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CHAPTER 11

CHAPTER 11. Rational and Irrational Numbers. Rational Numbers. 11-1 Properties of Rational Numbers. Rational Numbers. A real number that can be expressed as the quotient of two integers. Examples. 7 = 7/1 5 2/3 = 17/3 .43 = 43/100 -1 4/5 = -9/5. Write as a quotient of integers. 3 48%

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CHAPTER 11

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  1. CHAPTER 11 Rational and Irrational Numbers

  2. Rational Numbers 11-1 Properties of Rational Numbers

  3. Rational Numbers • A real number that can be expressed as the quotient of two integers.

  4. Examples • 7 = 7/1 • 5 2/3 = 17/3 • .43 = 43/100 • -1 4/5 = -9/5

  5. Write as a quotient of integers • 3 • 48% • .60 • - 2 3/5

  6. Which rational number is greater8/3 or 17/7

  7. Rules • a/c > b/d if and only if ad > bc. • a/c < b/d if and only if ad < bc

  8. Examples • 4/7 ? 3/8 • 7/9 ? 4/5 • 8/15 ? 3/4

  9. Density Property • Between every pair of different rational numbers there is another rational number

  10. Implication • The density property implies that it is possible to find an unlimited or endless number of rational numbers between two given rational numbers.

  11. Formula If a < b, then to find the number halfway from a to b use: a + ½(b – a)

  12. Example • Find a rational number between -5/8 and -1/3.

  13. Rational Numbers 11-2 Decimal Forms of Rational Numbers

  14. Forms of Rational Numbers • Any common fraction can be written as a decimal by dividing the numerator by the denominator.

  15. Decimal Forms • Terminating • Nonterminating

  16. Examples Express each fraction as a terminating or repeating decimal 5/6 7/11 3 2/7

  17. Rule • For every integer n and every positive integer d, the decimal form of the rational number n/d either terminates or eventually repeats in a block of fewer than d digits.

  18. Rule • To express a terminating decimal as a common fraction, express the decimal as a common fraction with a power of 10 as the denominator.

  19. Express as a fraction • .38 • .425

  20. Solutions • .38 = 38/100 or 19/50 • .425 = 425/1000= 17/40

  21. Express a Repeating Decimal as a fraction • .542 • let N = 0.542 • Multiply both sides of the equation by a power of 10

  22. Continued • Subtract the original equation from the new equation • Solve

  23. Rational Numbers 11-3 Rational Square Roots

  24. Rule If a2 = b, then a is a square root of b.

  25. Terminology • Radical sign is  • Radicand is the number beneath the radical sign

  26. Product Property of Square Roots For any nonnegative real numbers a and b: ab = (a) (b)

  27. Quotient Property of Square Roots For any nonnegative real number a and any positive real number b: a/b = (a) /(b)

  28. Examples • 36 • 100 • - 81/1600 • 0.04

  29. Irrational Numbers 11-4 Irrational Square Roots

  30. Irrational Numbers • Real number that cannot be expressed in the form a/b where a and b are integers.

  31. Property of Completeness • Every decimal number represents a real number, and every real number can be represented as a decimal.

  32. Rational or Irrational • 17 • 49 • 1.21 • 5 + 2 2

  33. Simplify • 63 • 128 • 50 • 6108

  34. Simplify • 63 = 9 7 = 37 • 128 = 64 2 = 82 • 50 = 25 5 = 55 • 6108= 636 3=36 3

  35. Rational Numbers 11-5 Square Roots of Variable Expressions

  36. Simplify • 196y2 • 36x8 • m2-6m + 9 • 18a3

  37. Solutions • 196y2 = ± 18y • 36x8 = ± 6x4 • m2-6m + 9 = ±(m -3) • 18a3 = ± 3a 2a

  38. Solve by factoring • Get the equation equal to zero • Factor • Set each factor equal to zero and solve

  39. Examples • 9x2 = 64 • 45r2 – 500 = 0 • 81y2 – 16= 0

  40. Irrational Numbers 11-6 The Pythagorean Theorem

  41. The Pythagorean Theorem In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. a2 + b2 = c2

  42. Example c a b

  43. Example c 8 15

  44. Solution a2 + b2 = c2 82 + 152 = c2 64 + 225 =c2 289 =c2 17 = c

  45. Example The length of one side of a right triangle is 28 cm. The length of the hypotenuse is 53 cm. Find the length of the unknown side.

  46. Solution a2 + b2 = c2 a2 + 282 = 532 a2 + 784 =2809 a2 =2025 a = 45

  47. Converse of the Pythagorean Theorem If the sum of the squares of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest, then the triangle is a right triangle. The right side is opposite the longest side.

  48. Radical Expressions 11-7 Multiplying, Dividing, and Simplifying Radicals

  49. Rationalization The process of eliminating a radical from the denominator.

  50. Simplest Form • No integral radicand has a perfect-square factor other than 1 • No fractions are under a radical sign, and • No radicals are in a denominator

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