{"id":1153,"date":"2017-12-12T19:52:20","date_gmt":"2017-12-12T19:52:20","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&p=1153"},"modified":"2017-12-30T01:15:36","modified_gmt":"2017-12-30T01:15:36","slug":"terminating-or-repeating","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/terminating-or-repeating\/","title":{"raw":"Terminating or Repeating?","rendered":"Terminating or Repeating?"},"content":{"raw":"[latexpage]\r\n

You\u2019ve seen that when you write a fraction as a decimal, sometimes the decimal terminates<\/i>, like: \\[\\frac 1 2 = 0.5 \\quad \\text{ and } \\quad \\frac {33} {1000} = 0.033.\\]<\/p>\r\n

However, some fractions have\u00a0 decimal representations that go on forever in a repeating pattern, like: \\[\\frac 1 3 = 0.33333\\dots \\quad \\text{ and } \\quad \\frac {6} {7} = 0.857142857142857142857142\\dots.\\]<\/p>\r\n

It\u2019s not totally obvious, but\u00a0 it is true:\u00a0Those are the only two things that can happen when you write a fraction as a decimal.<\/p>\r\n

Of course, you can imagine <\/i>(but never write down) a decimal that goes on forever but doesn\u2019t repeat itself, for example:\u00a0\u00a0\\[0.101001000100001000001\\dots \\quad \\text{ and } \\quad \\pi = 3.14159265358979\\dots.\\]<\/p>\r\n

But these numbers can never be written as a nice fraction $\\frac a b$ where $a$ and $b$ are whole numbers.\u00a0They are called irrational numbers<\/i>.\u00a0The reason for this name:\u00a0Fractions like $\\frac a b$ are also called ratios<\/i>.\u00a0 Irrational numbers cannot be expressed as a ratio<\/i> of two whole numbers.<\/p>\r\n

For now, we\u2019ll think about the question: Which fractions have decimal representations that terminate, and which fractions have decimal representations that repeat forever?\u00a0 We\u2019ll focus just on unit fractions<\/i>.<\/p>\r\n\r\n

\r\n

Definition<\/h3>\r\n

A unit <\/i>fraction<\/b> is a fraction that has 1 in the numerator. It looks like $\\frac 1 n$ for some whole number $n$.<\/p>\r\n\r\n<\/div>\r\n \r\n

\r\n

Think \/ Pair \/ Share<\/h3>\r\n
    \r\n \t
  • Which of the following fractions have infinitely long decimal representations and which do not? \\[\u00a0\\frac 12 \\qquad \\frac 13 \\qquad \\frac 14 \\qquad \\frac15 \\qquad \\frac16 \\qquad \\frac 17 \\qquad \\frac 18 \\qquad \\frac 19 \\qquad \\frac 1{10}.\\]<\/li>\r\n<\/ul>\r\n
      \r\n \t
    • Try some more examples on your own.\u00a0 Do you have a conjecture?<\/li>\r\n<\/ul>\r\n

      A fraction<\/i>\u00a0$\\frac 1 b$ has an infinitely long decimal expansion if:<\/i><\/p>\r\n

      ________________________________.<\/i><\/p>\r\n\r\n<\/div>\r\n \r\n

      \r\n

      Problem 7<\/h3>\r\n

      Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 2.\u00a0 (You may want to use a calculator to compute the decimal representations.\u00a0 The point is to look for and then explain a pattern, rather than to compute by hand.)<\/p>\r\n

      Try even more examples until you can make a conjecture:\u00a0 What is the decimal representation of the unit fraction $\\frac 1{2^n}$?<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Fraction<\/th>\r\nDenominator<\/th>\r\nDecimal<\/th>\r\n<\/tr>\r\n
      $\\frac 1 2$<\/td>\r\n$2$<\/td>\r\n$0.5$<\/td>\r\n<\/tr>\r\n
      $\\frac 1 4$<\/td>\r\n$2^2$<\/td>\r\n$0.25$<\/td>\r\n<\/tr>\r\n
      $\\frac 1 8$<\/td>\r\n$2^3$<\/td>\r\n$0.125$<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {16}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {32}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {64}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {128}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {256}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n
      \r\n

