CHAPTER FIVE NUMBER THEORY AND THE REAL NUMBER SYSTEM

Transcript

1 CHAPTER FIVE NUMBER THEORY AND THE REAL NUMBER SYSTEM Exercise Set.. Number theory is the study of numbers and their properties.. If a and b are factors of c, then c a is an integer and c b is an integer.. a) a divides b means that b divided by a has a remainder of zero. b) a is divisible by b means that a divided by b has a remainder of zero.. A prime number is natural number greater than that has exactly two factors (or divisors), itself and one.. A composite number is a natural number that is divisible by a number other than itself and. Any natural number that is not prime is composite.. Every composite number can be expressed as a unique product of prime numbers. 7. a) The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set. b) Determine the prime factorization of each number. Then find the product of the prime factors with the largest exponent in each of the prime factorizations. 8. a) The greatest common divisor (GCD) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set. b) Determine the prime factorization of each number. Then find the product of the prime factors with the smallest exponent that appears in each of the prime factorizations. c) 8 = = The prime factors with the smallest exponents that appear in each of the factorizations are. The GCD of and 0 is = Mersenne Primes are prime numbers of the form n - where n is a prime number.. A conjecture is a supposition that has not been proved nor disproved.. Goldbach s conjecture states that every even number greater than or equal to can be represented as the sum of two (not necessarily distinct) prime numbers.. Twin primes are of the form p, p+, where p is a prime number. An example is & 7.

2 CHAPTER Number Theory and the Real Number System. The prime numbers between and 0 are:,,, 7,,, 7, 9,, 9,, 7,,, 7,, 9,, 7, 7, 7, 79, 8, 89, 9, The prime numbers between and 0 are:,,, 7,,, 7, 9,, 9,, 7,,, 7,, 9,, 7, 7, 7, 79, 8, 87, 89, 9, 97,. True; since 9 =. True; since = 9 7. False; since is divisible by False; since is a multiple of. 9. False; since is divisible by True; since =.. True; If a number is divisible by, then it is also divisible by.. False; If a number is divisible by, then it is also divisible by.. False; If a number is divisible by, then the. True. sum of the number s digits is divisible by.. True; since =.. True; since =. 7. Divisible by,,,, 8 and Divisible by,,,,, 8, and. 9. Divisible by and. 0. Divisible by,,,,, 8, and.. Divisible by,,,,, 8, and.. Divisible by none of the numbers.. = 70. (other answers are possible). 9 = 00. (other answers are possible. 9 = =. = = = =

3 SECTION = 9. = 7. The prime factors of and 8 are: =, = a) The common factor is, thus, the GCD =. b) The factors with the greatest exponent that appear in either are,,. Thus, the LCM = = The prime factors of and are: =, = 7 a) The common factor is: ; thus, the GCD =. b) The factors with the greatest exponent that appear in either are:,, 7, ; thus, the LCM = 7 =. 7 = = 9 8. The prime factors of 0 and are: 0 = and = a) The common factor is ; thus, the GCD =. b) The factors with the greatest exponent that appear in either is, ; the LCM = = 80.. The prime factors of 0 and 900 are: 0 =, 900 = a) The common factors are:, ; thus, the GCD = = 0. b) The factors with the greatest exponent that appear in either are:,, ; thus, the LCM = = = = 9 9. The prime factors of 8 and are: 8 =, = a) The common factors are:,; thus, the GCD = =. b) The factors with the greatest exponent that appear in either are:, ; thus, the LCM = =. The prime factors of and 0 are: =, 0 = a) The common factors are:,, ; thus, the GCD = =. b) The factors with the greatest exponent that appear in either are:,, ; thus, the LCM = = 0. The prime factors of 9 and are: 9 =, = a) The common factors are: ; thus, the GCD = =. b) The factors with the greatest exponent that appear in either are:,, ; thus, the LCM = = 088. The prime factors of, 8, and 8 are: =, 8 =, 8 = 7 a) The common factors are: ; thus, the GCD = = 8. b) The factors with the greatest exponent that appear in either are: 7, ; thus, LCM = 7 = 8. The prime factors of 8, 78, and 98 are: 8 =, 78 =, 98 = a) The common factors are:, ; thus, the GCD = =. b) The factors with the greatest exponent that appear in either are:,,, ; thus, the LCM = = 7 7. Use the list of primes generated in exercise. The next two sets of twin primes are: 7, 9, 9,. 8. No. Any other two consecutive natural numbers will include an even number, and even numbers greater than two are composite.

4 CHAPTER Number Theory and the Real Number System 9. (a), Yes; (b), 0 No; (c), Yes; (d) 9, Yes n 0. Fermat number = +, where n is a natural number. + =, + = + = 7, 8 + = + = 7. These numbers are prime.. = +, = +, 8 = +, = + 7, = + 7, = 7 + 7, = +, 8 = +, 0 = + 7. Use the formula n, where n is a prime number. =, = 7, =, 7 = 7, = 89.. The gcd of 0 and 0 is 70 dolls.. The gcd of 88 and 9 is 9 cars.. The gcd of and 0 is 7 cards. The gcd of 0 and 80 is 0 trees. 7. The lcm of and 0 is 80 mins. 8. The lcm of 00 and 000 is 000 miles. 9. The least common multiple of and is 0. Thus, it will be 0 days before they both have the same night off again. 70. The least common multiple of and 8 is 90. Thus, it will be 90 days before he visits both on the same day again. 7. a) The possible committee sizes are:,,, 0, or. b) The number of committees possible are: committees of, 0 committees of, committees of, committees of 0, or committees of. 7. a) = 7 = + b) Conjecture: Every prime number greater than = = + differs by from a multiple of the number. 7 = 8 9 = 8 + = 9 = 0 c) The conjecture appears to be correct. 7. A number is divisible by if both and divide the number. 7. A number is divisible by if both and divide the number. 7. = with rem. =. = with rem. = 0. Thus, gcd of and is = with rem. =. 7 = with rem. =. = with rem. = 0. Thus, gcd of 0 and 7 is. 8. The proper factors of are:,,,, and = Thus, is not a perfect #. 8. The proper factors of 8 are:,,,,, 8,,, and = with rem. =. = with rem. =. = with rem. = 0. Thus, gcd of 8 and is = with rem. = = with rem. = 0. Thus, the gcd of 0 and 80 is The proper factors of 8 are:,,, 7, and = 8 Thus, 8 is a perfect number = 7 Thus, 8 is not a perfect # = with rem. = 0. = with rem. = 0. Thus, gcd of 8 and is = w/rem. = 0. 0 = w/rem. = = w/rem. = 0. Thus, gcd of and 0 is The proper factors of 9 are:,,,8,,,,, and = 9 Thus, 9 is a perfect #

