CHAPTER 9 Rational Numbers, Real Numbers, and Algebra 1 Problem. A man’s boyhood lasted 6 of his life, he then played soccer for 1 1 12 of his life, and he married after 8 more of his life. A daughter was born 9 years after his marriage, and her birth coincided with the halfway point of his life. How old was the man when he died? Strategy 14 – Solve an Equation. This strategy may be appropriate when A variable has been introduced. • The words is, is equal to, or equalsappear in a problem. • The stated conditions can easily be represented with an equation. • Let m = the man’s age when he died. 1 1 1 1 m + m + m + 9 = m 6 12 8 2 Multiply each side by 24. 4m + 2m + 3m + 216 = 12m 216 = 3m 72 = m The man was 72 when he died. 40 9.1. THE RATIONAL NUMBERS 41 9.1. The Rational Numbers Where we are so far: Each arrow represents “is a subset of.” The rational numbers are an extension of both the Fractions and the Integers. The need: 3 We can solve 5x = 3 using fractions to get x = , but what about 5x = 3? 5 − 3 We can solve x+7 = 0 using integers to get x = 7, but what about x+ = 0? − 2 So we need to extend both the fractions and the integers to take care of these problems. Two approaches: 1) Focusing on the property that every number has an opposite, wew take the fractions wioth their opposites as the rational numbers. Then we note the 12 7 integers are also included: 4 = and 7 = − , for instance. 3 − 1 2) Focusing on the property that every nonzero number has a recipracal, we form all possible “fractions” where the numerator is an integer and the denominator is a nonzero integer. We will follow the second approach: Definition (Rational Numbers). The set of rational numbers is the set a Q = a and b are integers, b = 0 . b| 6 n o 42 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Example. 3 7 3 6 0 0 , − , , − , , 4 2 5 4 6 4 Also, − − − 1 13 3 45 3 = − , 6 = − − 4 4 − 7 7 Each fraction can be written in this form and each integer n can be written as n . 1 We now have: Definition (Equality of Rational Numbers). a c a c Let and Then = if and only if ad = bc. b d b d Recall that “equal” refers to the abstract concept of quantity attached to a number, while “equivalent” refers to the various numerals used for a number. 1 2 3 1 2 3 = = = − = − = − 2 4 6 2 4 6 − − 1 − are all numerals representing the ration number . 2 2 2 2 6 12 2 = − = = = − = − −3 3 3 9 18 − 3 − − 2 − are all numerals representing the rational number . −3 9.1. THE RATIONAL NUMBERS 43 a Theorem. Let be any rational number and n any nonzero integer. b Then a an na = = . b bn nb a A rational number is in simplest form or lowest terms if a and b have no b common prime factors and b is positive. Example. 3 5 0 − , , are in lowest terms. 4 7 1 4 10 0 0 , − , , are not in lowest terms. 3 15 3 3 − − − Example. 96 = 108 −8 12 8 8 · = = − 9 12 9 9 − · − 60 − = 84 −5 12 5 5 − · = − = 7 12 7 7 − · − Definition (Addition of Rational Numbers). a c Let and be any rational numbers. Then b d a c ad + bc + = . b d bd Corollary. a c a + c + = . b b b 44 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Example. 3 5 ( 3)12 + 8 5 36 + 40 4 1 − + = − · = − = = 8 12 8 12 96 96 24 · 9 10 1 = − + = " 24 24 24# a a Consider − = since ( a)( b) = ab. Also, b b − − − a a a + a 0 − + = − = = 0, b b b b so a a = − . −b b a Theorem. Let be any rational number. Then b a a a = − = . −b b b − Fractional Number Line: 9.1. THE RATIONAL NUMBERS 45 Properties of Rational Number Addition a c e Let , , and be any rational numbers. b d f a c (Closure) + is a rational number. b d a c c a (Commutative) + = + . b d d b a c e a c e (Associative) + + = + + . b d f b d f a ⇣ a ⌘ a ⇣ 0 ⌘ (Identity) + 0 = = 0 + 0 = , m = 0 . b b b m 6 ⇣ a ⌘ (Additive Inverse) For every rational number , there exists a unique rational a b number such that −b a a a a + = 0 = + . b − b − b b Example. ⇣ ⌘ ⇣ ⌘ 2 3 7 + + = (mentally?) 9 5 9 2 ⇣7 3⌘ + + = 9 9 5 2 ⇣7 ⌘3 3 3 8 + + = 1 + = 1 = 9 9 5 5 5 5 ⇣ ⌘ 3 18 17 + − + = (mentally?) 