      Problem 8<\/h3>\r\n

      Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 5.\u00a0 (You may want to use a calculator to compute the decimal representations.\u00a0 The point is to look for and then explain a pattern, rather than to compute by hand.)<\/p>\r\n

      Try even more examples until you can make a conjecture:\u00a0 What is the decimal representation of the unit fraction $\\frac 1{5^n}$?<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Fraction<\/th>\r\nDenominator<\/th>\r\nDecimal<\/th>\r\n<\/tr>\r\n
      $\\frac 1 5$<\/td>\r\n$5$<\/td>\r\n$0.2$<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {25}$<\/td>\r\n$5^2$<\/td>\r\n$0.04$<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {125}$<\/td>\r\n$5^3$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {625}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {3125}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      $\\frac 1 {15625}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

      Marcus noticed a pattern in the table from Problem 7, but was having trouble explaining exactly what he noticed.\u00a0 Here\u2019s what he said to his group:<\/p>\r\n\r\n

      \r\n

      I remembered that when we wrote fractions as decimals before, we tried to make the denominator into a power of ten.\u00a0 So we can do this:<\/i><\/p>\r\n

      \\[\\frac12 = \\frac12 \\cdot \\frac55 = \\frac{5}{10} = 0.5.\\]<\/p>\r\n

      \\[\\frac14 = \\frac14 \\cdot \\frac{25}{25} = \\frac{25}{100} = 0.25.\\]<\/p>\r\n

      \\[\\frac18 = \\frac18 \\cdot \\frac{125}{125} = \\frac{125}{1000} = 0.125.\\]<\/p>\r\n

      When we only have 2\u2019s, we can always turn them into 10\u2019s by adding enough 5\u2019s.<\/i><\/p>\r\n<\/blockquote>\r\n \r\n

      \r\n

      Think \/ Pair \/ Share<\/h3>\r\n
        \r\n \t
      • Write out several more examples of what Marcus discovered.<\/li>\r\n \t
      • If Marcus had the unit fraction $\\frac 1 {2^n}$, what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?<\/li>\r\n \t
      • Now think about unit fractions with powers of 5 in the denominator. If Marcus had the unit fraction $\\frac 1{5^n}$, what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?<\/li>\r\n<\/ul>\r\n<\/div>\r\n

        Marcus had a really good insight, but he didn\u2019t explain it very well.\u00a0 He doesn\u2019t really mean that we \u201cturn 2\u2019s into 10\u2019s.\u201d\u00a0 And he\u2019s not doing any addition, so talking about \u201cadding enough 5\u2019s\u201d is pretty confusing.<\/p>\r\n\r\n

        \r\n

        Problem 9<\/h3>\r\n
          \r\n \t
        1. Complete the statement below by filling in the numerator of the fraction.\r\n
          The unit fraction $\\frac 1{2^n}$\u00a0has a decimal representation that terminates.\u00a0 The representation will have\u00a0$n$\u00a0decimal digits, and will be equivalent to the fraction \\[\\frac{?}{10^n}.\\]<\/blockquote>\r\n<\/li>\r\n \t
        2. Write a better version of Marcus\u2019s explanation to justify why this fact is true.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
          \r\n

          Problem 10<\/h3>\r\n

          Write a statement about the decimal representations of unit fractions $\\frac 1{5^n}$ and justify that your statement is correct.\u00a0 (Use the statement in Problem 9 as a model.)<\/p>\r\n\r\n<\/div>\r\n \r\n

          \r\n

          Problem 11<\/h3>\r\n

          Each of the fractions listed below has a terminating decimal representation.\u00a0 Explain how you could know this for sure, without actually calculating the decimal representation.<\/p>\r\n\\[\\frac 1{10} \\qquad\\quad \\frac 1{20} \\qquad\\quad \\frac 1{50} \\qquad\\quad \\frac 1{200} \\qquad\\quad \\frac 1{500} \\qquad\\quad \\frac 1{4000}.\\]\r\n\r\n<\/div>\r\n

          The Period of a Repeating Decimal<\/h1>\r\n

          If the denominator of a fraction can be factored into just 2\u2019s and 5\u2019s, you can always form an equivalent fraction where the denominator is a power of ten.<\/p>\r\n

