Lesson Properties of Real Numbers
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• Slide 1
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Properties of Real Numbers

Text • Slide 2
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What is a real number?

A real number is a value that represents a quantity along a continuous number line. • Question 3
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Q.

Which is not true about real numbers?

Real numbers can be ordered.

The symbol for the set of real numbers is R, which is the letter R in the typeface "blackboard bold".

They are not irrational numbers

• Slide 4
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• Real numbers can be ordered.
• The symbol for the set of real numbers is , which is the letter R in the typeface "blackboard bold".
• Slide 5
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The numbers of Real numbers

• counting (natural) numbers (){1, 2, 3, ... }
• whole numbers {0, 1, 2, 3, ... }
• integers (){... , -3, -2, -1, 0, 1, 2, 3, ...}
• rational numbers () (such as -½, 6.25, )
•  and irrational numbers

FYI: The series of three dots ( ... ), seen within the sets above, is called an ellipsis.

As used in mathematics, the ellipsis means "and so forth".

• Question 6
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Q.

The properties of the Real Number System will prove useful when working with equations, functions and formulas in Algebra.

True

False

• Question 7
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Q.

The properties of real numbers, does not allow the creation of equivalent expressions which will often aid in solving problems.

True

False

• Question 8
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Q.

In addition, they can be used to help explain or justify solutions.

True

False

• Slide 9
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• will prove useful when working with equations, functions and formulas in Algebra,
• as they allow for the creation of equivalent expressions which will often aid in solving problems.
• In addition, they can be used to help explain or justify solutions.
• Slide 10
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Property (a, b and c are real numbers, variables or algebraic expressions) • Slide 11
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1. Distributive Property

a • (b + c) = a • b + a • c

• 3 • (4 + 5) = 3 • 4 + 3 • 5

• Slide 12
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a + b = b + a

• "commute = to get up and move to a new location : switch places"

• 3 + 4 = 4 + 3

• Slide 13
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3. Commutative Property of Multiplication

a • b = b • a

• "commute = to get up and move to a new location: switch places"

• 3 • 4 = 4 • 3

• Question 14
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Q.

8 + 4 = 12, so 4+8 =

12

3

4

14

• Question 15
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Q.

Which of the following illustrates the Commutative Law of Multiplication?

3 x 4 = 6 x 2

3 x 4 = 12 x 1

3 x 4 = 2 x 6

3 x 4 = 4 x 3

• Question 16
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Q.

Which one of the following illustrates the Commutative Law of Addition?

5 + 7 = 7 + 5

5 + 7 = 4 + 8

5 + 7 = 3 + 9

5 + 7 = 10 + 2

• Question 17
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Q.

3 x ____ = 9 x 3

7

9

27

11

• Question 18
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Q.

What is the missing number?

6 + ____ = 7 + 6

4

13

7

9

• Slide 19
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+ (b + c) = (a + b) + c

• "regroup - elements do not physically move, they simply group with a new friend."

• 3 + (4 + 5) = (3 + 4) + 5

• Slide 20
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5. Associative Property of Multiplication

a • (b • c) = (a • b) • c

• "regroup - elements do not physically move, they simply group with a new friend."

• 3 • (4 • 5) = (3 • 4) • 5

• Question 21
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Q.

Which of the following illustrates the Associative Law of multiplication?

4 × (3 × 6) = (3 × 12) × 2

4 × (3 × 6) = (6 × 6) × 2

4 × (3 × 6) = (4 × 3) × 6

4 × (3 × 6) = (3 × 8) × 3

• Question 22
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Q.

Which one of the following illustrates the Associative Law of addition?

3 + ( 2 + 4) = (4 + 4) + 1

3 + (2 + 4) = (3 + 2) + 4

3 + (2 + 4) = (5 + 2) + 2

3 + (2 + 4) = (2 + 6) + 1

• Question 23
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Q.

5 × (2 × 9) is also equal to:

5 x (2 - 9)

5 x (2 + 9)

(5 x 2) x 9

(5 x 2) + 9

• Question 24
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Q.

What is the missing number?

5 × (__× 9) = (5 × 8) × 9

5

360

8

9

• Question 25
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Q.

What is the missing number?

4 + (____ + 5) = (4 + 7) + 5

4

5

16

7

• Question 26
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Q.

