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Chapter 2

Chapter 2. Working with Real Numbers. 2-1 Basic Assumptions. CLOSURE PROPERTIES. a + b and ab are unique 7 + 5 = 12 7 x 5 = 35. COMMUTATIVE PROPERTIES. a + b = b + a ab = ba. 2 + 6 = 6 + 2 2 x 6 = 6 x 2. ASSOCIATIVE PROPERTIES. (a + b) + c = a + (b +c) (ab)c = a(bc).

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Chapter 2

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  1. Chapter 2 Working with Real Numbers

  2. 2-1 Basic Assumptions

  3. CLOSURE PROPERTIES a + b and ab are unique 7 + 5 = 12 7 x 5 = 35

  4. COMMUTATIVE PROPERTIES a + b = b + a ab = ba 2 + 6 = 6 + 2 2 x 6 = 6 x 2

  5. ASSOCIATIVE PROPERTIES (a + b) + c = a + (b +c) (ab)c = a(bc) (5 + 15) + 20 = 5 + (15 +20) (5·15)20 = 5(15·20)

  6. Properties of Equality

  7. Reflexive Property - a = a • Symmetric Property – Ifa = b,thenb = a • Transitive Property – If a = b, and b = c, then a = c

  8. 2-2 Addition on a Number Line

  9. IDENTITY PROPERTIES There is a unique real number 0 such that: a + 0 = 0 + a = a -3 + 0 = 0 + -3 = -3

  10. PROPERTY OF OPPOSITES • For each a, there is a unique real number – a such that: a + (-a) = 0 and (-a)+ a = 0(-a) is called the opposite or additive inverse of a

  11. Property of the opposite of a Sum For all real numbers a and b: -(a + b) = (-a) + (-b) The opposite of a sum of real numbers is equal to the sum of the opposites of the numbers. -(8 +2) = (-8) + (-2)

  12. 2-3 Rules for Addition

  13. Addition Rules • If a and b are both positive, then a + b = a + b 3 + 7 = 10

  14. Addition Rules • If a and b are both negative, then a + b = -(a + b) (-6) + (-2) = -(6 +2) = -8

  15. Addition Rules • If a is positive and b is negative and a has the greater absolute value, then a + b = a - b 6 + (-2) = (6 - 2) = 4

  16. Addition Rules • If a is positive and b is negative and b has the greater absolute value, then a + b = -(b - a) 4 + (-9) = -(9 -4) = -5

  17. Addition Rules • If a and b are opposites, then a + b = 0 2 + (-2) = 0

  18. 2-4 Subtracting Real Numbers

  19. DEFINITION of SUBTRACTION For all real numbers a and b, a – b = a + (-b) To subtract any real number, add its opposite

  20. Examples • 3 – (-4) • -y – (-y + 4) • -(f + 8) • -(-b + 6 – a) • m – (-n + 3)

  21. 2-5 The Distributive Property

  22. DISTRIBUTIVE PROPERTY a(b + c) = ab + ac (b +c)a = ba + ca 5(12 + 3) = 5•12 + 5 •3 = 75 (12 + 3)5 = 12• 5 + 3 • 5 = 75

  23. Examples • 2(3x + 4) • 5n + 7(n – 3) • 2(x – 6) + 9 • 8 + 3(4 – y) • 8(k + m) - 15(2k + 5m)

  24. 2-6 Rules for Multiplication

  25. IDENTITY PROPERTY of MULTIPLICATION There is a unique real number 1 such that for every real number a, a · 1 = a and 1 · a = a

  26. MULTIPLICATIVE PROPERTY OF 0 For every real numbera, a · 0 = 0 and 0 · a = 0

  27. MULTIPLICATIVE PROPERTY OF -1 For every real numbera, a(-1) = -a and (-1)a = -a

  28. PROPERTY of OPPOSITES in PRODUCTS For all real numberaandb, -ab = (-a)(b) and -ab = a(-b)

  29. Examples • (-1)(3d – e + 8) • -6(7n – 6) • -[-4(x – y)]

  30. 2-7 Problem Solving: Consecutive Integers

  31. EVEN INTEGER An integer that is the product of 2 and any integer. …-6, -4, -2, 0, 2, 4, 6,…

  32. ODD INTEGER An integer that is not even. …-5, -3, -1, 1, 3, 5,…

  33. Consecutive Integers Integers that are listed in natural order, from least to greatest …,-2, -1, 0, 1, 2, …

  34. Example Three consecutive integers have the sum of 24. Find all three integers.

  35. CONSECUTIVE EVEN INTEGER Integers obtained by counting by twos beginning with any even integer. 12, 14, 16

  36. Example Four consecutive even integers have a sum of 36. Find all four integers.

  37. CONSECUTIVE ODD INTEGER Integers obtained by counting by twos beginning with any odd integer. 5,7,9

  38. Example There are three consecutive odd integers. The largest integer is 9 less than the sum of the smaller two integers. Find all three integers.

  39. 2-8 The Reciprocal of a Real Number

  40. PROPERTY OF RECIPROCALS For each a except 0, there is a unique real number 1/a such that: a · (1/a) = 1 and (1/a)· a = 11/a is called the reciprocal or multiplicative inverse of a

  41. PROPERTY of the RECIPROCAL of the OPPOSITE of a Number For each a except 0, 1/-a = -1/a The reciprocal of –a is -1/a

  42. PROPERTY of the RECIPROCAL of a PRODUCT For all nonzero numbers a and b, 1/ab = 1/a ·1/b The reciprocal of the product of two nonzero numbers is the product of their reciprocals.

  43. 2-9 Dividing Real Numbers

  44. DEFINITION OF DIVISION For every real number a and every nonzero real number b, the quotient is defined by: a÷b = a·1/b To divide by a nonzero number, multiply by its reciprocal

  45. The quotient of two positive numbers or two negative numbers is a positive number -24/-3 = 8and24/3 = 8

  46. The quotient of two numbers when one is positive and the other negative is a negative number. 24/-3 = -8and-24/3 = -8

  47. PROPERTY OF DIVISION For all real numbers a, b,andc such thatc 0, a + b = a + b and c c c a - b = a - b c c c

  48. Examples • 4 ÷ 16 • 8x 16 3.5x + 25 5

  49. The End

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