          For example, if we start with the fraction \\[\\frac 1 {2^a 5^b},\\]<\/p>\r\n

          we can form an equivalent fraction \\[\\frac 1 {2^a 5^b} \\ = \\ \\frac 1 {2^a 5^b} \\cdot \\frac {2^b 5^a} {2^b 5^a} \\ = \\ \\frac {2^b 5^a} {2^{a+b} 5^{a+b}} \\ = \\ \\frac {2^b 5^a} {10^{a+b}} .\\]<\/p>\r\n

          The denominator of this fraction is a power of ten, so the decimal expansion is finite with (at most)\u00a0$a+b$ places.<\/p>\r\n

          What about fractions where the denominator has other prime factors besides 2\u2019s and 5\u2019s?\u00a0 Certainly we can\u2019t<\/i> turn the denominator into a power of 10, because powers of 10 have just 2\u2019s and 5\u2019s as their prime factors.\u00a0 So in this case the decimal expansion will go on forever.\u00a0 But why will it have a repeating<\/i> pattern<\/i>?\u00a0 And is there anything else interesting we can say in this case?<\/p>\r\n\r\n

          \r\n

          Definition<\/h3>\r\n

          The period<\/b> of a repeating decimal is the smallest number of digits that repeat.<\/p>\r\n\r\n<\/div>\r\n

          For example, we saw that \\[\\frac 13 \\ = \\ 0.33333\\dots = 0.\\overline{3}.\\]<\/p>\r\n

          The repeating part is just the single digit 3, so the period of this repeating decimal is one.<\/p>\r\n

          Similarly, we know that \\[\\frac 67 \\ =\\ 0.857142857142857142857142\\dots\u00a0\\ = \\ 0.\\overline{857142}.\\phantom{857142857142857142}.\\]<\/p>\r\n

          The smallest repeating part is the digits\u00a0$857142$, so the period of this repeating decimal is 6.<\/p>\r\n

          You can think of it this way: the period<\/i> is the length of the string of digits under the vinculum (the horizontal bar that indicates the repeating digits).<\/p>\r\n\r\n

          \r\n

          Problem 12<\/h3>\r\n

          Complete the table below which shows the decimal expansion of unit fractions where the denominator has prime factors besides 2 and 5.\u00a0 (You may want to use a calculator to compute the decimal representations.\u00a0 The point is to look for and then explain a pattern, rather than to compute by hand.)<\/p>\r\n

          Try even more examples until you can make a conjecture:\u00a0 What can you say about the period of the fraction\u00a0$\\frac 1 n$\u00a0when\u00a0$n$\u00a0has prime factors besides 2 and 5?<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Fraction<\/th>\r\nDecimal<\/th>\r\nPeriod<\/th>\r\n<\/tr>\r\n
          $\\frac 1 3$<\/td>\r\n$0.\\overline{3}$<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
          $\\frac 1 6$<\/td>\r\n$0.1\\overline{6}$<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
          $\\frac 1 7$<\/td>\r\n$0.\\overline{142857}$<\/td>\r\n6<\/td>\r\n<\/tr>\r\n
          $\\frac 1 {9}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          $\\frac 1 {11}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          $\\frac 1 {12}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          $\\frac 1 {13}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          $\\frac 1 {14}$<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

          Imagine you are doing the \u201cDots & Boxes\u201d division to compute the decimal representation of a unit fraction like $\\frac 1 6$.\u00a0 You start with a single dot in the ones box:<\/p>\r\n

          \"\"<\/p>\r\n

          To find the decimal expansion, you \u201cunexplode\u201d dots, form groups of six, see how many dots are left, and repeat.<\/p>\r\n

          Draw your own pictures to follow along this explanation:<\/p>\r\nPicture 1:<\/b> When you unexplode the first dot, you get 10 dots in the\u00a0$\\frac 1{10}$\u00a0box, which gives one group of six with remainder of 4.\r\n\r\nPicture 2: <\/b>When you unexplode those four dots, you get 40 dots in the\u00a0$\\frac 1{100}$\u00a0box, which gives six group of six with remainder of 4.\r\n\r\nPicture 3:<\/b> Unexplode those 4 dots to get 40 in the next box to the right.\r\n\r\nPicture 4:<\/b> Make six groups of 6 dots with remainder 4.\r\n

