### Learning Outcomes

- Simplify expressions using multiplication of integers
- Simplify expressions using division of integers

### KEY words

**Product**: The result of multiplying two or more numbers**Quotient**: The result of dividing two numbers

## Multiplication of Integers

Since multiplication is mathematical shorthand for repeated addition, our money 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.

Since multiplication is repeated addition, [latex]a\cdot b[/latex] means adding [latex]a[/latex] to itself [latex]b[/latex] times. This means that if we multiply our $5 by 3, we add $5 to itself 3 times, which gives us $15. However, if we multiply a $5 debt by 3, we end up with a $15 debt. So, [latex]3\cdot\left(-5\right)=-15[/latex].

When we multiply two integers, the product is positive, when the signs are the same,and when the signs are different, the product is negative. * Product* is the name given to the result of multiplying two or more numbers together.

### MULTIPLICATION of Signed Numbers

The sign of the product of two numbers depends on their signs.

Same Signs | Quotient |
---|---|

(+)(+) | (+) |

(-)(-) | (+) |

Different Signs | |

(+)(-) | (-) |

(-)(+) | (-) |

### example

Multiply each of the following:

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

Solution:

1. | |

[latex]-9\cdot 3[/latex] | |

The signs are different, so the product is negative. | [latex]-27[/latex] |

2. | |

[latex]-2\left(-5\right)[/latex] | |

The signs are the same, so the product is positive. | [latex]10[/latex] |

3. | |

[latex]4\left(-8\right)[/latex] | |

The signs are different, so the product is negative. | [latex]-32[/latex] |

4. | |

[latex]7\cdot 6[/latex] | |

The signs are the same, so the product is positive. | [latex]42[/latex] |

### try it

Watch the following video for more examples of how to multiply integers.

### Properties of Multiplication

We know that [latex]2\cdot\,3=3\cdot\,2[/latex]. But is[latex]-2\cdot\,3=3\cdot\left(-2\right)[/latex]? Well[latex]-2\cdot\,3=-6[/latex] and [latex]3\cdot\left(-2\right)=-6[/latex]. So,[latex]-2\cdot\,3=3\cdot\left(-2\right)[/latex]. In fact, this is true for all integer values and is called the*commutative property of multiplication***.**The orderthat we multiply integersdoesn’t matter.

It is also true that multiplying any integer by [latex]1[/latex] has no effect on the integer. For example, [latex]-5\cdot\,1=-5[/latex]. Because [latex]1[/latex] does not change the identity of any integer it is multiplied onto. [latex]1[/latex] is called the* multiplicative identity*.

### COMMUTATIVE PROPERTY OF multiplication

For any integers [latex]a[/latex] and [latex]b[/latex], [latex]a\cdot\,b=b\cdot\,a[/latex].

The order in which we multiply integers does not matter.

### THE MULTIPLICATIVE IDENTITY

[latex]1\cdot\,a=a[/latex], for any integer [latex]a[/latex].

Multiplying an integer by [latex]1[/latex] does not change the value of the integer.

[latex]1[/latex] is the multiplicative identity.

What about multiplying by [latex]-1[/latex]?

[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 an integer by [latex]-1[/latex], we get its opposite.

### Multiplication by [latex]-1[/latex]

Multiplying an integer by [latex]-1[/latex] gives the opposite of the integer.

[latex]-1\cdot\,a=-a[/latex]

### example

Multiply each of the following:

- [latex]-1\cdot 7[/latex]
- [latex]-1\left(-11\right)[/latex]

Show Solution

Zero is another number that has the same result for all integers. Any integer multiplied by [latex]0[/latex] results in a product of [latex]0[/latex]. In other words, [latex]a\cdot\,0=0[/latex] for any integer [latex]a[/latex].

### Example

Find the product:

- [latex]-3(0)=0[/latex]
- [latex]-4(-7)(0)=0[/latex]
- [latex]5(-4)(-3)(7)(-2)(8)(0)=0[/latex]

### Multiplication by [latex]0[/latex]

Multiplying an integer by [latex]0[/latex] gives [latex]0[/latex].

[latex]0\cdot\,a=0[/latex]

### Multiplying more than two integers

When we multiply more than two integers, we multiply them two at a time.