5 SECTION a) 0 = Adding to each exponent and then multiplying these numbers, we get (+)(+)(+) = = divisors of No, and are not unique prime factors since =. Any number that divides, will also divide, but 8 does not divide all numbers that are divisible by. Some examples are:,, and The sum of the digits will be a number divisible by, thus the number is divisible by. 88. The sum of the groups which have the same three digits will always be divisible by three. (i.e. d + d + d = d and d) 89.,08 = (, );,000 8 =,000 and 8 8 = Thus, since and 8 8, =, = 7, =, 7 = 7 are prime numbers, but =,08 =,07; and since 89 =,07, 07 is not prime = ++, 9 = ++, = ++, = ++7, = ++, = ++7, = ++7, = ++7, = +7+7, 7 = ++7, 8 = ++, 9 = ++, 0 = (a) 00 = + 997; (b) 000 = + 997; (c) 000 = Exercise Set.. Begin at zero, draw an arrow to the value of the first number. From the tip of that arrow draw another arrow by moving a number of spaces equal to the value of the second number. Be sure to move left if the number is negative and move right if the number is positive. The sum of the two numbers is at the tip of the second arrow.. n; Additive Inverse = that number when added to n yields the Additive Identity (= 0); n + (- n) =0. To rewrite a subtraction problem as an addition problem, rewrite the subtraction sign as an addition sign and change the second number to its additive inverse.. The product of two numbers with like signs is a positive number, and the product of two numbers with unlike signs is a negative number.. The quotient of two numbers with like signs is a positive number, and the quotient of two numbers with unlike signs is a negative number.. If we set 0 = x and we cross multiply, we get the equation 0x =. Since 0 x = 0, we get = 0, which is a false statement, which means that there is no such number x. Therefore, division by 0 is not allowed = 8. + (-) = - 9. (-7) + 9 =. (-) + (-) = -. [+(- )]+0 = - +0 = -. (+)+( ) = 7+( ) =. [( )+( )]+9 = 7+9 =. [8+(- )]+(- ) = []+(- ) =. [(- )+(- 9)]+ = [- ]+ = -. [+(- )]+8 = [-8]+8 = 7. - = = = (-) = (- ) = - + = = - + (-) = = + (- 0) = (- ) = 8+ =

6 8 CHAPTER Number Theory and the Real Number System. [+(-)] - = - =+(-) = -. -(8+) = - = +(-) = = (-) = - 9. (-)(-) = 0. (-) = -. [(- 8)(- )] = = 9. ()(-)(-) = (- 0)(-) =. ( )(-) = (0)(-) = - 0. (-9)(-)(-) = (9)(-) = -8. [(-)(-)] [(-)(8)] = (8)(-0) = -70. [(-8)()()](-) = [(-)()](-) = [-0](-) = 0 7. (-) = 8. 8 = (-) = - 0. = -. 8 = 7. 7 =. =. 8 =. ( ) = 8. (- 900) (- ) = 7. True; every whole number is an integer. 8. False; Negative numbers are not natural numbers. 9. False; the difference of two negative integers may be positive, negative, or zero. 0. True.. True; the product of two integers with like signs is a positive integer.. False; the difference of a positive integer and a neg. integer may be +, - or zero.. False; the sum of a positive integer and a negative integer could be pos., neg., or zero.. True; the quotient of two integers with unlike signs is a negative number.. False; the product of two integers with unlike signs is always a negative integer.. False; the quotient of any two integers with like signs is a positive number. 7. ( + 7) = = 8. (-) [ (- 7)] = (-) [-] = 9. [(-)] - = - + (-) = [(-)(-)] - = 0+(-) = 7. ( - 8)() = (-)() = -. [8 (-)](-) = (-9)(-) = 7. [+(-7)] = [-] = -. ( 9) (-) = (-) (-) =. [(-)(-)] ( -) = (+ (- )) = (-) = -. [(-)] (-) = (-0) (-) = 7., -, -, 0,, 8. 0, -0, -, 0,, ,-, -, -, -, , - 7, - 7,, 7, 7. - (-79.8) = =.8 o F = + = pts.

7 SECTION = = pts = + + = + =. The Texans did make a first down = = 0 7.,9 - (-8) =,9 + 8 =,777 feet 77. a) + ( 8) = = 9. There is a 9 hr.time diff. b) ( 7) = + 7 =. There is a hr. time diff. 80. a) The next pentagonal numbers are,, and 70. The n th pentagonal. b) The number is obtained by adding the n th triangular # (see section.) to the n th square number (see section.) and subtracting n. For example, if n =, the th triangular number is 7. 8 ( 97) = =,79 feet a 78. b = a b = a b 80. b) continued: and the th square number is. The sum of and is and n = =, which is the th pentagonal #. The next pentagonal numbers are 9, 7,, 7, and. c) Since 70 is the 7 th pentagonal number and 9 is the 8 th pentagonal number, 7 cannot be a pentagonal number = (other answers are possible) 8. (a) + = + (b) = = = = 7 (c) = Exercise Set.. Rational numbers is the set of all numbers of the form p/q, where p and q are integers, and q 0.. a) Multiply and divide the number by the position value of the last nonzero digit to the right of the decimal point. 00(0.97) 97 b) a) Divide both the numerator and the denominator by their greatest common divisor. b) = = Divide the numerator by the denominator. The quotient is the integer part of the mixed number. The fraction part of the mixed number is the remainder divided by the divisor.. For positive mixed numbers, multiply the denominator of the fraction by the integer preceding it. Add this product to the numerator. This sum is the numerator of the improper fraction; the denominator is the same as the denominator of the mixed number. For negative mixed numbers, you can temporarily ignore the negative sign, perform the conversion described above, and then reattach the negative sign.. a) The product of two fractions is found by multiplying the numerators and multiplying the denominators.