          Since the remainder repeated (we got a remainder of 4 again), we can see that the process will now repeat forever:<\/p>\r\n\r\n

            \r\n \t
          • unexplode 4 dots to get 40 in the next box to the right,<\/li>\r\n \t
          • make six groups of 6 dots with remainder 4,<\/li>\r\n \t
          • unexplode 4 dots to get 40 in the next box to the right,<\/li>\r\n \t
          • make six groups of 6 dots with remainder 4,<\/li>\r\n \t
          • and so on forever...<\/li>\r\n<\/ul>\r\n

            On Your Own<\/h3>\r\n

            Work on the following exercises on your own or with a partner.<\/p>\r\n\r\n

              \r\n \t
            1. Use \u201cDots & Boxes\u201d division to compute the decimal representation of $\\frac 1{11}$.\u00a0 Explain how you know for sure the process will repeat forever.<\/li>\r\n \t
            2. Use \u201cDots & Boxes\u201d division to compute the decimal representation of $\\frac 1{12}$.\u00a0 Explain how you know for sure the process will repeat forever.<\/li>\r\n \t
            3. What are the possible remainders<\/i> you can get when you use division to compute the fraction $\\frac 1 7$?\u00a0 How can you be sure the process will eventually repeat?<\/li>\r\n \t
            4. What are the possible remainders<\/i> you can get when you use division to compute the fraction $\\frac 1 9$?\u00a0 How can you be sure the process will eventually repeat?<\/li>\r\n<\/ol>\r\n \r\n
              \r\n

              Problem 13<\/h3>\r\n

              Suppose that\u00a0$n$\u00a0is a whole number, and it has some prime factors besides 2\u2019s and 5\u2019s.\u00a0 Write a convincing argument that:<\/p>\r\n\r\n

                \r\n \t
              1. The decimal representation of\u00a0$\\frac 1 n$\u00a0will go on forever (it will not terminate).<\/li>\r\n \t
              2. The decimal representation of\u00a0$\\frac 1 n$\u00a0will be an infinite repeating<\/i> decimal.<\/li>\r\n \t
              3. The period of the decimal representation\u00a0 of\u00a0$\\frac 1 n$ will be less than $n$.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                \r\n

                Problem 14<\/h3>\r\n
                  \r\n \t
                1. Find the \u201cdecimal\u201d expansion for\u00a0$\\frac 1 2$\u00a0in the following bases.\u00a0 Be sure to show your work: \\[ \\text{two}, \\quad \\text{three}, \\quad \\text{four},\\quad \\text{five}, \\quad \\text{six}, \\quad \\text{seven}.\\]<\/li>\r\n \t
                2. Make a conjecture: If I write the decimal expansion of\u00a0$\\frac 1 2$\u00a0in base\u00a0$b$, when will that expansion be finite and when will it be an infinite repeating decimal expansion?<\/li>\r\n \t
                3. Can you prove your conjecture is true?<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"

                  You\u2019ve seen that when you write a fraction as a decimal, sometimes the decimal terminates<\/i>, like: <\/p>\n

                    <\/span>   <\/span>\"\[\frac 1 2 = 0.5 \quad \text{ and } \quad \frac {33} {1000} = 0.033.\]\"<\/p>\n

                  However, some fractions have\u00a0 decimal representations that go on forever in a repeating pattern, like: <\/p>\n

                    <\/span>   <\/span>\"\[\frac 1 3 = 0.33333\dots \quad \text{ and } \quad \frac {6} {7} = 0.857142857142857142857142\dots.\]\"<\/p>\n

                  It\u2019s not totally obvious, but\u00a0 it is true:\u00a0Those are the only two things that can happen when you write a fraction as a decimal.<\/p>\n

                  Of course, you can imagine <\/i>(but never write down) a decimal that goes on forever but doesn\u2019t repeat itself, for example:\u00a0\u00a0<\/p>\n

                    <\/span>   <\/span>\"\[0.101001000100001000001\dots \quad \text{ and } \quad \pi = 3.14159265358979\dots.\]\"<\/p>\n

