The Real Number Properties Algebra 1 Homework And Practice

A real number is a value that represents a quantity along a continuous number line. 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 real numbers include: 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 (such as ).

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".

The properties of the Real Number System 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)

Examples

Verbal hints

1.

Distributive Property
a • (b + c) = a • b + a • c

3 • (4 + 5) = 3 • 4 + 3 • 5

"multiplication distributes across addition"

2.

Commutative Property of Addition
a + b = b + a

3 + 4 = 4 + 3

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

3.

Commutative Property of Multiplication
a • b = b • a

3 • 4 = 4 • 3

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

4.

Associative Property of Addition
a + (b + c) = (a + b) + c

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

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

5.

Associative Property of Multiplication
a • (b • c) = (a • b) • c

3 • (4 • 5) = (3 • 4) • 5

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

6.

Additive Identity Property
a + 0 = a

4 + 0 = 4

"the value that returns the input unchanged"

7.

Multiplicative Identity Property
a • 1 = a

4 • 1 = 4

"the value that returns the input unchanged"

8.

Additive Inverse Property
a + (-a) = 0

4 + (-4) = 0

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

9.

Multiplicative Inverse Property

"the value that brings you back to the identity element under multiplication"

10.

Zero Property of Multiplication
a • 0 = 0

4 • 0 = 0

"zero times any value is 0"

11.

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"

12.

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"

13.

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."

14.

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."

15.

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."

16.

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."

17.

Substitution Property
If a = b, then a may be substituted for b, or conversely.

If x = 5, and x + y = z,
then 5 + y = z.

"a value may be substituted for its equal."

18.

Reflexive (or Identity) Property of Equality
a = a

12 = 12

"a real number is always
equal to itself"

19.

Symmetric Property of Equality
If a = b, then b = a.

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

20.

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"

21.

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)

FYI: Regarding the term "natural numbers", there is no universal agreement about whether to include zero in this set. Most mathematicians hold with the older traditional and define natural numbers to be the counting numbers [positive integers {1, 2, 3, ...}]. Computer scientists, set theorists, logicians and other mathematicians define natural numbers to be the whole numbers [non-negative integers {0, 1, 2, ...}].This site will use the term "natural numbers" to refer to the counting numbers {1, 2, 3, ...}.


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Properties Worksheets

Properties of Mathematics Worksheets

These dynamically created Properties Worksheets are great for learning and practicing the different properties of mathematics. These Properties Worksheets are ideal for 4th Grade, 5th Grade, 6th Grade, and 7th Grade students.

Click here for a Detailed Description of all the Properties Worksheets.



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Detailed Description for All Properties Worksheets

Definition for Properties of Mathematics Worksheet
This Properties Worksheet is a great handout for reinforcing the different properties of Mathematics. This handout include the Associative Property, Commutative Property, Distributive Property, Identity Property, Additive Inverse Property, Multiplicative Inverse Property, Addition Property of Zero, Multiplication Property of Zero, Property of Equality, Reflexive Property, Symmetric Property, and Transitive Property. This Properties Worksheet is ideal for 4th Grade, 5th Grade, 6th Grade, and 7th Grade students.

Identifying Properties of Mathematics Worksheets
These Properties Worksheets are great for testing students on identifying the different properties of mathematics, such as the Associative Property, Commutative Property, Distributive Property, Identity Property, Additive Inverse Property, Multiplicative Inverse Property, Addition Property of Zero, and Multiplication Property of Zero. These Properties Worksheets are ideal for 4th Grade, 5th Grade, 6th Grade, and 7th Grade students.

Working with Properties of Mathematics Worksheets
These Properties Worksheets are great for testing students their working knowledge of the different properties of mathematics, such as the Associative Property, Commutative Property, Distributive Property, Identity Property, Additive Inverse Property, Multiplicative Inverse Property, Addition Property of Zero, and Multiplication Property of Zero. These Properties Worksheets are ideal for 4th Grade, 5th Grade, 6th Grade, and 7th Grade students.


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