### Example

Multiply [latex]-2\cdot 4\cdot \left(-5\right)[/latex]

**Solution**

Multiplying from left to right: [latex]-2\cdot 4\cdot \left(-5\right)=-8\cdot \left(-5\right)=40[/latex]

### Try It

Multiply [latex]2\cdot\left(-10\right)\cdot\left(-7\right)\cdot\,5[/latex]

Show Answer

Another way to solve the problem [latex]2\cdot\left(-10\right)\cdot\left(-7\right)\cdot 5[/latex] is to notice that [latex]2\cdot 5=10[/latex] is part of problem. To use this requires reorganizing the problem using the commutative property to:[latex]2\cdot 5\cdot\left(-10\right)\cdot\left(-7\right)[/latex]. Then[latex]2\cdot 5=10[/latex] and[latex]\left(-10\right)\cdot\left(-7\right)=70[/latex]. Multiplying these products gives:[latex]10\cdot\left(70\right)=700[/latex]. The same answer as before. Regrouping the numbers is an example of the** associative property of multiplication**.

### THE ASSOCIATIVE PROPERTY OF multiplication

[latex]\left(a\cdot\,b\right)\cdot\,c=a\cdot\,\left(b\cdot\,c\right)[/latex] for any integers [latex]a, b, c.[/latex]

Regrouping the integers results in the same product.

### Try It

Multiply: [latex]-1\cdot 8\cdot\left(-12\right)\cdot 5[/latex]

Show Answer

## Division of Integers

Division is the inverse operation of multiplication. So, [latex]15\div 3=5[/latex] because [latex]5\cdot 3=15[/latex]. In words, this expression says that [latex]15[/latex] can be divided into [latex]3[/latex] groups of [latex]5[/latex] because adding five to itself three times gives [latex]15[/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

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

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

### Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Same Signs | Quotient |
---|---|

(+)(+) | (+) |

(-)(-) | (+) |

Different Signs | |

(+)(-) | (-) |

(-)(+) | (-) |

Remember, you can always check the answer to a division problem by multiplying.

### example

Divide each of the following and check the answer:

- [latex]-27\div 3[/latex]
- [latex]-100\div \left(-4\right)[/latex]

Solution

1. | |

[latex]-27\div 3[/latex] | |

The signs are different, so the quotient is negative. | [latex]-9[/latex] |

Check | [latex](-9)(3)=-27[/latex] |

2. | |

[latex]-100\div \left(-4\right)[/latex] | |

The signs are the same, so the quotient is positive. | [latex]25[/latex] |

Check | [layex]25(-4)=-100[/latex] |

### try it

Just as we saw with multiplication, when we divide an integer by [latex]1[/latex], the result is the same number. What happens when we divide an integer by [latex]-1?[/latex] Let’s divide a positive number and then a negative number by [latex]-1[/latex] to see what we get.

[latex]\begin{array}{cccc}8\div \left(-1\right)\hfill & & & -9\div \left(-1\right)\hfill \\ -8\hfill & & & 9\hfill \\ \hfill \text{-8 is the opposite of 8}\hfill & & & \hfill \text{9 is the opposite of -9}\hfill \end{array}[/latex]

When we divide a number by [latex]-1[/latex] we get its opposite.

### Division by [latex]-1[/latex]

[latex]a\div \left(-1\right)=-a[/latex], for any integer [latex]a[/latex]

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

### example

Divide each of the following:

- [latex]16\div \left(-1\right)[/latex]
- [latex]-20\div \left(-1\right)[/latex]

Show Solution

### try it

Dividing any number [latex]\text{(except 0)}[/latex] by itself, produces a quotient of [latex]1[/latex]. Also, any number divided by [latex]1[/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.

### Division Properties of One

Any number (except 0) divided by itself is one. | [latex]a\div a=1[/latex] |

Any number divided by one is the same number. | [latex]a\div 1=a[/latex] |

### example

Divide. Then check by multiplying:

- [latex]11\div 11[/latex]
- [latex]\frac{19}{1}[/latex]
- [latex]1\overline{)7}[/latex]

Show Solution

### try it

Suppose we have [latex]\text{\$0}[/latex], and want to divide it among [latex]3[/latex] people. How much would each person get? Each person would get [latex]\text{\$0}[/latex]. Zero divided by any number is [latex]0[/latex].

Now suppose that we want to divide [latex]\text{\$10}[/latex] by [latex]0[/latex]. That means we would want to find a number that we multiply by [latex]0[/latex], to get [latex]10[/latex]. This cannot happen, because [latex]0[/latex] times any number is [latex]0[/latex]. Division by zero is said to be ** undefined**.

These two ideas make up the Division Properties of Zero.

### Division Properties of Zero

Zero divided by any number is [latex]0[/latex]. | [latex]0\div a=0[/latex] |

Dividing a number by zero is undefined. | [latex]a\div 0 = [/latex] undefined |

### example

Divide. Check by multiplying:

- [latex]0\div 3[/latex]
- [latex]\frac{10}{0}[/latex]

Show Solution

### try it

### Try It

Find the quotient:

- [latex]0\div (-24)[/latex]
- [latex]45\div 0[/latex]

Show Answer

Watch the following video for more examples of how to divide integers with the same and different signs.