8 CHAPTER Number Theory and the Real Number System b) = a) The reciprocal of a number is divided by the number. b) The reciprocal of - is = 8. a) To divide two fractions, multiply the first fraction by the reciprocal of the second fraction b) = = a) To add or subtract two fractions with a common denominator, we add or subtract their numerators and keep the common denominator. 7 b) + = c) = a) First rewrite each fraction with a common denominator. Then add or subtract the fractions. 7 b) + = + = + = c) = = We can multiply a fraction by the number one in the form of c/c (where c is a nonzero integer) and the number will maintain the same value.. Yes GCD of and is GCD of and is GCD of and 9 is GCD of and is GCD of and 800 is GCD of and is GCD of and 7 is GCD of and is GCD of and 9 is GCD of and 8 is ( )( 7) + + = ( )( ) =. (()( ) + ) = + = =. ( 7)( ) + 7 = + = =

9 SECTION. 7. ( )( ) + = + = 79 = 8. ( )( ) = ( )( 8) = ( )( ) + 8+ =. ()() = ()( ) + + =. ()() = ( )( ) + 0+ =. ( 7 ) 7 + = ( ) = ( )( ) + = = 7. = = ( 8)( ) + 8 = 8 =.0 0. = = = = ( )( ) = 8. = = = =.. = = = = = = = = Let n= 0., n=. n =. 9n n = 0. = n 9 9 9n =.0 0. Let n= 0., n=. n =. 9n n = 0. n 9 9 9n =.0. Let n=.9, n= 9.9 n = 9.9 n =.9 9n = 8.0 9n 8 = n 9 9

10 CHAPTER Number Theory and the Real Number System. Let n= 0., 0n=. 0n =. n = 0. 99n =.0 99n 7 = n Let n=., 0n=. 0n =. n =. 99n =.0 99n = n Let n=., 00n=. 00n =. n =. 999n =.0 999n = n Let n=.0, 0n=. 0n =. n =. 90n = n 9 = n Let n=.9, 0n= 9.9 0n = 9.9 n =.9 90n =.0 90n = n Let n =.78, 00n = n = n = n =.0 990n 7 = n Let n=.9, 00n= n = 9.9 n =.9 990n = n = n = = = = = 7 7 = = = = = 7 7. = = = = = The lcm of and is. + = + = + = 80. The lcm of and 8 is. 0 7 = = = The lcm of and is = + = + = 8 8. The lcm of and is. + = + = + = 8. The lcm of 9 and is = = = 9 9

11 SECTION. 8. The lcm of 0 and is = = = The lcm of,,and 7 is = = + + = The lcm of, 8, and 7 is. + + = = + + = 87. The lcm of 0,0,and 0 is = = = The lcm of, 0, and 0 is = = = = = = = = = = 9 = = = + = + = = = = = = = = + + = + + = = + = + = = + = + = = = = = The LCM of,, is. + + = + + = + + = musk thistles

12 CHAPTER Number Theory and the Real Number System inches =.7" =.7 oz. = oz = = cups of snipped parsley 8 = = tsp of pepper 8 8 = = cups of sliced carrots 7. The LCM of,, is = = + + = = = = = She must proofread.0 of the book or = 7 pages.. 8+ = = a) 7 inches 7 + b) 7 = = 7 " = = 9 = 0 = 9 = 9 " = + = = = Student tutors represent 0. of the budget. 7 = 8 cups. ( ) = = 8 inches = in = 90 + = 9 ". 8 ft = in. = in () The length of each piece is 8 in original area = 8 9 = 78 sq. in.; new area = 8 = 87 sq. in area increase = = 8 78 = 8 = 8 sq. in

13 SECTION. 7. width = 8 ft. in. = 9 in. + in. = 99 in.; length = ft. 8 in. = in. + 8 in. = 8 in. " a) perimeter = L + W = (8) + (99) = in = 7 ft. = 7ft.in. "/ft. b) width = 8ft. in. = 8 ft. = 8 ft. = ft.; 8 length = ft. 8in. = ft. = ft. = ft Area = L w = 88 sq.ft 7. c) Volume = L W H = = 8080 = 80.7 cu. ft a) = = 9 in. 8 b) + = in. c) + = + = in (.7) =.90 =.0. = 9.7 = (.789) =.9 =.70 = 0.8 = 0. = 7.79 = = 8. + = = = = = = = + = = = = =. + = + = = = + = = 7. a) Water (or milk): + = + = 8 8 Oats: + = = cup cup;

14 CHAPTER Number Theory and the Real Number System. a) b) 0.9 c) = 0., = 0., + = =, = d) 0.9 =. a) 8 b) c) times d) times Exercise Set.. A rational number can be written as a ratio of two integers, p/q, with q not equal to zero. Numbers that cannot be written as the ratio of two integers are called irrational numbers.. The principal square root of a number n written n, is the positive number that when multiplied by itself gives n.. A perfect square number is any number that is the square of a natural number.. The product rule for radical numbers: a b = a b a 0, b 0 a a The quotient rule for radical numbers: a 0, b 0 b = b. a) To add or subtract two or more square roots with the same radicand, add or subtract their coefficients and then multiply by the common radical. b) + 9 = 8 9 = =. A rationalized denominator contains no radical expressions. 7. a) Multiply both the numerator and denominator by the same number that will result in the radicand in the denominator becoming a 8. (a) [# ] [ Enter] perfect square. (b) 7 =.7 =. b) = 9. 8 = 9 = irrational 9. = rational. rational. Irrational; non-terminating, non-repeating decimal. Irrational; non-terminating, non-repeating decimal. Irrational; π is non-terminating, non-repeating.. Rational; quotient of two integers. Rational; terminating decimal 7. Irrational; non-terminating, non-repeating decimal 8. Rational; = is an integer. 9. = 8 0. =. 0 =. =. 9 =. =. =. = 7. 0 = 8. = 9., rational, integer, natural 0., rational, integer. =, rat l, integer., nat l. rational. rational. rational. rational

