Numbers 1: A Functional History of Numbers 2: Viewing numbers in an abstract way

Questions Why numbers?

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

Natural Numbers N N = {0,1,2,…} N* = {1,2,3,…} Natural numbers are the counting numbers. You may have called these “Whole” numbers.

Identities and Inverses Addition  What is the Identity for Addition?  For a given number a, what is the additive inverse?

Integers Z = {…,-2,-1,0,1,2,…} The set of integers includes the Natural numbers and their “additive inverses”, or their negatives

Identities and Inverses Multiplication  What is the Identity for Multiplication?  For a given number a, what is the multiplicative inverse?

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

Rational Numbers, cont. Rational numbers can also be described as the solution set to a certain type of equation

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

Irrational Numbers, cont. Irrational numbers cannot be expressed as ratios Examples:  Roots  Transcendental numbers

Real Numbers R includes all Rational and Irrational Numbers (including “Transcendental” numbers)

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?

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?

Homework TBD

Numbers Notation Radicals

Operating with Irrational Numbers: Radicals Reducing radicals FOIL with Radicals Rationalization of denominators

Operations, Skills Reducing/Simplifying Radicals FOIL with radicals Rationalizing Denominators

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!

Numbers Day 2 Numbers Review Radicals (Rationalizing Denominators) Equations

Problem of the Day Describe and give examples Simplify

Rationalizing Denominators Using the properties of the difference of squares to get a rational denominator (with no radical)

Homework Exercise 2.1:  5 (a, e) Exercise 2.2.1:  1, 2, 3 (a, c, e on each)

Numbers Review Techniques “Showing”

Problem of the Day Solve for x in each of the following equations

Today Next Class: Quiz Algebraic Technique Absolute Value Diagnostic Test*  Not for punitive purposes (within reason)

Absolute Value, Modulus “Distance from zero” Maps R  R+

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

Absolute Value Examples

Homework Exercise  3-6 (left column only) Next Class: Quiz  Numbers, Radicals, Solving one-variable equations and inequalities, Absolute Value equations and inequalities