                  But these numbers can never be written as a nice fraction \"\frac a b\" where \"a\" and \"b\" are whole numbers.\u00a0They are called irrational numbers<\/i>.\u00a0The reason for this name:\u00a0Fractions like \"\frac a b\" are also called ratios<\/i>.\u00a0 Irrational numbers cannot be expressed as a ratio<\/i> of two whole numbers.<\/p>\n

                  For now, we\u2019ll think about the question: Which fractions have decimal representations that terminate, and which fractions have decimal representations that repeat forever?\u00a0 We\u2019ll focus just on unit fractions<\/i>.<\/p>\n

                  \n

                  Definition<\/h3>\n

                  A unit <\/i>fraction<\/b> is a fraction that has 1 in the numerator. It looks like \"\frac 1 n\" for some whole number \"n\".<\/p>\n<\/div>\n

                   <\/p>\n

                  \n

                  Think \/ Pair \/ Share<\/h3>\n
                    \n
                  • Which of the following fractions have infinitely long decimal representations and which do not?\n

                      <\/span>   <\/span>\"\[ \frac 12 \qquad \frac 13 \qquad \frac 14 \qquad \frac15 \qquad \frac16 \qquad \frac 17 \qquad \frac 18 \qquad \frac 19 \qquad \frac 1{10}.\]\"<\/p>\n<\/li>\n<\/ul>\n

                      \n
                    • Try some more examples on your own.\u00a0 Do you have a conjecture?<\/li>\n<\/ul>\n

                      A fraction<\/i>\u00a0\"\frac 1 b\" has an infinitely long decimal expansion if:<\/i><\/p>\n

                      ________________________________.<\/i><\/p>\n<\/div>\n

                       <\/p>\n

                      \n

                      Problem 7<\/h3>\n

                      Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 2.\u00a0 (You may want to use a calculator to compute the decimal representations.\u00a0 The point is to look for and then explain a pattern, rather than to compute by hand.)<\/p>\n

                      Try even more examples until you can make a conjecture:\u00a0 What is the decimal representation of the unit fraction \"\frac 1{2^n}\"?<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
                      Fraction<\/th>\nDenominator<\/th>\nDecimal<\/th>\n<\/tr>\n
                      \"\frac 1 2\"<\/td>\n\"2\"<\/td>\n\"0.5\"<\/td>\n<\/tr>\n
                      \"\frac 1 4\"<\/td>\n\"2^2\"<\/td>\n\"0.25\"<\/td>\n<\/tr>\n
                      \"\frac 1 8\"<\/td>\n\"2^3\"<\/td>\n\"0.125\"<\/td>\n<\/tr>\n
                      \"\frac 1 {16}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {32}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {64}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {128}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {256}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                       <\/p>\n

                      \n

                      Problem 8<\/h3>\n

                      Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 5.\u00a0 (You may want to use a calculator to compute the decimal representations.\u00a0 The point is to look for and then explain a pattern, rather than to compute by hand.)<\/p>\n

                      Try even more examples until you can make a conjecture:\u00a0 What is the decimal representation of the unit fraction \"\frac 1{5^n}\"?<\/p>\n\n\n\n\n\n\n\n\n\n
                      Fraction<\/th>\nDenominator<\/th>\nDecimal<\/th>\n<\/tr>\n
                      \"\frac 1 5\"<\/td>\n\"5\"<\/td>\n\"0.2\"<\/td>\n<\/tr>\n
                      \"\frac 1 {25}\"<\/td>\n\"5^2\"<\/td>\n\"0.04\"<\/td>\n<\/tr>\n
                      \"\frac 1 {125}\"<\/td>\n\"5^3\"<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {625}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {3125}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                      \"\frac 1 {15625}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                      Marcus noticed a pattern in the table from Problem 7, but was having trouble explaining exactly what he noticed.\u00a0 Here\u2019s what he said to his group:<\/p>\n

                      \n

                      I remembered that when we wrote fractions as decimals before, we tried to make the denominator into a power of ten.\u00a0 So we can do this:<\/i><\/p>\n

                      \n

                        <\/span>   <\/span>\"\[\frac12 = \frac12 \cdot \frac55 = \frac{5}{10} = 0.5.\]\"<\/p>\n

