Properties of Multiplication

The properties of multiplication define certain relationships that always hold true when multiplying real numbers. You may be familiar with some of these concepts already, but it can be helpful to understand why math rules work and what they are called.

Importantly, some math textbooks only list a few properties of multiplication, while others may use slightly different names than you'll see below. Regardless, understanding all of the properties of multiplication listed below will help you deepen your understanding of mathematics.

Properties of multiplication: Identity property

The identity property states that for all real numbers a, multiplying by 1 gives you the same number a. Put another way, the following is true:

$\left(1\right)a=a$

$a\left(1\right)=a$

For example, $5\times 1=5$ and $1\times 7=7$ . Since multiplying any real number by 1 gets you the number you started with, one is considered the identity element of multiplication.

Properties of multiplication: Inverse property

The inverse property states that for all non-zero real numbers a, multiplying a by the fraction 1a yields a product of 1. Expressed in mathematical terms, we get:

$a\times \frac{1}{a}=1$

$\frac{1}{a}\left(a\right)=1$

For example, $5\times \frac{1}{5}=1$ and $\frac{1}{7}\left(7\right)=1$ . The inverse property means that $\frac{1}{a}$ is considered the reciprocal of a and that $\frac{1}{a}$ is the multiplicative inverse of a. We need to specify non-zero because zero is a real number but you cannot divide by zero.

Properties of multiplication: Multiplicative property of zero

The multiplicative property of zero states that any real number multiplied by zero produces a product of zero. Expressed mathematically, we get:

$a\left(0\right)=0$

$\left(0\right)a=0$

So, $100\times 0=0$ and $7936980259123\left(0\right)=0$ . This makes sense as any number taken zero times will always be zero no matter how big it might be.

Properties of multiplication: Commutative property of multiplication

The commutative property of multiplication states that for all real numbers a and b, the order you multiply them does not change the result. In mathematical terms,

$a\times b=b\times a$

For instance, $3\times 5=15$ and $5\times 3=15$ . If you're working with negative numbers, $-3\times 5=-15$ and $5\times \left(-3\right)=-15$ . Addition has a commutative property too, but you don't have to worry about mixing them up because they are the same.

Properties of multiplication: Associative property of multiplication

The associative property of multiplication states that when you multiply any three real numbers a, b, and c, the order you multiply them does not change the result. We can express this mathematically like this:

$\left(a\times b\right)\times c=a\times (b\times c)$

For example, $(2\times 3)\times 4=24$ and $2\times (3\times 4)=24$ . Once again, working with negative numbers doesn't change how the associative property of multiplication works, and there is an associative property of addition that works the same way.

Properties of multiplication: Multiplicative property of -1

The multiplicative property of -1 states that multiplying any real non-zero number a by -1 gets you a product of -a. Here's what it looks like in mathematical terms:

$a\left(-1\right)=-a$

$\left(-1\right)\left(a\right)=-a$

For example, $8\times \left(-1\right)=-8$ and $-1\times 9=-9$ . This holds true for negative numbers as well since $-3\times \left(-1\right)=3$ and $-\left(-3\right)$ cancels out the minus sign. The multiplicative property of -1 is similar to the identity property except that you add a minus sign.

Properties of multiplication: Property of opposites in products

The property of opposites in products states that multiplying a positive number by a negative number produces a negative product while multiplying two negative numbers produces a positive product whenever you're working with real numbers. We need three equations to express this one mathematically:

$a\times \left(-b\right)=-ab$

$b\times \left(-a\right)=-ab$

$-a\times \left(-b\right)=ab$

For instance, $4\times \left(-3\right)=-12$ , $-4\times 3=-12$ , but $-4\times \left(-3\right)=12$ . Watch those signs!

Properties of multiplication practice questions

a. $a\times 1=?$

$a$

b. What is the reciprocal of a?

$\frac{1}{a}$

c. $7093958003857\times 0=?$

$0$

d. $10\times 9=90$ and $9\times 10=90$ . Which property of multiplication does this illustrate?

Commutative property of multiplication

e. $\left(5\times 3\right)\times 8=5\times \left(3\times 8\right)$ . Which property of multiplication does this illustrate?

Associative Property of Multiplication

f. what is $-1\left(467892\right)$ ?

$-467892$

g. Assuming real positive numbers a and b, would $-a\left(b\right)$ be positive or negative? Why?

Negative, the property of opposites in products

Topics related to the Properties of Multiplication

Factorials

Basic Operations

Commutative Property

Flashcards covering the Properties of Multiplication

3rd Grade Math Flashcards

Basic Arithmetic Flashcards

Practice tests covering the Properties of Multiplication

Common Core: 3rd Grade Math Diagnostic Tests

3rd Grade Math Practice Tests

Varsity Tutors offers assistance with the properties of multiplication

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