf{\text{5}},\mathbf{\text{3}}[/latex] times and the result was [latex]\mathbf{\text{15}}[/latex]. So we found that:

[latex]\begin{array}{ccc}5\cdot 3=15\hfill & & -5\left(3\right)=-15\hfill \\ 5\left(-3\right)=-15\hfill & & \left(-5\right)\left(-3\right)=15\hfill \end{array}[/latex]

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

multiplication of signed numbers

Same Signs

  • Two positives: Product is positive
  • Two negatives: Product is positive

Different Signs

  • Positive and negative: Product is negative
  • Negative and positive: Product is negative
Multiply each of the following:

  1. [latex]-9\cdot 3[/latex]
  2. [latex]-2\left(-5\right)[/latex]
  3. [latex]4\left(-8\right)[/latex]
  4. [latex]7\cdot 6[/latex]

When we multiply a number by [latex]1[/latex], the result is the same number.

What happens when we multiply a number by [latex]-1?[/latex]

Let’s multiply a positive number and then a negative number by [latex]-1[/latex] to see what we get.

[latex]\begin{array}{ccc}\hfill -1\cdot 4\hfill & & \hfill -1\left(-3\right)\hfill \\ \hfill -4\hfill & & \hfill 3\hfill \\ \hfill -4\text{ is the opposite of }\mathbf{\text{4}}\hfill & & \hfill \mathbf{\text{3}}\text{ is the opposite of }-3\hfill \end{array}[/latex]

Each time we multiply a number by [latex]-1[/latex], we get its opposite.

multiplying by [latex]-1[/latex]

Multiplying a number by [latex]-1[/latex] gives its opposite.

 

[latex]-1a=-a[/latex]