Transcript Objective
Honors Algebra 2 Chapter 1 Real Numbers, Algebra, and Problem Solving 1.1 Real Numbers and Operations There is exactly one real number for each point on a number line. Objective: Show that a number is rational and distinguish between rational and irrational numbers. If a real number cannot be expressed as a ratio of 2 integers then it is called irrational. Absolute Value of a Number • The absolute value of a number is the distance on a number line the number is from 0. Distance from 0 is 2 |-2| = 2 Objective: Add positive and negative numbers. Objective: Subtract positive and negative numbers. The additive inverse of a number is the number added to it to get 0. Objective: Add positive and negative numbers. Objective: Subtract positive and negative numbers. Objective: Divide positive and negative numbers. The reciprocal/Multiplicative Inverse of a number is the number we multiply it by to get 1. Objective: Divide positive and negative numbers. Objective: Recognize division by zero as impossible Thus we cannot define and must exclude division by 0. Zero is the only real number that does not have a reciprocal. Is it sometimes, always, or never true that, if x is a real number, (- x)( - x) is negative? Is it sometimes, always, or never true that, if x is a real number, (x)( - x) is negative? Is it sometimes, always, or never true that, if x is a real number, (x)(x) is negative? 1.3 Algebraic Expressions and Properties of Real Numbers Variable: Any symbol that is used to represent various numbers Constant: Any symbol used to represent a fixed number Objective: Use number properties to write equivalent expressions. Equivalent Expressions: Expressions that have the same value for all acceptable replacements. Objective: Use number properties to write equivalent expressions. Objective: Use number properties to write equivalent expressions. Additive Identity: The number 0. Multiplicative Identity: The number 1 For Exercises 1-4, use the properties of real numbers to answer each question. 1. If m + n = m, what is the value of n? 2. If m - n = 0, what is the value of n? What is n called with respect to m? 3. If mn = 1, what is the value of n? What is n called with respect to m? 4. If mn = m, what is the value of n? Suppose we define a new operation @ on the set of real numbers as follows: a @ b = 4a - b. Thus 9 @ 2 = 4(9) - 2 = 34. Is @ commutative? That is, does a @ b = b @ a for all real numbers a and b? 1.4 The Distributive Property Objective: Use the distributive property to multiply. Objective: Use the distributive property to factor expressions. Factoring: The reverse of multiplying. To factor an Expression: To find an equivalent expression that is a product. Objective: Collect like terms. Like Terms: Terms whose variables are the same Objective: Write the inverse of a sum. 1.5 Solving Equations Objective: Solve equations using the addition and multiplication properties. A mathematical sentence A = B says that the symbols A and B are equivalent. Such a sentence is an equation. The set of all acceptable replacements is the replacement set. The set of all solutions is the solution set. Objective: Solve equations using the addition and multiplication properties. Objective: Prove Identities Identity: An equation that is true for all acceptable replacements. To Prove an Identity: • Pick one side of the equation and manipulate it using properties of real numbers to show that it can be transformed so that it is exactly the same as the other side. HW #1.1-5 Pg 8 35-38 Pg 13 53-57 Pg 19 52-53 Pg 25 77-80 Pg 29 54-56 HW Quiz #1.1-5 Monday, March 28, 2016 Pg 8 38 Pg 13 56 Pg 25 78 Pg 29 56 Pg 8 36 Pg 13 54 Pg 2580 Pg 29 54 Row 1, 3, 5 Row 2, 4, 6 1.6 Writing Equations Objective: Become familiar with and solve simple algebraic problems At 6:00 AM the Wong family left for a vacation trip and drove south at an average speed of 40 mph. Their friends, the Heisers, left two hours later and traveled the same route at an average speed of 55 mph. At what time could the Heisers expect to overtake the Wongs? Objective: Become familiar with and solve simple algebraic problems It has been found that the world record for the men's 10,000-meter run has been decreasing steadily since 1950.The record is approximately 28.87 minutes minus 0.05 times the number of years since 1950. Assume the record continues to decrease in this way. Predict what it will be in 2010. Objective: Become familiar with and solve simple algebraic problems An insecticide originally contained ½ ounce of pyrethrins. The 5 new formula contains oz of pyrethrins. What percent of the 8 pyrethrins of the original formula does the new formula contain? 1.7 Exponential Notation Objective: Simplify expressions with integer exponents. Objective: Simplify expressions with integer exponents. Thus we can say that bn and b-n are reciprocals 1.8 Properties of Exponents Objective: Multiply or divide with exponents. Objective: Multiply or divide with exponents. We do not define 0°. Notice the following. 00 011 01 01 01 0 1 Undefined 0 0 Objective: Use exponential notation in raising powers to powers. Objective: Use exponential notation in raising powers to powers. Objective: Use the rules for order of operations to simplify expressions. 1.10 Field Axioms Objective: Use the definition of a field Field: Any number system with two operations defined in which all of the axioms of real numbers hold The set of Real numbers forms a field with Addition and Multiplication Objective: Use the definition of a field Objective: Use the definition of a field Objective: Write Column Proofs Objective: Write Column Proofs Objective: Write Column Proofs HW #1.6-10 Pg 34 23-24 Pg 37 37-41 Pg 43 58-65 Pg 52-53 12-25 The End Chapter 1