Be prepared to take notes when the bell rings. P.1 Real Numbers Be prepared to take notes when the bell rings.

Real Numbers Set of numbers formed by joining the set of rational numbers and the set of irrational numbers Subsets: (all members of the subset are also included in the set) {1, 2, 3, 4, …} natural numbers {0, 1, 2, 3, …} whole numbers {…-3, -2, -1, 0, 1, 2, 3, …} integers

Rational and Irrational Numbers A real number that can be written as the ratio 𝑝 𝑞 of two integers, where q ≠0 Example: 1 3 =0.3333 Repeats 1 8 = 0.125 Terminates 125 111 =1.126126=1. 126 A real number that cannot be written as the ratio of two integers *infinite non-repeating decimals Example: 2 =1.4142135… 𝜋=3.1415926…

Non-Integer Fractions Real Numbers Irrational Numbers Rational Numbers Non-Integer Fractions Integers Negative Integers Whole Numbers Natural Numbers Zero

Real Number Line Negative Positive Origin Coordinate: Positive Negative Coordinate: Every point on the real number line corresponds to exactly one real number called its coordinate

Ordering Real Numbers Inequalities <𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 >𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 <𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 >𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 ≤𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 ≥𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 a b Example: a < b − 14 3 _______− 26 >

Describe the subset of real numbers represented by each inequality.

Interval: subsets of real numbers used to describe inequalities Notation 𝑎,𝑏 𝑎, 𝑏 Interval Type Closed Open Inequality 𝑎≤𝑥≤𝑏 𝑎<𝑥<𝑏 𝑎≤𝑥<𝑏 𝑎<𝑥≤𝑏 The endpoints of a closed interval ARE included in the interval. The endpoints of an open interval are NOT included in the interval. *Unbounded intervals using infinity can be seen on page 4

Properties of Absolute Value 𝑎 ≥0 −𝑎 = 𝑎 𝑎𝑏 = 𝑎 𝑏 𝑎 𝑏 = 𝑎 𝑏 , 𝑏≠0

Absolute value is used to define the distance (magnitude) between two points on the real number line Let a and b be real numbers. The distance between a and b is: 𝑑 𝑎,𝑏 = 𝑏−𝑎 = 𝑎−𝑏 The distance between -3 and 4 is: −3−4 = −7 =7

Algebraic Expressions Variables: letter that represents an unknown quantity Constant: Real number term in an algebraic expression Algebraic Expression: Combination of variables and real numbers (constants) combined using the operations of addition, subtraction, multiplication and division Examples of algebraic expressions: 5𝑥, 2𝑥−3, 4 𝑥 2 +2 , 7𝑥+𝑦 Terms: Parts of an algebraic expression separated by addition i.e. 𝑥 2 −5𝑥+8= 𝑥 2 + −5𝑥 +8 𝑥 2 , −5𝑥:𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑡𝑒𝑟𝑚𝑠 8:𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚 Coefficient: Numerical factor of a variable term Evaluate: Substitute numerical values for each variable to solve an algebraic expression

Examples of Evaluation Expression −3𝑥+5 3 𝑥 2 +2𝑥−1 Value of Variable 𝑥=3 𝑥=−1 Substitute −3(3)+5 3 (−1) 2 +2(−1)−1 Value of Expression −9+5=−4 3−2−1=0 Used Substitution Principle: If a=b, then a can be replaced by b in any expression involving a.

Basic Rules of Algebra 4 Arithmetic operations: Addition, + Subtraction, - Division, / ÷ Multiplication, × ∙ Addition and Multiplication are the primary operations. Subtraction is the inverse of Addition and Division is the inverse of Multiplication.

Basic Rules of Algebra 𝑎−𝑏=𝑎+ −𝑏 Subtraction: add the opposite of b Division: multiply by the reciprocal of b; if b≠0, then 𝑎−𝑏=𝑎+ −𝑏 −𝑏 is called the additive inverse (opposite of a real number) 𝑎 𝑏 =𝑎 1 𝑏 = 𝑎 𝑏 1 𝑏 is called the multiplicative inverse (reciprocal of a real number) 𝑎 𝑏 : 𝑎 𝑖𝑠 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟, 𝑏 𝑖𝑠 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

Let a, b and c be real numbers, variables or algebraic expressions. Commutative Property of Addition 𝑎+𝑏=𝑏+𝑎 Commutative Property of Multiplication 𝑎𝑏=𝑏𝑎 Associative Property of Addition 𝑎+𝑏 +𝑐=𝑎+ 𝑏+𝑐 Associative Property of Multiplication 𝑎𝑏 𝑐=𝑎 𝑏𝑐 Distributive Property 𝑎 𝑏+𝑐 =𝑎𝑏+𝑎𝑐 Additive Identity Property 𝑎+0=𝑎 Multiplicative Identity Property 𝑎∗1=𝑎 Additive Inverse Property 𝑎+ −𝑎 =0 Multiplicative Inverse Property 𝑎 1 𝑎 =1

Let a, b and c be real numbers, variables or algebraic expressions. Properties of Negation and Equality −1 𝑎=−𝑎 − −𝑎 =𝑎 −𝑎 𝑏=− 𝑎𝑏 =𝑎 −𝑏 −𝑎 −𝑏 =𝑎𝑏 −(𝑎+𝑏)=(−𝑎)+(−𝑏) 𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎+𝑐=𝑏+𝑐 𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐=𝑏𝑐 𝐼𝑓 𝑎+𝑐=𝑏+𝑐, 𝑡ℎ𝑒𝑛 𝑎=𝑏 𝐼𝑓 𝑎𝑐=𝑏𝑐, 𝑡ℎ𝑒𝑛 𝑎=𝑏 𝑖𝑓 𝑐≠0

Let a, b and c be real numbers, variables or algebraic expressions. Properties of Zero 𝑎+0=𝑎, 𝑎−0=𝑎 𝑎∗0=0 0 𝑎 =0, 𝑖𝑓 𝑎≠0 𝑎 0 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝐼𝑓 𝑎𝑏=0, 𝑡ℎ𝑒𝑛 𝑎=0 𝑜𝑟 𝑏=𝑜

Homework Problems Page 9 #’s 1-25 odd, 29, 33-39 odd, 43-47, 51-55, 59, 89- 93, 99-107, 111-115