# Basic Number Properties Unit Three

## Properties of Multiplication

Multiplication is counting by groups. If I add 2 + 2 + 2 or add two, three times, the result is 2 * 3 or 6. Numbers have certain properties when they are being multiplied.

Remember in algebra we never really divide. We always multiply by the a number's reciprocal.

Remember we talked about inverse. The multiple inverse of a number is it's reciprocal.

Look

You will need to remember the definition of reciprocal for the Multiplication properties.

All multiplication properties apply to division when you multiply by the reciprocal

Definition of division is multiplying a number by the reciprocal of the divisor.

12 divided by 3 is 4.

12 times 1/3 is 4.

### Introduction

In this unit we will explore the number properties of Multiplication.

These properties do not typically have numbers assigned to them.

This page refers to each property by number only for organizational purposes.

### Multiplication Property 1

Zero Property of Multiplication.

If I had a million dollars, zero times how much money would I have?

I would have nothing! Therefore the zero property of multiplication is any number times zero is zero or

n * 0 = 0

### Identity Property of Multiplication

Identity Property of Multiplication: Do you remember what identity means in math?

What can we multiply by any number to get the same number?

If you said one you would be correct!

n * 1 is always n.

Therefore the Identity property of multiplication says any number times 1 is that number or

n * 1 = n

### Inverse Property of Multiplication

Any number divided by itself is one correct?

How do you think this applies to your new definition of devision?

Consider the reciprocal example

gives the result of 6/6 which is 1

8 * 1/8 = 1

4/5 * 5/4 = 1

Therefore any number multiplied by its multiplicative inverse (reciprocal)is equal to 1

a/b * b/a = 1

a * 1/a = 1

### Commutative Property of Multiplication

Do you remember the definition of commute?

If you walk or ride a bus or drive a car, it means you commute.

So if we move the numbers around in a multiplication expression we always get the same result.

2 * 5 = 10 and 5 * 2 = 10

or for any numbers a and b

a * b = b * a

### Associative Property of Multiplication

How about the definition of associate?

In English the word association deals with a group.

For example your group of friends would be considered your associates.

Just like addition in multiplication regardless of what numbers are associated with each other the result is always the same

2 * (5 * 3) = 2 * 15 = 30

(2 * 5) * 3 = 10 * 3 = 30

See? Same answer therefore the Associative property of multiplication can be represented for any three numbers a, b and c.

a * (b * c) = (a * b) * c

### Distributive Property

What does it mean to distribute flyers? a test?

When a teacher distributes a test, she gives one test it each student.

In math when we distribute a number we multiply it by every term with in an association.

Consider:

2*(3 + 5)= 2 * 8 = 16

if we distribute the 2 we get:

(2 * 3) + (2 * 5) which is the same as 6 + 10 = 16

2 * 3 = 6 : 2 * 5 = 10

Notice that 2*(3 + 5) = (2 * 3) + (2 * 5) = 16

Therefore the Distribute Property says

a*(b + c) = a*b + a*c

a*(b - c) = a*b - a*c

### Multiplying and Dividing Positive and Negative Numbers

When you multiply two numbers with the same sign you always get a positive answer.

2 * 2 = 4 and -3 * -5 = 15

When you multiply two numbers with different signs you always get a negative answer.

-2 * 2 = -4 and 3 * -5 = -15

This also can be combined with the identity property such that any number * -1 is its additive inverse.

This is know as the multiplication property of -1

a * -1 = -a

-a * -1 = a

### Quick Check

Right click on the quick check below and open the link in a new window

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