Introduction: Connecting her Learning
Do you remember your an initial cell phone? How about your first GPS (Global placing System) or your first game system? even if it is you space thinking around the Droid in her pocket or the Nintendo v those clunky plastic cartridges the you had actually to blow dust the end of to acquire the video game to work, friend probably had actually a similar experience v the new technology. You take it time to discover every information of how your brand-new gadget worked. You may know exactly how to "turtle tip" in Mario brothers or download an app onto your tablet because of all the moment spent trying out those items.Mathematics is no different. When you have gotten in this world, you have to explore and also discover the attributes (properties) that make operations prefer addition, subtraction, multiplication, and department work. In this lesson, you will examine several of the nature of real numbers.
You are watching: Is the set of real numbers closed under addition
focusing Your Learning
Lesson Objectives
By the finish of this lesson, you must be able to:
identify the straightforward properties of real numbers.Presentation
Basic properties of real Numbers
The basic properties of actual numbers are provided to determine the order in which you simplify math expressions. The an easy properties of real numbers include the following:
The Closure home The Commutative building The Associative building The Distributive buildingTake a closer look at at every property.
The Closure Properties
actual numbers room closed under addition, subtraction, and multiplication.
That way if a and b are actual numbers, climate a + b is a distinct real number, and also a ⋅ b is a distinctive real number.
For example:
3 and also 11 are genuine numbers.
3 + 11 = 14 and also 3 ⋅ 11 = 33
notification that both 14 and also 33 are actual numbers.Any time you add, subtract, or multiply two actual numbers, the result will it is in a genuine number.
return this home seems obvious, part collections space not closed under certain operations.
Here room some examples.
Example 1
Real numbers space not close up door under department since, although 5 and 0 are real numbers, and also
are not actual numbers. (You have the right to say the is undefined, which means has actually no meaning. Likewise, is 2 since you have the right to multiply 3 by 2 to get 6. Over there is no number you can multiply 0 by to acquire 5.)Example 2
Natural numbers room not close up door under subtraction. Return 8 is a organic number, 8 − 8 is not. (8 − 8 = 0, and also 0 is not a herbal number.)
Watch the following video clip for second explanation and examples of the Closure Property.
Math video clip Toolkit Closure Property |
The Commutative Properties
The commutative nature tell you that 2 numbers can be added or multiplied in any kind of order without affecting the result.
allow a and also b represent real numbers.
Commutative property of Addition | Commutative home of Multiplication |
a + b = b + a | a ⋅ b = b ⋅ a |
Commutative Properties: Examples | |
3 + 4 = 4 + 3 | Both same 7 |
5 + 7 = 7 + 5 | Both represent the exact same sum |
4 ⋅ 8 = 8 ⋅ 4 | Both equal 32 |
y7 = 7y | Both represent the exact same product |
5 (3+1) = (3+1) 5 | Both stand for the exact same product |
(9 + 4) (5 + 2) = (5 + 2) (9 + 4) | Both represent the exact same product |
Watch the adhering to videos because that a detailed explanation the the Commutative Properties.
Math video Toolkit Commutative legislation of Addition Commutative legislation of Multiplication |
exercise Exercise: Commutative properties
It is time to practice what you have learned. You will require a piece of a document and a pencil to finish the complying with activity. create down the ideal number or letter the goes in the parentheses to do the declare true. Use the commutative properties. As soon as you space done, make sure to inspect your answers come see exactly how well girlfriend did.
Practice Exercise
6 + 5 = ( ) + 6 m + 12 = 12 + ( )
9 ⋅ 7 = ( ) ⋅ 9
6a = a ( )
4 (k − 5) = ( ) 4
(9a −1)( ) = (2b + 7)(9a − 1)
Check Answers
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 right into a much easier form): 5y6b8ac4
According to the commutative property of multiplication, you deserve to reorder the variables and numbers to and get all the numbers together and also all the letter together.
5⋅6⋅8⋅4⋅y⋅b⋅a⋅c | main point the numbers |
960ybac | |
960abcy | By convention, once possible, compose all letter in alphabetical order |
Use the example above to complete the following practice exercise.
Practice
leveling each that the following quantities.
3a7y9d
6b8acz4 ⋅ 5
4p6qr3 (a + b)
Check Answers
189ady 960abcz 72pqr (a + b)
The Associative Properties
The associative properties tell you that you may group together the quantities in any method without affecting the result.
(Let a, b, and c represent actual numbers.)
Associative property of Addition | Associative residential property of Multiplication |
(a + b) + c = a + (b + c) | (ab) c = a (bc) |
The following examples present how the Associative nature of enhancement and multiplication have the right to be used.
Associative building of Addition | |||
(2 + 6) + 1 | = | 2 + (6 + 1) | |
8 + 1 | = | 2 + 7 | |
9 | = | 9 | both equal 9 |
Associative residential or commercial property of Multiplication | |||
(2 ⋅ 3) ⋅ 5 | = | 2 ⋅ (3 ⋅ 5) | |
6 ⋅ 5 | = | 2 ⋅ 15 | |
30 | = | 30 | both equal 30 |
Watch the complying with videos because that a thorough explanation of the Associative Properties.
Math video clip Toolkit: Associative law of Addition Associative law of Multiplication |
exercise Exercise: Associative nature
It is time to practice what you have actually learned about the Associative Properties. You will need to gain out the a item of a paper and a pencil to finish the adhering to activity. Create down the ideal number or letter the goes in the parentheses to make the declare true. Use the Associative Properties. Once you are done make certain to inspect your answers come see just how well you did.
