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 Addition When 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.

Equality Property

Equality Property For 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 Multiplication The 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

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

Identify which property that justifies each of the following. 4  (8  2) = (4  8)  2 Associative Property of Multiplication

Identify which property that justifies each of the following. 6 + 8 = 8 + 6

Identify which property that justifies each of the following. 6 + 8 = 8 + 6 Commutative Property of Addition

Identify which property that justifies each of the following. 12 + 0 = 12

Identify which property that justifies each of the following. 12 + 0 = 12 Additive Identity Property

Identify which property that justifies each of the following. 5(2 + 9) = (5  2) + (5  9)

Identify which property that justifies each of the following. 5(2 + 9) = (5  2) + (5  9) Distributive Property

Identify which property that justifies each of the following. m + 12 + (-12) = 5 + (-12)

Identify which property that justifies each of the following. m + 12 + (-12) = 5 + (-12) Equality Property of Addition

Identify which property that justifies each of the following. 5 + (2 + 8) = (5 + 2) + 8

Identify which property that justifies each of the following. 5 + (2 + 8) = (5 + 2) + 8 Associative Property of Addition

Identify which property that justifies each of the following.

Identify which property that justifies each of the following. Multiplicative Inverse Property

Identify which property that justifies each of the following. 5  24 = 24  5

Identify which property that justifies each of the following. 5  24 = 24  5 Commutative Property of Multiplication

Identify which property that justifies each of the following. x • 0 = 0

Identify which property that justifies each of the following. x • 0 = 0 Zero Property of Multiplication

Identify which property that justifies each of the following. 18 + -18 = 0

Identify which property that justifies each of the following. 18 + -18 = 0 Additive Inverse Property

Identify which property that justifies each of the following. -34 1 = -34

Identify which property that justifies each of the following. -34 1 = -34 Multiplicative Identity Property

Identify which property that justifies each of the following. -64 (-1) = 64

Identify which property that justifies each of the following. -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.

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 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 10 + 4x3

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

Reciprocals Two numbers whose product is 1 are reciprocals of each other. Examples of Reciprocals: and 5 -3 and

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:

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.

Adding Two Fractions 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.