Real Numbers and Equality Properties

Properties of Real Numbers

What is a real number?

A real number is a value that represents a quantity along a continuous number line.

Multiple Choice

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

Concepts about real numbers

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

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".

Multiple Choice

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

• True

• False

Multiple Choice

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

• True

• False

Multiple Choice

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

• True

• False

Concepts about real numbers

• 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.

Property (a, b and c are real numbers, variables or algebraic expressions)

1. Distributive Property

a • (b + c) = a • b + a • c

• "multiplication distributes across addition"

• 3 • (4 + 5) = 3 • 4 + 3 • 5

2. Commutative Property of Addition

a + b = b + a

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

• 3 + 4 = 4 + 3

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

Multiple Choice

8 + 4 = 12, so 4+8 =

• 12

• 3

• 4

• 14

Multiple Choice

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

Multiple Choice

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

Multiple Choice

3 x ____ = 9 x 3

• 7

• 9

• 27

• 11

Multiple Choice

What is the missing number?

6 + ____ = 7 + 6

• 4

• 13

• 7

• 9

4. Associative Property of Addition

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

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

Multiple Choice

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

Multiple Choice

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

Multiple Choice

5 × (2 × 9) is also equal to:

• 5 x (2 - 9)

• 5 x (2 + 9)

• (5 x 2) x 9

• (5 x 2) + 9

Multiple Choice

What is the missing number?

5 × (__× 9) = (5 × 8) × 9

• 5

• 360

• 8

• 9

Multiple Choice

What is the missing number?

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

• 4

• 5

• 16

• 7

Multiple Choice

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

• 10

• 12

• 13

• 14

6. Additive Identity Property

a + 0 = a

• "the value that returns the input unchanged"

• 4 + 0 = 4

7. Multiplicative Identity Property

a • 1 = a

• "the value that returns the input unchanged"

• 4 • 1 = 4

Multiple Select

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

• Multiplication

• Addition

• Division

• Subtraction

• Mouctar

Multiple Choice

When using identity property the identity number always changes ?

• True

• False

• Split

• ...

Multiple Choice

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

Multiple Choice

Identify the property

a + 0 = a

• identity property of addition

• identity property of multiplication

• inverse property of addition

• multiplication property of zero

Multiple Choice

Identify the property
a + b = b + a

• Identity of addtion
• commutative of addition
• associative of addition
• symmetric

Closure Property of Addition

a + b is a real number

10 + 5 = 15 (a real number)

"the sum of any two real numbers

is another real number"

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"

Addition Property of Equality

If a = b, then a + c = b + c.

If x = 10,

then x + 3 = 10 + 3

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

Subtraction Property of Equality

If a = b, then a - c = b - c.

If x = 10,

then x - 3 = 10 - 3

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

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."

Division Property of Equality

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

If x = 10,

then x / 3 = 10 / 3

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

Reflexive (or Identity) Property of Equality

a = a

12 = 12

"a real number is always

equal to itself"

Reflexive (or Identity) Property of Equality

If a = b, then b = a.

"quantities that are equal can be read forward or backward"

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"

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 b, a is either equal to b, greater than b, or less than b." (common sense)

Reflexive (or Identity) Property of Equality

a = a

12 = 12

"a real number is always

equal to itself"

Quiz

Multiple Choice

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

• Associative Property

• Inverse Property

• Identity Property

• Distributive Property

Multiple Choice

It refers to "multiplication distributes across addition".

• Distributive Property

• Commutative Property of Addition

• Commutative Property of Multiplication

• Associative Property of Addition

Multiple Choice

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

• Distributive Property

• Commutative Property

• Inverse Property

• Associative Property

Multiple Choice

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

• Distributive Property

• Commutative Property of Addition

• Commutative Property of Multiplication

• Associative Property of Addition

Multiple Choice

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

• Inverse Property

• Inverse Property

• Identity Property

• Commutative Property

Multiple Choice

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

Multiple Choice

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

Multiple Choice

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

• Transitive Property

• Symmetric Property

• Law of Trichotomy

• Substitution Property

Multiple Choice

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

• Transitive Property

• Symmetric Property

• Law of Trichotomy

• Substitution Property

Multiple Choice

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

Questions?