                      \n

                        <\/span>   <\/span>\"\[\frac14 = \frac14 \cdot \frac{25}{25} = \frac{25}{100} = 0.25.\]\"<\/p>\n

                      \n

                        <\/span>   <\/span>\"\[\frac18 = \frac18 \cdot \frac{125}{125} = \frac{125}{1000} = 0.125.\]\"<\/p>\n

                      When we only have 2\u2019s, we can always turn them into 10\u2019s by adding enough 5\u2019s.<\/i><\/p>\n<\/blockquote>\n

                       <\/p>\n

                      \n

                      Think \/ Pair \/ Share<\/h3>\n
                        \n
                      • Write out several more examples of what Marcus discovered.<\/li>\n
                      • If Marcus had the unit fraction \"\frac 1 {2^n}\", what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?<\/li>\n
                      • Now think about unit fractions with powers of 5 in the denominator. If Marcus had the unit fraction \"\frac 1{5^n}\", what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?<\/li>\n<\/ul>\n<\/div>\n

                        Marcus had a really good insight, but he didn\u2019t explain it very well.\u00a0 He doesn\u2019t really mean that we \u201cturn 2\u2019s into 10\u2019s.\u201d\u00a0 And he\u2019s not doing any addition, so talking about \u201cadding enough 5\u2019s\u201d is pretty confusing.<\/p>\n

                        \n

                        Problem 9<\/h3>\n
                          \n
                        1. Complete the statement below by filling in the numerator of the fraction.
                          \n

                          The unit fraction \"\frac 1{2^n}\"\u00a0has a decimal representation that terminates.\u00a0 The representation will have\u00a0\"n\"\u00a0decimal digits, and will be equivalent to the fraction <\/p>\n

                            <\/span>   <\/span>\"\[\frac{?}{10^n}.\]\"<\/p>\n<\/blockquote>\n<\/li>\n

                        2. Write a better version of Marcus\u2019s explanation to justify why this fact is true.<\/li>\n<\/ol>\n<\/div>\n

                           <\/p>\n

                          \n

                          Problem 10<\/h3>\n

                          Write a statement about the decimal representations of unit fractions \"\frac 1{5^n}\" and justify that your statement is correct.\u00a0 (Use the statement in Problem 9 as a model.)<\/p>\n<\/div>\n

                           <\/p>\n

                          \n

                          Problem 11<\/h3>\n

                          Each of the fractions listed below has a terminating decimal representation.\u00a0 Explain how you could know this for sure, without actually calculating the decimal representation.<\/p>\n

                            <\/span>   <\/span>\"\[\frac 1{10} \qquad\quad \frac 1{20} \qquad\quad \frac 1{50} \qquad\quad \frac 1{200} \qquad\quad \frac 1{500} \qquad\quad \frac 1{4000}.\]\"<\/p>\n<\/div>\n

                          The Period of a Repeating Decimal<\/h1>\n

                          If the denominator of a fraction can be factored into just 2\u2019s and 5\u2019s, you can always form an equivalent fraction where the denominator is a power of ten.<\/p>\n

                          For example, if we start with the fraction <\/p>\n

                            <\/span>   <\/span>\"\[\frac 1 {2^a 5^b},\]\"<\/p>\n

                          we can form an equivalent fraction <\/p>\n

                            <\/span>   <\/span>\"\[\frac 1 {2^a 5^b} \ = \ \frac 1 {2^a 5^b} \cdot \frac {2^b 5^a} {2^b 5^a} \ = \ \frac {2^b 5^a} {2^{a+b} 5^{a+b}} \ = \ \frac {2^b 5^a} {10^{a+b}} .\]\"<\/p>\n

                          The denominator of this fraction is a power of ten, so the decimal expansion is finite with (at most)\u00a0\"a+b\" places.<\/p>\n

                          What about fractions where the denominator has other prime factors besides 2\u2019s and 5\u2019s?\u00a0 Certainly we can\u2019t<\/i> turn the denominator into a power of 10, because powers of 10 have just 2\u2019s and 5\u2019s as their prime factors.\u00a0 So in this case the decimal expansion will go on forever.\u00a0 But why will it have a repeating<\/i> pattern<\/i>?\u00a0 And is there anything else interesting we can say in this case?<\/p>\n