Practice Exercise
(9 + 2) + 5 = 9 + ( )
x + (5 + y) = ( )+ y
(11a) 6 = 11 ( )
Check Answers
(9 + 2) + 5 = 9 + (2 + 5) x + (5 + y) = (x + 5) + y (11a) 6 = 11 (a ⋅ 6)
The Distributive Properties
once you were first introduced to multiplication, you most likely known that the was occurred as a description for repeated addition.
take into consideration this: 4 + 4 + 4 = 3 ⋅ 4
notice that there room three 4s; the is, 4 shows up three times. Hence, 3 time 4. Algebra is generalised arithmetic, and you have the right to now make crucial generalization.
when the number a is included repeatedly, an interpretation n times, we have a + a + a + ⋯ + a (a appears n times)
Then, using multiplication together a summary for repeated addition, you have the right to replace a + a + a + ⋯ + a with n (a).
Example 1: x + x + x + x can be composed as 4x since x is repeatedly added 4 times.
x + x + x + x = 4x
Example 2: r + r can be created as 2r because r is repeatedly included 2 times.
r + r = 2r
The distributive property entails both multiplication and also addition. Take a look at the explanation below.
Rewrite 4(a + b).
STEP 1: You proceed by analysis 4(a + b) together multiplication: 4 time the quantity (a + b).
This directs you come 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 home of addition to collect all the a′s together and all the b′s together.
This directs you to write:4(a + b) = a + a + a + a + b + b + b + b
4a′s + 4b′s
STEP 3: Now, you use multiplication as a description for recurring addition.
This directs us to write:4(a + b) = 4a + 4b
friend have dispersed the 4 end the amount to both a and b.The Distributive Property
The Distributive Property | |
a (b + c) = a ⋅ b + a ⋅ c | (b + c) a = b ⋅ a + c ⋅ a |
Because the the commutative property and the convention of composing the variables in alphabetical order, girlfriend can additionally write the following:
b ⋅ a + c ⋅ a as a ⋅ b + a ⋅ c, so (b + c )a = a ⋅ b + a ⋅ c too.
The distributive building is beneficial when you can not or carry out not wish to perform operations inside parentheses.
Examples use the distributive building to rewrite every of the complying with quantities.
2( 5 + 7) =
6 ( x + 3) =
(z + 5) y =
Watch the adhering to videos for a comprehensive explanation that the Distributive Property.
Math video clip Toolkit: Distributive Property |
practice Exercise: Distributive properties
Use the distributive home to rewrite every of the following quantities there is no the parentheses. Once you perform operations using the distributive property, that 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)
Check Answers
The identity Properties
Additive identity
The number 0 is dubbed the additive identity because when it is included to any kind of real number, the preserves the identity of the number. Zero is the just additive identity.
for example: 6 + 0 = 6
Multiplicative Identity
The number 1 is dubbed the multiplicative identity because when 1 is multiplied by any kind of real number, that preserves the identity of the number. One is the only multiplicative identity.
for example: 6 ⋅ 1 = 6.
The identification properties space summarized as follows.
Additive identity Property | Multiplicative identity Property |
If a is a genuine number, climate a + 0 = a and also 0 + a = a | If a is a genuine number, climate a ⋅ 1 = a and 1 ⋅ a = a |
Watch the adhering to Khan Academy videos for secondary explanation and examples that the identification Property.
Math video Toolkit: Additive identity Property that 0 Multiplicative identity Property of 1 |
The station Properties
Additive Inverses
when two numbers are included together and the result is the additive identity, 0, the numbers are dubbed additive inverses of every other.
Example when 3 is included to −3, the an outcome is 0: the is 3 + (−3) = 0.
The number 3 and also −3 are additive inverses of every other.
What is the additive inverse of −15?
Answer: 15
For a an ext in-depth explanation that additive inverses, clock the following video clip by khan Academy.
Inverse building of Addition |
Multiplicative Inverses
as soon as two numbers space multiplied together and also the result is the multiplicative identity, 1, the number are dubbed multiplicative inverses of every other.
Example once 6 and also are multiplied together, the an outcome is 1: the is, 6 ⋅ = 1.
The number 6 and are multiplicative inverses of each other.
What is the multiplicative station of
?Answer:
The train station properties room as follows.
The Inverse nature
If a is any type of real number, climate there is a distinct real number −a, such the a + (−a ) = 0 and −a + a = 0 |
The number a and also −a are called additive inverses of every other. |
If a is any nonzero genuine number, then there is a distinctive real number such the a ⋅ = 1 and ⋅ a = 1 |
The numbers a and are called multiplicative inverses of every 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 adhering to exercise to practice what you have actually learned about the Additive and Multiplicative Inverses by picking the connect below. Additive and also Multiplicative Inverses practice when you have actually completed the practice, girlfriend can select the following connect to see exactly how you did. check Additive and Multiplicative Inverses answer |
Summarizing Your discovering
Did you recognize there to be so numerous kinds of nature for actual numbers? You have to now be familiar with closure, commutative, associative, distributive, identity, and inverse properties. Literal explanations were had to do the symbolic explanations simpler to interpret. Take a look in ~ the complying with Web site for added explanations of the nature of actual numbers.
Properties of real Numbers
Assessing your Learning
Now the you have actually read end the class carefully and also attempted the exercise questions, that is time because that a expertise check. Keep in mind that this is a graded part of this module therefore be certain you have prepared yourself prior to starting. |
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