### Learning Outcomes

- Identify the multiplication and division properties of zero

## Use the Properties of Zero

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

## Multiplication by Zero

What happens when you multiply a number by [latex]0?[/latex] Multiplying by [latex]0[/latex] makes the product equal zero. The product of any real number and [latex]0[/latex] is [latex]0[/latex].

### Multiplication by Zero

For any real number [latex]a[/latex],

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

### Exercises

Simplify:

1. [latex]-8\cdot 0[/latex]

2. [latex]\Large\frac{5}{12}\normalsize\cdot 0[/latex]

3. [latex]0\left(2.94\right)[/latex]

Solution:

1. | |

[latex]-8\cdot 0[/latex] | |

The product of any real number and 0 is 0. | [latex]0[/latex] |

2. | |

[latex]\Large\frac{5}{12}\normalsize\cdot 0[/latex] | |

The product of any real number and 0 is 0. | [latex]0[/latex] |

3. | |

[latex]0\left(2.94\right)[/latex] | |

The product of any real number and 0 is 0. | [latex]0[/latex] |

### TRY IT

## Dividing with Zero

What about dividing with [latex]0?[/latex] Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are [latex]0[/latex] cookies to share, so each person gets [latex]0[/latex] cookies.

[latex]0\div 3=0[/latex]

Remember that we can always check division with the related multiplication fact. So, we know that

[latex]0\div 3=0\text{ because }0\cdot 3=0[/latex]

**Tip**: If we think of 0/4 in terms of pizza, where we have 4 slices and you get 0. This could happen, but you would get 0!

### Division of Zero

For any real number [latex]a[/latex], except [latex]0,\Large\frac{0}{a}\normalsize =0[/latex] and [latex]0\div a=0[/latex].

Zero divided by any real number except zero is zero.

### Exercises

Simplify:

1. [latex]0\div 5[/latex]

2. [latex]\Large\frac{0}{-2}[/latex]

3. [latex]0\div\Large\frac{7}{8}[/latex]

### TRY IT

Now let’s think about dividing a number *by* zero. What is the result of dividing [latex]4[/latex] by [latex]0?[/latex] Think about the related multiplication fact. Is there a number that multiplied by [latex]0[/latex] gives [latex]4?[/latex]

[latex]4\div 0=[/latex]means [latex]x\cdot {0}=4[/latex]

Since any real number multiplied by [latex]0[/latex] equals [latex]0[/latex], there is no real number that can be multiplied by [latex]0[/latex] to obtain [latex]4[/latex]. We can conclude that there is no answer to [latex]4\div 0[/latex], and so we say that division by zero is undefined.

**Tip**: If we think of 3/0 in terms of pizza, where we have 0 slices and you want 3. This is like arguing with a three year old! I know you want 3 slices, but we don’t have any pizza. Therefore, this can NOT happen! So 3/0 is undefined.

### Division by Zero

For any real number [latex]a,\Large\frac{a}{0}[/latex], and [latex]a\div 0[/latex] are undefined.

Division *by* zero is undefined.

### Exercises

Simplify:

1. [latex]7.5\div 0[/latex]

2. [latex]\Large\frac{-32}{0}[/latex]

3. [latex]\Large\frac{4}{9}\normalsize\div 0[/latex]

### TRY IT

Below we summarize the properties of zero.

### Properties of Zero

**Multiplication by Zero:** For any real number [latex]a[/latex],

[latex]\begin{array}{c}a\cdot 0=0\text{ The product of any number and 0 is 0.}\hfill \end{array}[/latex]

**Division by Zero:** For any real number [latex]a,a\ne 0[/latex]

[latex]{\Large\frac{0}{a}}=0[/latex] Zero divided by any real number, except itself, is zero.

[latex]{\Large\frac{a}{0}}[/latex] is undefined. Division by zero is undefined.

Watch the following video for more examples of using the multiplication and division property of zero.