## Multiplying Whole Numbers: Properties of Multiplication

### Learning Outcomes

• Identify and use the multiplication property of zero
• Identify and use the identity property of multiplication
• Identify and use the commutative property of multiplication
• Multiply multiple-digit whole numbers using columns that represent place value

## Multiply Whole Numbers

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.

Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers. Start with memorizing the multiplication facts through $9 \times 9$. Knowing the times tables up to $12 \times 12$ will allow your brain to focus on problem solving in future math questions so you’re not stuck on the arithmetic.

$×$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
$1$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$
$2$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $20$ $22$ $24$
$3$ $0$ $3$ $6$ $9$ $12$ $15$ $18$ $21$ $24$ $27$ $30$ $33$ $36$
$4$ $0$ $4$ $8$ $12$ $16$ $20$ $24$ $28$ $32$ $36$ $40$ $44$ $48$
$5$ $0$ $5$ $10$ $15$ $20$ $25$ $30$ $35$ $40$ $45$ $50$ $55$ $60$
$6$ $0$ $6$ $12$ $18$ $24$ $30$ $36$ $42$ $48$ $54$ $60$ $66$ $72$
$7$ $0$ $7$ $14$ $21$ $28$ $35$ $42$ $49$ $56$ $63$ $70$ $77$ $84$
$8$ $0$ $8$ $16$ $24$ $32$ $40$ $48$ $56$ $64$ $72$ $80$ $88$ $96$
$9$ $0$ $9$ $18$ $27$ $36$ $45$ $54$ $63$ $72$ $81$ $90$ $99$ $108$
$10$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$ $110$ $120$
$11$ $0$ $11$ $22$ $33$ $44$ $55$ $66$ $77$ $88$ $99$ $110$ $121$ $132$
$12$ $0$ $12$ $24$ $36$ $48$ $60$ $72$ $84$ $96$ $108$ $120$ $132$ $144$

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.

### Multiplication Property of Zero

The product of any number and $0$ is $0$.

$\begin{array}{}\\ a\cdot 0=0\hfill \\ 0\cdot a=0\end{array}$

### example

Multiply:

1. $0\cdot 11$
2. $\left(42\right)0$

Solution:

 1. $0\cdot 11$ The product of any number and zero is zero. $0$ 2. $\left(42\right)0$ Multiplying by zero results in zero. $0$

### try it

What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and $1$ is called the multiplicative identity.

### Identity Property of Multiplication

The product of any number and $1$ is the number.

$\begin{array}{c}1\cdot a=a\\ a\cdot 1=a\end{array}$

### example

Multiply:

1. $\left(11\right)1$
2. $1\cdot 42$

### try it

Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that $8+9=17$ is the same as $9+8=17$.

Is this also true for multiplication? Let’s look at a few pairs of factors.

$4\cdot 7=28\quad 7\cdot 4=28$
$9\cdot 7=63\quad 7\cdot 9=63$
$8\cdot 9=72\quad 9\cdot 8=72$

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

### Commutative Property of Multiplication

Changing the order of the factors does not change their product.

$a\cdot b=b\cdot a$

### example

Multiply:

$8\cdot 7$
$7\cdot 8$

### try it

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

$\begin{array}{c}\hfill 27\\ \hfill \underset{\text{___}}{\times 3}\end{array}$

We start by multiplying $3$ by $7$.

$3\times 7=21$

We write the $1$ in the ones place of the product. We carry the $2$ tens by writing $2$ above the tens place.

Then we multiply the $3$ by the $2$, and add the $2$ above the tens place to the product. So $3\times 2=6$, and $6+2=8$. Write the $8$ in the tens place of the product.

The product is $81$.

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.

### example

Multiply: $15\cdot 4$

### example

Multiply: $286\cdot 5$

### try it

When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.

### Multiply two whole numbers to find the product

1. Write the numbers so each place value lines up vertically.
2. Multiply the digits in each place value.
• Work from right to left, starting with the ones place in the bottom number.
• Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
• If a product in a place value is more than $9$, carry to the next place value.
• Write the partial products, lining up the digits in the place values with the numbers above.
• Repeat for the tens place in the bottom number, the hundreds place, and so on.
• Insert a zero as a placeholder with each additional partial product.

### example

Multiply: $62\left(87\right)$

### example

Multiply:

1. $47\cdot 10$
2. $47\cdot 100$

Multiply:

### example

Multiply: $\left(354\right)\left(438\right)$

### example

Multiply: $\left(896\right)201$

### try it

When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

 to multiply $8\cdot 3\cdot 2$ first multiply $8\cdot 3$ $24\cdot 2$ then multiply $24\cdot 2$ $48$

In the video below, we summarize the concepts presented on this page including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.