Properties of Real Numbers. COMMUTATIVE PROPERTY.

Properties of Real Numbers

COMMUTATIVE PROPERTY

Commutative Property of Addition

When adding two numbers, the order of the numbers does not matter.

Examples:

a + b = b + a

2 + 3 = 3 + 2

(-5) + 4 = 4 + (-5)

Commutative Property of Multiplication

When multiplying two numbers, the order of the numbers does not matter.

Examples:

a • b = b • a

2 • 3 = 3 • 2

(-3) • 24 = 24 • (-3)

ASSOCIATIVE PROPERTY

Associative Property of AdditionWhen three numbers are added, it makes no difference which two numbers are added first.

Examples:

a + (b + c) = (a + b) + c

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

(4 + 2) + 6 = 4 + (2 + 6)

Associative Property of Multiplication

When three numbers are multiplied, it makes no difference which two numbers are multiplied first.

Examples:

a(bc) = (ab)c

2 • (3 • 5) = (2 • 3) • 5

(4 • 2) • 6 = 4 • (2 • 6)

DISTRIBUTIVE PROPERTY

Distributive Property

Multiplication distributes over addition.

Examples:

a(b + c) = ab + ac

2 (3 + 5) = (2 • 3) + (2 • 5)

(4 + 2) • 6 = (4 • 6) + (2 • 6)

IDENTITY PROPERTY

Additive Identity Property

The additive identity property states that if 0 is added to a number, the result is that number.

Example:

a + 0 = a

3 + 0 = 0 + 3 = 3

Multiplicative Identity Property

The multiplicative identity property states that if a number is multiplied by a 1, the result is that number.

Example:

a • 1= a

3 • 1= 3

INVERSE OPERATIONS PROPERTY

Inverse Operations Property

An operation that reverses the effect of another operation.

Examples: addition and subtraction

multiplication and division

squares and square roots

Additive Inverse Property

The additive inverse property states that opposites add to zero.

Examples:

k + (- k) = 0

7 + (-7) = 0

4x – 8 = 0

+ 8 = +8

Multiplicative Inverse Property

The multiplicative inverse property states that reciprocals multiply to 1.

515

1

23

32

1

EQUALITY PROPERTY

Equality PropertyFor all operations: The property that states that if you perform an operation with the same number to both sides of an equation, the sides remain equal (i.e., the equation continues to be true.)

2 Most Common:

Multiplicative Property of Equality

Additive Property of Equality

Multiplicative Property of Equality

The two sides of an equation remain equal if they are multiplied by the same number. That is: for any real numbers a, b, and c, if a = b, then ac = bc.

x = y; and x • 8 = y • 8

m=20 ; and m • 2 = 20 • 2

Additive Property of Equality• Additive Property of Equality: The two

sides of an equation remain equal if they are increased by the same number.

• That is: for any real numbers a, b, and c, if a = b, then a + c = b + c.

x = y; and x + 6 = y + 6

m=15 ; and m + 5 = 15 + 5

ZERO PROPERTY OF MULTIPLICATION

Zero Property of MultiplicationThe product of 0 and any number results in 0.That is, for any real number a, a × 0 = 0.

Examples:

0 • (-4) = 0

m • 0 = 0

(27x2y4) • 0 = 0

MULTIPLICATION PROPERTY OF NEGATIVE ONE

-1

Multiplication Property of Negative One

When you multiply a number by −1 it becomes the opposite of what you multiplied by −1.

Examples:

−1 • 5= −5

m • (− 1) = (− m)

-(7 – 3x) = -7 + 3x

PRACTICE

Identify which property that justifies each of the following.

4 (8 2) = (4 8) 2


4 (8 2) = (4 8) 2

Associative Property of Multiplication


6 + 8 = 8 + 6


6 + 8 = 8 + 6

Commutative Property of Addition


12 + 0 = 12


12 + 0 = 12

Additive Identity Property


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


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

Distributive Property


m + 12 + (-12) = 5 + (-12)


m + 12 + (-12) = 5 + (-12)

Equality Property of Addition


5 + (2 + 8) = (5 + 2) + 8


5 + (2 + 8) = (5 + 2) + 8

Associative Property of Addition


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95

1


Multiplicative Inverse Property

59

95

1


5 24 = 24 5


5 24 = 24 5

Commutative Property of Multiplication


x • 0 = 0


x • 0 = 0

Zero Property of Multiplication


18 + -18 = 0


18 + -18 = 0

Additive Inverse Property


-34 1 = -34


-34 1 = -34

Multiplicative Identity Property


-64 (-1) = 64


-64 (-1) = 64

Multiplication Property of Negative 1

CONCEPT REVIEW

Roots

Inverse operation of exponential form. Also used to isolate the variable and solve equations.

162 x

162 x

Combine Like Terms

Like terms are monomials that are of the same category. That is they are both constants, or they contain the same variables raised to the same powers. They can be combined to form a single term or used to simplify expressions & equations.

Combine Like TermsCombine like terms to simplify the expression.

10 + 4x3 + 7x4 – x2 + 5x3 +4x2 – 2x4 – 6

Hint: Remember to only combine terms that are all from the same number family (constants with constants, x2 with other

x2, x3 with x3, and x4 with x4)

Combine Like Terms

10 + 4x3 + 7x4 – x2 + 5x3 +4x2 – 12x4 – 6

+ 7x4 – 12x4 + 4x3 + 5x3 – x2 +4x210 – 6

+ 7x4– 12x4 + 4x3+ 5x3 +4x2 – x2 – 6 + 10

-5x4 + 9x3 + 2x2 + 4

Opposites

Two real numbers that are the same distance from the origin of the real number line are opposites of each other.

Examples of opposites:

2 and -2 -100 and 100 and 15 15

Reciprocals

Two numbers whose product is 1 are reciprocals of each other.

Examples of Reciprocals:

and 5 -3 and 13

54

45

and5

1

Absolute Value

The absolute value of a number is its distance from 0 on the number line. The absolute value of x is written .

Examples of absolute value:

x

5 5 37

37

Least Common Denominator

The least common denominator (LCD) is the smallest number divisible by all the denominators.

Example: The LCD of is 12 because 12 is the smallest number into which 3 and 4 will both divide.

23

54

and

Adding Two Fractions

34

56

912

1012

1912

To add two fractions you must first find the LCD. In the problem below the LCD is 12. Then rewrite the two addends as equivalent expressions with the LCD. Then add the numerators and keep the denominator.

Properties of Real Numbers. COMMUTATIVE PROPERTY.

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