7 + (4 + 2) = 13, so (7 + 4) + 2

10

12

13

14

• Slide 27
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a + 0 = a

• "the value that returns the input unchanged"

• 4 + 0 = 4

• Slide 28
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7. Multiplicative Identity Property

a • 1 = a

• "the value that returns the input unchanged"

• 4 • 1 = 4

• Question 29
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Q.

Which operation doesn't fully apply to the Identity Property ?

Multiplication

Division

Subtraction

Mouctar

• Question 30
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Q.

When using identity property the identity number always changes ?

True

False

Split

...

• Question 31
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Q. Pick the answer that is an example of the identity property of addtion
a + (-a) = 0
1 + 1 = 2
a + 0 = a
a + b = b + a
• Question 32
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Q.

Identify the property

a + 0 = a

identity property of multiplication

multiplication property of zero

• Question 33
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Q. Identify the property
a + b = b + a
symmetric
• Slide 34
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a + b is a real number

10 + 5 = 15 (a real number)

"the sum of any two real numbers

is another real number"

• Slide 35
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Closure Property of Multiplication

a • b is a real number

10 • 5 = 50 (a real number)

"the product of any two real numbers is another real number"

• Slide 36
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If a = b, then a + c = b + c.

If x = 10,

then + 3 = 10 + 3

"adding the same value to both sides of an equation will not change the truth value of the equation."

• Slide 37
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Subtraction Property of Equality

If a = b, then a - c = b - c.

If x = 10,

then - 3 = 10 - 3

"subtracting the same value from both sides of an equation will not change the truth value of the equation."

• Slide 38
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Multiplication Property of Equality

If a = b, then a • c = b • c.

If x = 10,

then x • 3 = 10 • 3

"multiplying both sides of an equation by the same value will not change the truth value of the equation."

• Slide 39
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Division Property of Equality

If a = b, then a / c = b / c, assuming c ≠ 0.

If x = 10,

then / 3 = 10 / 3

"dividing both sides of an equation by the same non-zero value will not change truth value of the equation."

• Slide 40
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Reflexive (or Identity) Property of Equality

a = a

12 = 12

"a real number is always

equal to itself"

• Slide 41
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Reflexive (or Identity) Property of Equality

If a = b, then b = a.

"quantities that are equal can be read forward or backward" • Slide 42
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Transitive Property of Equality

If a = b and b = c,then a = c.

If 2a = 10 and 10 = 4b,

then 2a = 4b.

"if two numbers are equal to the same number, then the two numbers are equal to each other"

• Slide 43
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Law of Trichotomy

Exactly ONE of the following holds:

a < b, a = b, a > b

If 8 > 6, then 8  6 and

8 is not < 6.

"for two real numbers a and ba is either equal to b, greater than b, or less than b." (common sense)

• Slide 44
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Reflexive (or Identity) Property of Equality

a = a

12 = 12

"a real number is always

equal to itself"

• Slide 45
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Quiz

Text • Question 46
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Q.

It refers to "the value that brings you back to the identity element under addition"

Associative Property

Inverse Property

Identity Property

Distributive Property

• Question 47
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Q.

It refers to "multiplication distributes across addition".

Distributive Property

Commutative Property of Multiplication

• Question 48
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Q.

It refers to "commute = to get up and move to a new location : switch places"

Distributive Property

Commutative Property

Inverse Property

Associative Property

• Question 49
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Q.

It refers to"regroup - elements do not physically move, they simply group with a new friend."

Distributive Property

Commutative Property of Multiplication

• Question 50
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Q.

It refers to "the value that returns the input unchanged"

Inverse Property

Inverse Property

Identity Property

Commutative Property

• Question 51
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Q.

It refers to "for two real numbers a and b, a is either equal to b, greater than b, or less than b."

Transitive Property

Symmetric Property

Law of Trichotomy

Substitution Property

• Question 52
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Q.

It refers to "if two numbers are equal to the same number, then the two numbers are equal to each other"

Transitive Property

Symmetric Property

Law of Trichotomy

Substitution Property

• Question 53
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Q.

It refers to "quantities that are equal can be read forward or backward"

Transitive Property

Symmetric Property

Law of Trichotomy

Substitution Property

• Question 54
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Q.

It refers to "a value may be substituted for its equal."

Transitive Property

Symmetric Property

Law of Trichotomy

Substitution Property

• Question 55
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Q.

It refers to "the product of any two real numbers is another real number"

Multiplicative Identity Property

Multiplicative Inverse Property

Closure Property of Multiplication

Multiplication Property of Equality

• Slide 56
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Questions?

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