1. Exploring Real Numbers Objectives: (1)To classify numbers (2)To compare numbers

2. Number Groups Natural Numbers – 1, 2, 3, … Whole Numbers – 0, 1, 2, … Integers – … -3, -2, -1, 0, 1, 2, 3, … Rational Numbers: – Integers, fractions, finite decimals, repeating decimals Irrational Numbers: – Infinite, non-repeating decimals Real Numbers: – All rational and irrational numbers

3. How are repeating decimals rational? We use the 10x – x rule. Ex: x = 0.99999…. 10x = 9.999999999… -x = - 0.999999999… 9x = 9 9 9 x = 1

4. How are repeating decimals rational? Wait a minute… If x = 0.999… and x = 1, then ???? 0.999… = 1

5. Definitions Counterexample: An example that proves a statement false. Inequality: a mathematical sentence that compares the value of two expressions using an inequality symbol, such as, or ≠ Opposites: two numbers that have the same distance from zero Absolute Value: a number’s distance from zero

6. Example #1: Classifying Numbers a.- (17/31) * Rational b.23 * Natural, Whole, Integers, Rational c.0 * Whole, integers, rational d.4.581 * rational

7. Example #2: Using Counterexamples Is each statement true or false? If it is false, give a counterexample. a.All whole number are rational numbers. - True, the easiest way to turn a whole number into a fraction is to put it over 1. b. The square of a number is always greater than the number. - False: 0.5 2 = 0.25. 0.5 is our counterexample

8. Example #3: Ordering Fractions Write -3/8, -1/2, and -5/12 in order from least to greatest. 1 st Step: write each fraction as a decimal -3/8 = -0.375 -1/2 = -0.5 -5/12 = -0.41666… 2 nd Step: order the decimals from least to greatest -0.5, -0.41666…, -0.375 3 rd Step: replace decimals with their fraction equivalents -1/2, -5/12, -3/8

9. Example #4: Finding Absolute Value Find each absolute value: a.  12   12 b.  -5.6   5.6 c.  5 – 8    -3  33