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Exploring Real Numbers

Exploring Real Numbers. Objectives: To classify numbers To compare numbers. 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:

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Exploring Real Numbers

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  1. Exploring Real Numbers Objectives: To classify numbers 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 • - (17/31) * Rational • 23 * Natural, Whole, Integers, Rational • 0 * Whole, integers, rational • 4.581 * rational

  7. Example #2: Using Counterexamples Is each statement true or false? If it is false, give a counterexample. • 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: 0.52 = 0.25. 0.5is our counterexample

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

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

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