15 SECTION. 7. rational 7. rational 8. irrational 9. 8 = 9 = 0. 0 = =. 8 = =. 0 = =. = 9 7= 7. 7= =. 80= =. 90= 9 = 7. = 8 = = 0 = 9. + =(+) = =(+) 7= = = = = + 0= + = + = = 7 9 = 7 = 8 = = = 7+ 7=(+) 7 = +7 = + = 7 = + = + 0 =(+ ) = =(+ 0) = = = +9 = = = = + = = = = ( +) = 9 = = =. = = =. = 8 = 9 =. 0= 00 = 0 =. = =. 8 = = 9= 8 8. = 7 8

16 8 CHAPTER Number Theory and the Real Number System 9. = 70. = = = = = = = 8 8 = = is between and since 7 is between = and 9 =. 7 is between. and since 7 is closer to 9 than to. Using a calculator is between and 7 since 7 is between = and 9 = 7. 7 is between and. since 7 is closer to than to 9. Using a calculator is between and since 7 is between 0 = and =. 7 is between and. since 7 is closer to 0 than to. Using a calculator is between and since is between = and =. is between. and since is closer to than to. Using a calculator is between and since 70 is between 9 = and 9 =. 70 is between and. since 70 is closer to 9 than to 9. Using a calculator is between and since 00 is between 9 = and =. 00 is between and. since 00 is closer to 9 than to. Using a calculator False. p is an irrational number for any prime number p. 8. False. The result may be a rational number or an irrational number. 87. True 88. True 89. False. The result may be a rational number or an irrational number. 90. False. The result may be a rational number or an irrational number. 9. +( ) = = 9. =

17 SECTION = 9= 9. No..7 since is an irrational number and.7 is a rational number. 9. = 7 7 is irrational and is irrational, therefore is irrational. Because.7 is rational, then No.. and are rational numbers, π is an 7 irrational number = = 8 = T = π = π = π π 7 π 7 = π 7 7. a) b) s 0 = mph 0.0 s 00 = 0 mph 0.0 c) s 00 = 0 mph 0.0 d) s 00 = 80 mph 0.0. a) b) c) d) 0 t =. sec 00 0 t = sec t = 7. sec 00 0 t = sec. a) The number is rational if the result on the calculator is a terminating or repeating decimal number. Otherwise, the number is irrational. b) Using a calculator, 0.0 = 0. a terminating decimal and thus it is rational. c) Using a calculator, 0.07 = 0.7K, thus it is irrational.. No. The sum of two irrational numbers may not be irrational. (i.e. + = 0). a) ( ) = ( ) = = b) ( ) + = + = c) = + = d) ( + ) + = ( 8) + = + = 8 Exercise Set.. The set of real numbers is the union of the rational numbers and the irrational numbers.. All real numbers = R. If the given operation is preformed on any two elements of the set and the result is an element of the set, then the set is closed under the given operation.. The order in which two numbers are multiplied does not make a difference in the result. Ex. =. The order in which two numbers are added does not make a difference in the result. Ex. a+b = b+a

18 0 CHAPTER Number Theory and the Real Number System. The associative property of addition states that when adding three real numbers, parentheses may be placed around any two adjacent numbers. (a+b)+c = a+(b+c) 7. The associative property of multiplication states that when multiplying three real numbers, parentheses may be placed around any two adjacent numbers. Ex. ( ) = ( ). 8. The distributive property of multiplication over addition allows you to either add first and then multiply, or multiply first and then add. a(b+c) = ab + ac 9. Closed. The sum of two natural numbers is a natural number.. Not closed. (i.e. = is not a natural number).. Not closed. (i.e. 0. is not a natural number).. Closed. The product of two natural numbers is a natural number.. Closed. The difference of two integers is an integer.. Closed. The sum of two integers is an integer.. Not closed. (i.e. 0.= 0. is not an integer).. Closed. The product of two integers is an integer. 7. Closed 8. Closed 9. Closed 0. Not closed. Not closed. Not closed. Not closed. Not closed. Closed. Closed 7. Not closed 8. Closed 9. Commutative property. The order is changed from ( x) + ( + ) = ( + ) + x ( + ) = + ( + ) ; Commutative because the only thing that has changed is the order of and.. ( ) ( ) = 0= ( ) ( ). ( ) + ( ) = = ( ) + ( ). No. =, but =. [( ) ( ] ) ( 7) = ( ) ( 7) = ( ) [( ] ) ( 7 )] = ( ) ( ) =. No. =, but =. [( ) + ( ] ) + ( 7) = ( 8) + ( 7) = ( ) + [( ] ) + ( 7 )] = ( ) + ( ) = 7. No. ( 8 ) = =, but 8 ( ) = 8 = 8. No. ( 8 7) = =, but 8 ( ) = 8 + = 9. No. ( 8 ) = =, but 8 ( ) = 8 = 0. No. + ( ) = + =, but ( + ) ( + ) = = 0. Commutative property of addition. ( x+ ) = x + Distributive property. ( 7 ) = 7 ( ) Associative property of multiplication. ( + 7) + = + ( 7+ ) Associative property of addition. v + w= w+ v Commutative property of addition. ( x) = ( ) x Associative property of multiplication

19 SECTION. 7. 7= 7 Commutative property of multiplication = Associative property of addition 9. 8( 7+ ) = Distributive property 0. = Commutative property of multiplication. Commutative property of addition. Commutative property of addition. Distributive property. Commutative property of multiplication. Commutative property of addition. Commutative property of multiplication 7. ( c+ 7) = c + 8. ( d ) = d + 9. ( x ) = x = x 0. ( ) 0 k + 8 = k + = k = x x + = + = x +. x x = = 8x 8 8. x x = = x. 0x 0 x = = x. ( ) =. ( ) 7 + = 7. ( + ) = + = + 9. a) Distributive property b) Associative property of addition c) Combine like terms 7. a) Distributive property b) Associative property of addition; c) Commutative property of addition d) Associative property of addition e) Combine like terms 7. a) Distributive property b) Commutative property of addition; c) Associative property of addition d) Combine like terms e) Commutative property of addition 7. Yes. You can either lock your door first or put on your seat belt first. + = + = ( ) = a) Distributive property b) Associative property of addition; c) Combine like terms 7. a) Distributive property b) Associative property of addition; c) Commutative property of addition d) Associative property of addition e) Combine like terms 7. a) Distributive property b) Commutative property of addition; c) Associative property of addition d) Combine like terms e) Commutative property of addition 7. Yes. Can be done independently; no order needed

