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Published byMerryl McDonald Modified over 8 years ago
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Numbers 1: A Functional History of Numbers 2: Viewing numbers in an abstract way
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Questions Why numbers?
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Questions…some answers? Why numbers? Counting Measurement Comparison Architecture, engineering Trade, accounting and finance Medicine So that Jamie can have a job Patterns, predicting phenomena
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Natural Numbers N N = {0,1,2,…} N* = {1,2,3,…} Natural numbers are the counting numbers. You may have called these “Whole” numbers.
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Identities and Inverses Addition What is the Identity for Addition? For a given number a, what is the additive inverse?
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Integers Z = {…,-2,-1,0,1,2,…} The set of integers includes the Natural numbers and their “additive inverses”, or their negatives
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Identities and Inverses Multiplication What is the Identity for Multiplication? For a given number a, what is the multiplicative inverse?
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Rational Numbers Q = The set of numbers that can be expressed as a ratio of two Integers Includes all Integers, and therefore all Natural numbers In addition, this set includes fractions, terminating decimals, repeating decimals
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Rational Numbers, cont. Rational numbers can also be described as the solution set to a certain type of equation
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Irrational Numbers Give the side of a square that has an area of 2 square meters Give the hypotenuse of a right triangle with legs measuring 5 feet and 6 feet
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Irrational Numbers, cont. Irrational numbers cannot be expressed as ratios Examples: Roots Transcendental numbers
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Real Numbers R includes all Rational and Irrational Numbers (including “Transcendental” numbers)
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Assignment 1 Draw a Venn Diagram that shows the relationship between: Natural numbers, Integers, Rational numbers, Irrational numbers and Real numbers N (N, Z, Q,, and R) Challenge for homework: where do complex numbers (C) fit in this diagram?
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Assignment 2 Using your Venn Diagram for the Number System, write the numbers on the right in the appropriate space Challenge: where do the positive and negative multiples of 3 (or “3Z”) fit?
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Homework TBD
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Numbers Notation Radicals
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Operating with Irrational Numbers: Radicals Reducing radicals FOIL with Radicals Rationalization of denominators
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Operations, Skills Reducing/Simplifying Radicals FOIL with radicals Rationalizing Denominators
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Homework Exercise 2.1 (page 20) 1, 2 (left column of each) Consult the notation table on page 16) 3 4: a, b Study N, N*, Z, Q, Q(“bar”), R, C Yes, I do want you to write the letters correctly Get Supplies!
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Numbers Day 2 Numbers Review Radicals (Rationalizing Denominators) Equations
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Problem of the Day Describe and give examples Simplify
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Rationalizing Denominators Using the properties of the difference of squares to get a rational denominator (with no radical)
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Homework Exercise 2.1: 5 (a, e) Exercise 2.2.1: 1, 2, 3 (a, c, e on each)
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Numbers Review Techniques “Showing”
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Problem of the Day Solve for x in each of the following equations
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Today Next Class: Quiz Algebraic Technique Absolute Value Diagnostic Test* Not for punitive purposes (within reason)
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Absolute Value, Modulus “Distance from zero” Maps R R+
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Absolute Value, Modulus Inequalities Keep in mind the statements below Any step involving multiplication/division by a negative number “switches” the direction of the inequality
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Absolute Value Examples
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Homework Exercise 2.2.2 3-6 (left column only) Next Class: Quiz Numbers, Radicals, Solving one-variable equations and inequalities, Absolute Value equations and inequalities
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