We’re Open - Call Now!

Call Now to Set Up Tutoring

(888) 736-0920

We’re Open - Call Now!

Call Now to Set Up Tutoring

(888) 736-0920

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.

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.

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.

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.

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.

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.

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.

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!

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

Common Core: 3rd Grade Math Diagnostic Tests

The properties of multiplication above are essential tools in the study of algebra, especially once students begin working with polynomials and other algebraic equations. If the student in your life is having a tough time memorizing them all, a 1-on-1 math tutor can provide fresh explanations and examples until the information finally clicks. Reach out to the Educational Directors at Varsity Tutors right now to get connected with a tutor and learn more about what makes private instruction such a powerful educational tool.

Subjects Near Me

- Topology Tutors
- Elementary School Science Tutors
- WEST-E Test Prep
- Utah Bar Exam Test Prep
- Series 86 Test Prep
- IB Biology SL Tutors
- Missouri Bar Exam Test Prep
- API - Associate in Personal Insurance Test Prep
- Series 66 Courses & Classes
- Louisiana EOC Courses & Classes
- PS Exam - Professional Licensed Surveyor Principles of Surveying Exam Test Prep
- 4th Grade Spanish Tutors
- Risk Management Tutors
- IB Global Politics HL Tutors
- Microbiology Tutors
- Series 28 Test Prep
- PCAT Reading Comprehension Tutors
- ARDMS - American Registry for Diagnostic Medical Sonography Test Prep
- General Relativity Tutors
- Biochemistry Tutors

Popular Cities