20 CHAPTER Number Theory and the Real Number System 77. No. The clothes must be washed first before 78. No. The PC must be turned on first before you can being dryed. type a term paper. 79. Yes. Can be done in either order; either fill the 80. No. The lamp must be turned on first before car with gas or wash the windshield reading a book. 8. Yes. The order of events does not matter. 8. No. The book must be read first, then write a report, then make a presentation. 8. Yes. The order does not matter. 8. Yes. The order does not matter. 8. Yes. The order does not matter 8. No. The egg cannot be poured before it is cracked. 87. Yes. The final result will be the same regardless of 88. Yes. The meatloaf will taste the same regardless the order of the events. of the order the items are mixed. 89. Baking pizzelles: mixing eggs into the batter, or mixing sugar into the batter.; Yard work: mowing the lawn, or trimming the bushes 90. Washing siding/washing windows/washing the car Writing letters to spouse, parents or friends 9. No. 0 a = 0 but a 0 is undefined. 9. a) No. (Man eating) tiger is a tiger that eats men, and man (eating tiger) is a man that is eating a tiger. b) No. (Horse riding) monkey is a monkey that rides a horse, and horse (riding monkey) is a horse that rides a monkey. c) Answers will vary. Exercise Set.. is the base and is the exponent or power.. b n is b multiplied by itself n times. n b = b b bl b n factors of b. a) If m and n are natural numbers and a is any real number, then a m a n = a m+n + 7 b) = = = 8. a) If a is any real number except 0, then a 0 =. 0 b) 7 =. a) If m and n are natural numbers and a is any real number except 0, then b) = = = a a m n = a m n. a) If n is a natural number and a is any real number except 0, then a b) 8 n = n a..

21 SECTION. 7. a) If m and n are natural numbers and a is any real m n mn number, then ( a ) = a 8. Since raised to any power equals, 00 =. = = = b) ( ) a) Since raised to any exponent equals +, then ( )( ) ( )( ) = = = b) Since raised to an even exponent equals, then number ( ) ( ) c) In 0 ( ) () 00 0 = = = 0 - is not raised to the 0 st 0 0 power, but + is; so ( )( ) ( )( ) d) Since is raised to a negative exponent is -, then ( ) 0 = = = =. a) Move the decimal point in the original number to the right or left until you obtain a number greater or equal to and less than. Count the number of places the decimal was moved. If it was moved to the left the count is a positive number and if it was moved to the right the count is a negative number. Multiply the number obtained in the first step by raised to the count number. b) =.. note: the count number is. a) If the exponent is positive, move the decimal point in the number to the right the same number of places as the exponent adding zeros where necessary. If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent adding zeros where necessary. b).7 = a) The number is greater than or equal to. b) The number is greater than or equal but <. c) The number is less than.. = =. = = 8. ( ) = ( ) ( ) ( ) ( ) =. = ( ) ( ) ( ) ( ) = 7. = ( ) ( ) = 9 8. ( ) ( ) ( ) = = 9 9. = = = = ( ) ( ) ( ) = =. ( ) = ( ) ( ) =. ( ) ( ) ( ) = =. ( ) ( ) ( ) ( ) ( ) = = 7. = 9

22 CHAPTER Number Theory and the Real Number System. 7 7 = = = = + 7. = = = = 7 =7 = ( )( )( )( )( ) 7 = = 8 9. ( ) 0 = 0. ( ) ( )( )( )( ) = = 8. = ( )( )( )( ) = 8. ( )( )( )( ) = = ( ) = = = 09. ( ) = = = 7. + = = = 8. = = = = = ( ) ( ) ( ). ( ) ( ) ( ) = =. ( ) = = = 09 ( ) = =.. ( ) + = = =. x =.97 x 7. =. x =. x =. x =.7 x =.9 x =. x =.8 x =.0 x =. x. 000 =. x 7. 7 = 7. x =.0 x =. x =. x. 8. x = x = x = x = x = x = x = x = x = x = x = 7..0 x = x = x = x = x = 0.00

23 SECTION. 77. (.0x )(.0x ) = 8.0x = ( )( ) 79. (. x )(.0 x ) =. x = ( )( ) x.0x = 8.x = 8.. x.0 x =. x = x. x 0.0 x x =.0 x = x 8. x.x x 8..0 x.0 x x.0 x.0 x 0.0 x 8..0 x.0 x x ( 00000)( ) = (.0 x )(.0 x ) =.0 x.0 x. x 7. x ( )( ) 88.. x.0 x =. x =. x. x.0 x =.9 x ( )( ) = 90. ( )( ) 9.. x 0.x.0x 7.0 x 9..0 x x.0x.0 x.0 9. =.0 x = 0. x =.0 x = 0. x 9 =.0 x = 0. = x,. x,.,.8 x x,. x, 8. x,. x x ; ;. x ; 0,000 ; Note: = 7.9 x, 0,000 = x 0.,9,000;.79 x ;. x 7 ; 7,000,000. $. 8.0 = a) $,90.8 b).908 GDP/person. $ = 0.08 a) $,.97 b) $. GDP/person = a) $,9,000,000,000,000,000 b) $ 0.x seconds =.98 people 8 d. t =.8 a) 8,000 hrs. b).8 x hrs r.. t = d r = 9000 mi 0000 mph =.9 a).9 hrs b).9 x hrs

