## Introduction: Connecting her Learning Mathematics is no different. Once you have entered this world, you need to explore and discover the features (properties) that make operations favor addition, subtraction, multiplication, and division work. In this lesson, you will certainly examine few of the nature of actual numbers.

You are watching: Is the set of real numbers closed under subtraction

## Lesson Objectives

By the finish of this lesson, you should be may be to:

determine the basic properties of actual numbers.

## Presentation

### Basic properties of real Numbers

The straightforward properties of actual numbers are supplied to recognize the order in i beg your pardon you leveling math expressions. The straightforward properties of actual numbers include the following:

The Closure building The Commutative residential property The Associative residential or commercial property The Distributive residential or commercial property

Take a closer look at every property.

### The Closure Properties

genuine numbers room closed under addition, subtraction, and also multiplication.

That means if a and b are real numbers, then a + b is a distinctive real number, and a b is a unique real number.

For example:

3 and also 11 are real numbers.

3 + 11 = 14 and 3 ⋅ 11 = 33 notice that both 14 and 33 are actual numbers.

Any time you add, subtract, or multiply two actual numbers, the result will be a real number.

although this building seems obvious, some collections room not closed under particular operations.

Here room some examples.

Example 1

Real numbers are not closeup of the door under division since, although 5 and also 0 are actual numbers, and room not actual numbers. (You can say the is undefined, which method has actually no meaning. Likewise, is 2 because you have the right to multiply 3 by 2 to acquire 6. There is no number you have the right to multiply 0 by to obtain 5.)

Example 2

Natural numbers are not closeup of the door under subtraction. Although 8 is a natural number, 8 − 8 is not. (8 − 8 = 0, and 0 is no a natural number.)

Watch the following video for second explanation and also examples that the Closure Property.

 Math video clip Toolkit Closure Property

### The Commutative Properties

The commutative nature tell friend that 2 numbers deserve to be added or multiply in any order without affecting the result.

let a and b represent actual numbers.

 Commutative property of Addition Commutative building of Multiplication a + b = b + a a ⋅ b = b ⋅ a

 Commutative Properties: Examples 3 + 4 = 4 + 3 Both equal 7 5 + 7 = 7 + 5 Both represent the exact same sum 4 ⋅ 8 = 8 ⋅ 4 Both equal 32 y7 = 7y Both stand for the same product 5 (3+1) = (3+1) 5 Both stand for the exact same product (9 + 4) (5 + 2) = (5 + 2) (9 + 4) Both stand for the same product

Watch the following videos for a detailed explanation the the Commutative Properties.

 Math video clip Toolkit Commutative legislation of Addition Commutative legislation of Multiplication

## exercise Exercise: Commutative nature

It is time to exercise what you have learned. You will require a piece of a file and a pencil to finish the complying with activity. write down the appropriate number or letter the goes in the clip to do the statement true. Use the commutative properties. Once you room done, make certain to check your answers to see exactly how well friend did.

Practice Exercise

6 + 5 = ( ) + 6 m + 12 = 12 + ( )

9 ⋅ 7 = ( ) ⋅ 9

6a = a ( )

4 (k − 5) = ( ) 4

(9a −1)( ) = (2b + 7)(9a − 1)

6 + 5 = (5) + 6 m + 12 = 12 + (m) 9 ⋅ 7 = (7) ⋅ 9 6a = a(6) 4(k - 5) = (k - 5)4 (9a - 1)(2b + 7) = (2b + 7)(9a - 1)

Example

Simplify (rearrange into a much easier form): 5y6b8ac4

According to the commutative residential property of multiplication, you can reorder the variables and also numbers to and get every the numbers together and also all the letter together.

 5⋅6⋅8⋅4⋅y⋅b⋅a⋅c multiply the numbers 960ybac 960abcy By convention, when possible, write all letter in alphabetical order

Use the example over to finish the adhering to practice exercise.

Practice

leveling each that the adhering to quantities.

3a7y9d

6b8acz4 ⋅ 5

4p6qr3 (a + b)

189ady 960abcz 72pqr (a + b)

### The Associative Properties

The associative properties tell you the you may group together the amounts in any way without affect the result.

(Let a, b, and also c represent genuine numbers.)

 Associative residential or commercial property of Addition Associative property of Multiplication (a + b) + c = a + (b + c) (ab) c = a (bc)

The complying with examples show how the Associative properties of enhancement and multiplication deserve to be used.

 Associative home of Addition (2 + 6) + 1 = 2 + (6 + 1) 8 + 1 = 2 + 7 9 = 9 both equal 9
 Associative building of Multiplication (2 ⋅ 3) ⋅ 5 = 2 ⋅ (3 ⋅ 5) 6 ⋅ 5 = 2 ⋅ 15 30 = 30 both same 30

Watch the complying with videos for a thorough explanation that the Associative Properties.

 Math video Toolkit: Associative legislation of Addition Associative legislation of Multiplication

## practice Exercise: Associative nature

It is time to exercise what you have actually learned about the Associative Properties. Girlfriend will require to acquire out the a piece of a record and a pencil to finish the following activity. Compose down the ideal number or letter the goes in the clip to do the explain true. Usage the Associative Properties. As soon as you are done make certain to check your answers to see just how well you did.