11 66 23 3 ⇣ 18 17⌘ + − + = 11 66 23 3 ⇣ 3 17⌘ 17 17 + − + = 0 + = 11 11 23 23 23 ⇣ ⌘ 46 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Theorem (Additive Cancellation). a c e Let , , and be any rational numbers. Then b d f a e c e a c If + = + , then = . b f d f b d Theorem (Opposite of the Opposite). a Let be any rational number. Then b a a = . − − b b ⇣ ⌘ Definition (Subtraction of Rational Numbers). a c Let and be any rational numbers. Then b d a c a c = + . b − d b − d Also, ⇣ ⌘ a c a c a c a + ( c) a c = + = + − = − = − b − b b − b b b b b and ⇣ ⌘ ⇣ ⌘ a c ad bc ad bc = = − . b − d bd − bd bd Example. 2 7 2(12) 9( 7) 24 ( 63) 24 + 63 87 29 − = − − = − − = = = 9 − 12 108 108 108 108 36 ⇣ ⌘ 2 7 = + 9 12 8 21 29 = + = 36 36 36 9.1. THE RATIONAL NUMBERS 47 3 3 ( 3)4 7(3) 24 ( 63) 12 21 12 + ( 21) 33 − = − − = − − = − − = − − = − 7 − 4 7(4) 108 28 28 28 12 21 33 33 = − = − = 28 − 28 28 −28 33 inverse or opposite of |{z} 28 Definition (Multiplication of Rational Numbers). a c Let and be any rational numbers. Then b d a c ac = . b · d bd Properties of Rational Number Multiplication a c e Let , , and be any rational numbers. b d f a c (Closure) is a rational number. b · d a c c a (Commutative) = . b · d d · b a c e a c e (Associative) = . b · d · f b · d · f ⇣ ⌘ a ⇣a ⌘ a m (Multiplicative Identity) 1 = = 1 1 = , m = 0 . b · b · b m 6 ⇣ a⌘ (Multiplicative Inverse) For every nonzero rational number , there exists a b b unique rational number such that a a b = 1. b · a Note. The multiplicative inverse of a rational number is also called the reciprocal of the number. 48 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Distributive Property of Multiplication over Addition a c e Let , , and be any rational numbers. Then b d f a c e a c a e + = + . b d f b · d b · f ⇣ ⌘ Example. 9 23 7 − − = (mentally?) 7 · 27 · 9 ⇣ 23 − 9⌘ ⇣ 7⌘ 23 9 7 23 23 23 − − = − − = 1 = = − 27 · 7 · 9 27 · 7 · 9 27 · 27 27 ⇣− ⌘ ⇣ ⌘ − ⇣ ⌘ − − 3 11 3 11 + − = (mentally?) 7 · − 21 7 − 21 11⇣ 3⌘ ⇣ 11⌘⇣ 3⌘ 11 3 3 11 + − = + − = 0 = 0 − 21 · 7 − 21 7 − 21 7 7 −21 · ⇣ ⌘ ⇣ ⌘ ⇣a ⌘⇣c ⌘ ⇣ ⌘h ⇣ ⌘i c Definition. Let and be any rational numbers where is nonzero. b d d Then a c a d = . b ÷ d b ⇥ c Common Denominator Division: a c a b a = = , so b ÷ b b ÷ c c a c a = or a c. b ÷ b c ÷ a A third method since a b = : ÷ b a ⇣ c a d a ⌘d a d a c a c = = = = = ÷ , b ÷ d b · c c · c c · c b ÷ d b d so ÷ a c a c = ÷ . b ÷ d b d ÷ 9.1. THE RATIONAL NUMBERS 49 Example. (1) 40 10 − − = 27 ÷ 9 40 ( 10) 4 − ÷ − = . 27 9 3 ÷ (2) 47 3 − = 49 ÷ 49 47 47 = − . 3 3 − (3) 3 5 − = 8 ÷ 6 3 3 ◆6◆ 9 − = − . ◆8◆ · 5 20 4 50 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Ordering Rational Numbers Three equivalent ways (as we did with fractions): 1) Number-Line Approach: a c c a < or > if and only if b d d b a ⇣ c ⌘ is to the left of on the rational number line. b d 2) Common-Positive-Denominator Approach: a c < if and only if a < c and b > 0. b c 3)Addition approach: a c p < if and only if the is a positive rational number such that b d q a p c + = , or equivalently, b q d a c c a < if and only if is positive. b d d − b 5 11 Example. Compare and . −6 −12 First way: 11 5 11 5 11 10 1 = + = + = . Then −12 − − 6 −12 6 −12 12 −12 ⇣5 ⌘ 11 1 11 5 = , so < . − 6 − − 12 12 − 12 − 6 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ 9.1. THE RATIONAL NUMBERS 51 Second way: 5 10 11 5 = . Since 11 < 10 and 12 > 0, < . −6 −12 − − −12 − 6 ⇣ ⌘ Cross Multiplication Inequality a a Let and be any rational numbers, where b > 0 and d > 0. Then b b a c < if and only if ad < bc. b d 5 11 Example. Compare − and − . 6 12 Third way: Note that the denominators are positive. Since 11 5 ( 5)12 = 60 and 6( 11) = 66, and 66 < 60, − < − . − − − − − − 12 6 Properties of Order for Rational Numbers Transitive Less than and addition Less than and multiplication by a positive Less than and multiplication by a negative Density Property – between any two rational numbers there exists at least one rational number Similar Properties hold for >, , .