24 CHAPTER Number Theory and the Real Number System 7. (00,000)(0,000,000,000) = ( x )( x ) = 0 x = x a) 0,000,000,000,000,000 drops b).0 x drops 8. (0)(,800,000) = ( x )(.8 x ) = 9 x 7 =.9 x 8 a) 90,000,000 cells b).9 x 8 cells =.8 x a) 8,000 times b).8 x times. a) (0,000 cu.ft./sec) (0 sec/min) (0 min/hr) ( hr) = 8,0,000,000 ft b) 8. x 9 cu ft. $. 7.0 = a) $,.97 8,09.00 = $,.78. a) 8 billion = 8,000,000,000 =.8 diapers b) () (.8) ( )=. x =. x or,,000 miles. a) (0.0) (,00,000,000) = $70,000,000 b) (0.) (,00,000,000) = $00,000,000 c) (0.) (,00,000,000) = $,000,000 d) (0.0) (,00,000,000) = $0,000,000. a) (0.0) (,00,000,000) = $,0,000,000 b) (0.0) (,00,000,000) = $,0,000,000 c) (0.) (,00,000,000) = $0,000,000 d) (0.) (,00,000,000) = $0,000,000.,000 times, since meter = millimeters =,000 millimeters. Since gram = milligrams and gram = - kilograms, - kilograms = milligrams Kilograms milligrams =, Thus, kilogram = milligrams = 0. =, times 8. a) () ( billion) = billion =,000,000,000 people b),000,000,000 =,000,000,000 = 9,7 people per day ()() ,000,000,000,000, = 0,000 seconds or about. days,900,000,000,000.9

25 SECTION. 7. a),000,000 =.0 x ;,000,000,000 =.0 9 ;,000,000,000,000 =.0.0 b) =.0 days or,000 days =.7 years c) =.0 days or,000,000 days =,79.7 years d) =.0 days or,000,000,000 days =,79,7.0 years.0 9 billion.0 e).0 =,000 times greater million.0. a) (.8 x mi/sec) (0 sec/min) (0 min/hr) ( hr/day) ( days/yr) ( yr) = (.8 x )( x )( x )(. x )(. x ) = 8.9 x =.89 x miles b) t = d 9. = r.8 7 =.0 x = 00 seconds or 8 min. 0 sec.. a) E(0) = x 0 = x = bacteria b) E(/) = x / =. = 8. bacteria Exercise Set.7. A sequence is a list of numbers that are related to each other by a given rule. One example is,,, 8.. The terms of the sequence.. a) An arithmetic sequence is a sequence in which each term differs from the preceding term by a constant amount. One example is,, 7,,... b) A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant. One example is,, 9, 7,.... a) d = +, b) r =. a) a n = n st term of the sequence b) a = st term of a sequence c) d = common difference in a sequence d) s n = the sum of the st n terms of the arithmetic sequence. a) a n = n st term of the sequence b) a = st term of a sequence c) r = common ratio between consecutive terms d) s n = the sum of the st n terms of the arithmetic sequence 7. a =, d =,, 7, 9, 8. a =, d =,, 7,, 9. a = -, d = - -, -,,, 7. a = -, d = - -, -, -,, 9.,,,,., 7,,, 9. /,, /,, /. /,, /,, 7/. a, a =, d =,,, 8,, 7 a = 7. a 9, a =, d = -,,, -, -, -, -7, -9, -, - a 9 = -

26 8 CHAPTER Number Theory and the Real Number System 7. a, a = -, d =, -, -, -,,,, 7, 9,, a = 8. a = 7 + ( ) ( ) = 7+() ( ) = 7 = a 0 = + ( 9)( ) = 9 = = 0. ( )( ) 7 + = 8 = =. a = ( ) + = + = 9. a = ( ) 8 + = +. a n = n a n = + (n - ) = + n = n. a n = n - a n = + (n - ) = + n - = n. a n = n a n = + (n ) = + n - = n.,, -, - a n = + (n - )(-) = - n + a n = - n + n = + n = n 7. an = ( ) 9 + n = + n = n 9. an = ( ) 8. an = - + (n - )() = n an = - + (n - )() = n - 8. s n = ( + ) ( + ) ( ) na a n = = 7 ( )( ). s n = ( + ) ( ) ( )( ) = 0. s n =. s 8 = ( + ) 0( + 99) 0( 0) 9 ( 8) ( )( 0) = 00 9 ( ). s 9 = ( ) 8 + ( ) 8 ( ) s 8 = = = = s 8 = = s 8. n- 9. a =, r =, ( ) a = ar = n n,,,, 8. a =, r = -, ( ) ( ) a = = =, -, 8, -, 0. a =, r =, ( ) ( ) a = = 8= 8, 8,,, 8. 8,,,,.,,,,.,,, 8, 9., 8,,, ,, 9/, 7/, 8/,, 7. a = () = ()() = a = () = ()() =

27 SECTION a = = = = = 8. a = ( ) 0. a 7 = ( ) = (79) = 87. a a =, r =, ( ) a = =,777,. a = ( )( ) 9 = 9,. a 8 a = -, r = -, ( )( ) 7 n.,,, 8 ( ) n a = =.,,, a = ( ) n n n a = =,0 8 7., -,, - ( ) a = n n 8. a n = n n ar = a r = n n 9. a n = ( ) n. a n = ar ( 9) = n a ( r ) ( ) ( ). s r 7 7 a ( r ) ( ) ( 8). s = 7 r = 7,0 a r = n n 0. a n = ( )( ) n. a n = ar ( ) = n a ( r ) ( ) ( ). s s. s 9 r a ( 9 9 r ) ( ) ( 9) = 9 r =,,8 7. s a ( r ) 7( ) 7( 77) = r = 0,0 8. n =, a = -, r = s a n ( r ) ( ) = r = 78 = 77 ( ) ( )( 78) 9. n =, a = -, r = s ( ) ( ) ( ) ( ) ( )( 79) ( )( ) ( 0)( + 0) ( 0)( ) ( ) s = 0 = 00 0 ( 0)( + 99) ( 0)( 00) ( ) s = 0 00 = n =, a =, r = ½ 7. s ( ) ( ) = = ( 0)( + 00) ( 0)( 0) ( ) s = 0 0 = ( 0)( + 0) ( 0)( ) ( ) s = = 8 0