Practice Exercise

(9 + 2) + 5 = 9 + ( )

x + (5 + y) = ( )+ y

(11a) 6 = 11 ( )

(9 + 2) + 5 = 9 + (2 + 5) x + (5 + y) = (x + 5) + y (11a) 6 = 11 (a ⋅ 6)

### The Distributive Properties

when you were first introduced come multiplication, you most likely recognized that that was arisen as a description for recurring addition.

consider this: 4 + 4 + 4 = 3 ⋅ 4

an alert that there are three 4s; the is, 4 shows up three times. Hence, 3 times 4. Algebra is generalised arithmetic, and also you can now make critical generalization.

when the number a is included repeatedly, definition n times, we have a + a + a + ⋯ + a (a shows up n times)

Then, making use of multiplication as a summary for repetitive addition, you have the right to replace a + a + a + ⋯ + a with n (a).

Example 1: x + x + x + x can be written as 4x since x is repeatedly added 4 times.

x + x + x + x = 4x

Example 2: r + r deserve to be created as 2r since r is repeatedly included 2 times.

r + r = 2r

The distributive property involves both multiplication and addition. Take a look at the explanation below.

Rewrite 4(a + b).

STEP 1: You continue by analysis 4(a + b) as multiplication: 4 time the amount (a + b).

This directs you to write:

4(a + b) = (a + b) + (a + b) + (a + b) + (a + b) = a + b +a + b + a + b + a + b

STEP 2: currently you usage the commutative property of enhancement to collection all the a′s together and also all the b′s together.

This directs you come write:

4(a + b) = a + a + a + a + b + b + b + b

4a′s + 4b′s

STEP 3: Now, you use multiplication together a description for recurring addition.

This directs united state to write:

4(a + b) = 4a + 4b

friend have dispersed the 4 over the sum to both a and also b. The Distributive Property

 The Distributive Property a (b + c) = a ⋅ b + a ⋅ c (b + c) a = b ⋅ a + c ⋅ a

Because that the commutative property and also the convention of creating the variables in alphabet order, friend can also write the following:

ba + c a as ab + a c, so (b + c )a = a b + ac too.

The distributive building is valuable when you cannot or perform not great to execute operations within parentheses.

Examples usage the distributive building to rewrite every of the complying with quantities.

2( 5 + 7) = 6 ( x + 3) = (z + 5) y = Watch the following videos because that a in-depth explanation the the Distributive Property.

 Math video Toolkit: Distributive Property

## practice Exercise: Distributive nature

Use the distributive building to rewrite each of the complying with quantities without the parentheses. Once you execute operations utilizing the distributive property, the is often called expanding the expression.

Practice Exercise

3 (2 + 1)

(x + 6) 7

4 (a + y)

(9 + 2) a

a (x + 5)

1 (x + y)

### The identification Properties

The number 0 is dubbed the additive identity due to the fact that when it is included to any kind of real number, it preserves the identity of that number. Zero is the only additive identity.

because that example: 6 + 0 = 6

Multiplicative Identity

The number 1 is called the multiplicative identity since when 1 is multiply by any type of real number, it preserves the identification of the number. One is the only multiplicative identity.

for example: 6 ⋅ 1 = 6.

The identity properties are summarized as follows.

 Additive identity Property Multiplicative identification Property If a is a real number, climate a + 0 = a and 0 + a = a If a is a actual number, then a ⋅ 1 = a and also 1 ⋅ a = a

Watch the complying with Khan Academy videos for an additional explanation and also examples that the identification Property.

 Math video Toolkit: Additive identity Property of 0 Multiplicative identification Property that 1

### The train station Properties

as soon as two number are included together and also the result is the additive identity, 0, the numbers are called additive inverses of every other.

Example as soon as 3 is added to −3, the result is 0: that is 3 + (−3) = 0.

The number 3 and −3 are additive inverses of every other.

What is the additive train station of −15?

For a an ext in-depth explanation the additive inverses, clock the following video by khan Academy.

Multiplicative Inverses

when two numbers are multiplied together and the an outcome is the multiplicative identity, 1, the numbers are dubbed multiplicative inverses of every other.

Example once 6 and room multiplied together, the an outcome is 1: that is, 6 ⋅ = 1.

The number 6 and space multiplicative inverses of each other.

What is the multiplicative train station of ? The inverse properties room as follows.

The Inverse properties

 If a is any real number, climate there is a distinct real number −a, such that a + (−a ) = 0 and also −a + a = 0 The numbers a and also −a are dubbed additive inverses of each other. If a is any type of nonzero real number, then there is a distinctive real number such the a ⋅ = 1 and also ⋅ a = 1 The numbers a and also are dubbed multiplicative inverses of each other.

For a more in-depth explanation that multiplicative inverses watch the following video clip by cannes Academy.

 Inverse residential property of Multiplication

## Exercise: Additive and also Multiplicative Inverses Complete the following exercise to exercise what you have actually learned around the Additive and Multiplicative Inverses by picking the attach below.See more: How Do Sponges Use Water To Carry Out Essential Functions, Phylum Porifera Additive and Multiplicative Inverses exercise when you have completed the practice, you can select the following link to see just how you did. inspect Additive and also Multiplicative Inverses answer Now the you have read over the lesson carefully and attempted the practice questions, it is time because that a expertise check. Note that this is a graded component of this module so be sure you have actually prepared yourself prior to starting.