28 0 CHAPTER Number Theory and the Real Number System 7. a) Using the formula a n = a + (n )d, we get a 8 = 0,00 + (8 ) (0) = $8,00 b) 8( ) = 8(8800) 77. a = 7 + ()( ) = 7 0 = in. = $9,00 7. a) a = 9 + ()( ) = 9 = in. b) ( )( 9) [(9 + )] ( )( 9) = 9 in. ( ) ( + ) 78. s 78 times 79.,,,. n= ( + ) ( ) s = ( ) = 9 PCs 80. a n = a r n ( )( ) 9 a = = 99 students 8. a = 00(0.8) = 00(0.) =.88 g 8. a = a r = () =,78 layers 8. a = 0,000(.0) = $,8 8. a = 0(0.8) =.88 ft. 8. This is a geometric sequence where a =000 and r =. In ten years the stock will triple its value times. a = a r - = 000() = $8, The sequence of bets during a losing streak is geometric. n a a) a = a r n- = () - ( r ) ( ) = () = $ s $ r n a b) a = a r n- = () - ( r ) ( ) ( ) = () = $0 s $ r n a c) a = a r n- = () - ( r ) ( ) ( ) = () = $,0 s $, 0 r n a d) a = a r n- = () - ( r ) ( ) ( ) = () = $,0 s $,0 r e) If you lose too many times in a row, then you will run out of money [ (/ ) ] (/ ) = 8[ (/ )] / = 8 = The arithmetic sequence 80 0, 0 0, 0 0, 70 0,.has a common difference of 80. Thus, a n = 80(n ) = 80n 0, n 89., 8,,,08 is an arithmetic sequence with a = and d =. Using the expression for the n th term of an arithmetic sequence an = a + (n )d or 08 = + (n ) and dividing both sides by gives 8 = + n or n = Since a = a r and a = a r, a /a = r. Thus r = 8/ = 7 or r =. Then = a = a r = a () or a = / = 8.

29 SECTION.7 9. The total distance is 0 plus twice the sum of the terms of the geometric sequence having a = (0) (0.8) = [ (0.8) ] [ 0.78] and r = 0.8. Thus s = (0.7) = ( 0.8) So the total distance is 0 + (80.78) = 9.8 ft. 9. The sequence of bets during a losing streak is geometric. n a a) a = a r n- = () - ( r ) ( ) = () = $ s $ r n a b) a = a r n- = () - ( r ) ( ) ( ) = () = $0 s $ r n a c) a = a r n- = () - ( r ) ( ) ( ) = () = $,0 s $, 0 r n a d) a = a r n- = () - ( r ) ( ) ( ) = () = $,0 s $,0 r e) If you lose too many times in a row, then you will run out of money. Exercise Set.8. Begin with the numbers,, then add and to get and continue to add the previous two numbers in the sequence to get the next number in the sequence.. a),,,,8,,,,,89 b).7.9 = c) = d) = e)..7 = f).8. = +. a) Golden number = b).8 = golden ratio When a line segment AB is divided at a point C, such that the ratio of the whole, AB, to the larger part, AC, is equal to the ratio of the larger part, AC, to the smaller part, CB, then each AB AC of the ratios and is known as the golden ratio. AC CB AB AC c) The golden proportion is: = AC CB d) The golden rectangle: L a+ b a + golden number W a b. All are essentially the same number when rounded.. a) Flowering head of a sunflower b) Great Pyramid. a) Petals on daisies b) Parthenon in Athens + 7. a) = b) = c) Differ by 8. 89, =.09, part of Fibonacci sequence 89

30 CHAPTER Number Theory and the Real Number System 9. / =, / =, / =., / =., 8/ =., /8 =., / =., / =.9, / =.7 89/ =.88. The consecutive ratios alternate increasing then decreasing about the golden ratio.. The ratio of the second to the first and the fourth to the third estimates the golden ratio.. Fib. No. prime factors Fib. No. prime factors prime 89 prime prime prime prime prime. If the first ten are selected; = =. If is selected the result is () 8 = 8 = which is the second number preceding.. If,,, and 8 are selected the result is = 8 9 = =. Answers will vary.. / =. which is a little <.. 7. Answers will vary. 8. Answers will vary. 9. Answers will vary. 0. Answers will vary.. Answers will vary.. Answers will vary.. Fibonacci type; + 8 = = 7. Not Fibonacci. Each term is not the sum of the two preceding terms.. Not Fibonacci. Each term is not the sum of the two preceding terms.. Fibonacci type: + =, + = Each term is the sum of the two preceding terms. 7. Fibonacci type; 0 + = ; + = Fibonacci type; + = ; + = 9. Fibonacci type; = -; 0 + (-) = - 0. Fibonacci type; 7 + = 0; + 0 =. a) If and are selected the sequence is,,,,, 8,, b) / =., / =.00, / =., / =., 8/ =.9, /8 =.8,. a) If and 7 are selected the sequence is, 7,, 9,, 0, 8, b) 7/ =., /7 =.7, 9/ =.8, /9 =., 0/ =., 8/0 =.,. a) If, 8, and are selected the result is 8 ()() = =. b) If,, and are selected the result is ()() = =. c) The square of the middle term of three consecutive terms in a Fibonacci sequence differs from the product of the st and nd term by.

31 SECTION.8. The sum of the numbers along the diagonals parallel to the one shown is a Fibonacci number.. a) Lucas sequence:,,, 7,, 8, 9, 7, b) 8 + = 9; + = 7 c) The first column is a Fibonacci-type sequence.., x, - + x, - + x, -0 + x, -0 + x, x, x, -0 + x, - + x a) -,,,, 8,, 8, 8,, 7 b),,, 0,,,,,, 0 c),,,,, 0,,,, d), 7,,,,,,, 7, 8 e), 8,,,,,,, 8, f) Yes, because each multiple causes the x term to be greater than the number term. 7. ( a+b) = a Let x = a b = + b = a + = x a b b a x a b x x x + = x x x = 0 a =, b = -, c = - b± b ac ± ()( ) ± Solve for x using the quadratic formula, x = = = a () 8. ( ) multiply by x x + = x ( x) x = x x = x x = x x + = 0 a =, b = -, c = x ()() 7 Solve for x using the quadratic formula, x = b ± b ac ± ± ± = i a () 9. Answers will vary. {,, ] {, 0, ] {, 08, } {7,, } 0. a) reflections, paths b) reflections, 8 paths c) reflections, paths Review Exercises. Use the divisibility rules in section.. 70,90 is divisible by,,,,, and 9.. Use the divisibility rules in section.. 00, is divisible by,,,, and 9. 7 = = = 7

32 CHAPTER Number Theory and the Real Number System = 7. =, 0 = 7. 7 = ; gcd = ; lcm = = 0 8. =, 0 = gcd = lcm = 0 9. =, 8 = 9 gcd = 9; lcm = = 7, 0 = ; gcd = = 0; lcm = 7 = =, 0 =, 9 = ; gcd = = ; lcm = = 80. =, 8 =, = ; gcd =. =, 9 = ; lcm = = ; lcm = = =. In days the train will stop in both cities.. + =. + (- 7) = = + ( 8) = 8. ( ) + ( ) = 9. = + ( ) = 9 0. ( ) = + =. ( + 7) = + ( ) = 0. + (9 ) = + =. ( )( ) =. ( )(9) =. ( ) =. -/-7 = 7. /- = - 8. [8 ( )]( ) = ( )( ) = 9. [( )( )] = = 0. [ 0 ()] ) = ( ) =. / = 0.. / = 0.. /0 = /8 = 0.7. / =.. /7 = / = /8 = /8 = /7 = = = n = n = 7.7 n =.7 99n =.00 99n n n =.. n =. 9n = 9 9 n = 0. 9n =.0 n = = = = n =.. 0n =. 90n n =. n n = = 7 7 =

33 Review Exercises 9... (( )( )) ( )( 7) (( )( )) = ( 7)( ) = + = + =. 7 = 7 = 7 = = + = + = 8. 0 = = 9 7 =. 7 7 = 7 = = + 8+ = = = 8 8 = 8 = 0. + = + = + = + = = = 7 = 8 8. ( ) 7 teaspoons. 0 = = =. 00 = 0 = 0 = = 8 8. = = + = = = = + = 8 7. = 8 = 9 = 9 = 7. 8 = 8= = = 7. 8= = 8 9 = 77. = 78. ( + 7) = (+ )= + 8= ( + )= 8+ = + 8. x + = + x Commutative property of addition 8. m = m Commutative property of multiplication

34 CHAPTER Number Theory and the Real Number System 8. Associative property of addition 8. Distributive property 8. Commutative property of addition 8. Commutative property of addition 87. Associative property of multiplication 88. Commutative property of multiplication 89. Distributive property 90. Commutative property of multiplication 9. Natural numbers closed for addition + = 7 9. Whole numbers not closed for subtraction = - 9. Not closed; is not an integer 9. Closed 9. Not closed; = is not irrational 9. Not closed; 0 is undefined 97. = = = = 9 = 9 = 8 0. = =. 7 0 =.. ( ) = = =. ( ) = = = 8. 0,000 =. x =.8 x =.7 x 8.,90,000 =.9 x 9.. x 7 =,000, x = x = x = 0,000. a) (7 x )( x ) () x - =. x. a) ( x )(. x ) ()(.) x ( ).0 x =.0 x = =.... = = 0. = a) (,000,000)(,000) = (.0 x )(.0 x ) = ()() x = 8.0 x 9 b) 8.0 E a) (,000)(0.0000) = (. x )(.0 x ) = (.)() x = 7.0 x = 0.7 b) 7.0 E = 9. =. x =, = = = times. 0, 000, =, =$,.. Arithmetic, 7. Geometric 8,. Arithmetic, 8. Geometric /, / 7. Arithmetic, 9 8. Geometric, 9., 7,, a = 0., -, -, -8, -, -, -0, - a 8 = , -, -, -, 0,,,, 0 a =.,,,, 8 a = 8

35 Review Exercises 7. a = (/) - = (/) = (/) = /. a = () - = () = (8) = 8. s 0 = 7. s 8 = ( + ) 0 89 ( + ) = ()(9) =. s 8 = (8)(8) 8. s 0 = ( + ) 8 ( (8)( ) ( + ) (0)(.7) s = ( ) ()( 8) ()( 80) = s = ( ) ()( 8) ()( 80) = 80. s = ( ) ( ) ()( + ) ()() = ( ) ( ) ( ) ()( ) ()( ). s = = ( ). Arithmetic: an = n +. Arithmetic: an = + (n ) = + n = n. Arithmetic: an = (/)n + (/). Geometric: an = () n 7. Geometric: an = ( ) n 8. Geometric: an = (/) n 9. Yes;, 0. Yes; 7, 8. No;,,,-,-,-. No Chapter Test. 8, is divisible by:,,,, 9, = 7. [( ) + ( 9)] + 8 = + 8 = 7. 7 = 0. [( 70)( )] (8 ) = 0 [8 + ( )] =0 ( ) = 7. ( )( ) = ( 9)( 9) = = 9 = = = + =

36 8 CHAPTER Number Theory and the Real Number System.. 9 = = = = = + = + = 9. The integers are closed under multiplication since the product of two integers is always an integer.. Associative property of addition. Distributive property 7. = = = 8. = = = =0.8 = an = n +. ( ) ( ) a = () = =. ( ) ( ). an = () n-.,,,,, 8,,,, Group Projects. In this exercise, you may obtain different answers depending upon how you work the problem. ) a) servings Rice: / cup, Salt: / tsp., Butter: tsp. b) serving Rice: / cup, Salt: /8 tsp., Butter: / tsp. c) 9 servings Rice: cup, Salt: 7/8 tsp., Butter: 7 / tsp. =. a) Area of triangle = A bh ( )( ) Area of triangle = A bh ( )( ) Area of rectangle = A = bh= R ( )( ) = = 0 = + = + = b) Area of trapezoid = A h( b b ) ( )( ) c) Yes, same T

37 Group Projects 9. Co-pay for prescriptions = 0% Co=pay for office visits = $ Co-pay for medical tests = 0% 0/: $ +.0 ($) = $.00 0/7: $ +.0 (8) = $7.0 0/9: $ +.0 (8) +.0 (7) = $7.0 a) Total = $97.0 b).0 () +.80 (88) +.80 (8) +,0 (7) = $88.80 c) $ = $0.80. a) branch b) 8 branches c) branches d) Yes