                          \n

                          Definition<\/h3>\n

                          The period<\/b> of a repeating decimal is the smallest number of digits that repeat.<\/p>\n<\/div>\n

                          For example, we saw that <\/p>\n

                            <\/span>   <\/span>\"\[\frac 13 \ = \ 0.33333\dots = 0.\overline{3}.\]\"<\/p>\n

                          The repeating part is just the single digit 3, so the period of this repeating decimal is one.<\/p>\n

                          Similarly, we know that <\/p>\n

                            <\/span>   <\/span>\"\[\frac 67 \ =\ 0.857142857142857142857142\dots \ = \ 0.\overline{857142}.\phantom{857142857142857142}.\]\"<\/p>\n

                          The smallest repeating part is the digits\u00a0\"857142\", so the period of this repeating decimal is 6.<\/p>\n

                          You can think of it this way: the period<\/i> is the length of the string of digits under the vinculum (the horizontal bar that indicates the repeating digits).<\/p>\n

                          \n

                          Problem 12<\/h3>\n

                          Complete the table below which shows the decimal expansion of unit fractions where the denominator has prime factors besides 2 and 5.\u00a0 (You may want to use a calculator to compute the decimal representations.\u00a0 The point is to look for and then explain a pattern, rather than to compute by hand.)<\/p>\n

                          Try even more examples until you can make a conjecture:\u00a0 What can you say about the period of the fraction\u00a0\"\frac 1 n\"\u00a0when\u00a0\"n\"\u00a0has prime factors besides 2 and 5?<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
                          Fraction<\/th>\nDecimal<\/th>\nPeriod<\/th>\n<\/tr>\n
                          \"\frac 1 3\"<\/td>\n\"0.\overline{3}\"<\/td>\n1<\/td>\n<\/tr>\n
                          \"\frac 1 6\"<\/td>\n\"0.1\overline{6}\"<\/td>\n1<\/td>\n<\/tr>\n
                          \"\frac 1 7\"<\/td>\n\"0.\overline{142857}\"<\/td>\n6<\/td>\n<\/tr>\n
                          \"\frac 1 {9}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                          \"\frac 1 {11}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                          \"\frac 1 {12}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                          \"\frac 1 {13}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                          \"\frac 1 {14}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                          Imagine you are doing the \u201cDots & Boxes\u201d division to compute the decimal representation of a unit fraction like \"\frac 1 6\".\u00a0 You start with a single dot in the ones box:<\/p>\n

                          \"\"<\/p>\n

                          To find the decimal expansion, you \u201cunexplode\u201d dots, form groups of six, see how many dots are left, and repeat.<\/p>\n

                          Draw your own pictures to follow along this explanation:<\/p>\n

                          Picture 1:<\/b> When you unexplode the first dot, you get 10 dots in the\u00a0\"\frac 1{10}\"\u00a0box, which gives one group of six with remainder of 4.<\/p>\n

                          Picture 2: <\/b>When you unexplode those four dots, you get 40 dots in the\u00a0\"\frac 1{100}\"\u00a0box, which gives six group of six with remainder of 4.<\/p>\n

                          Picture 3:<\/b> Unexplode those 4 dots to get 40 in the next box to the right.<\/p>\n

                          Picture 4:<\/b> Make six groups of 6 dots with remainder 4.<\/p>\n

                          Since the remainder repeated (we got a remainder of 4 again), we can see that the process will now repeat forever:<\/p>\n

                            \n
                          • unexplode 4 dots to get 40 in the next box to the right,<\/li>\n
                          • make six groups of 6 dots with remainder 4,<\/li>\n
                          • unexplode 4 dots to get 40 in the next box to the right,<\/li>\n
                          • make six groups of 6 dots with remainder 4,<\/li>\n
                          • and so on forever…<\/li>\n<\/ul>\n

                            On Your Own<\/h3>\n

                            Work on the following exercises on your own or with a partner.<\/p>\n

                              \n
                            1. Use \u201cDots & Boxes\u201d division to compute the decimal representation of \"\frac 1{11}\".\u00a0 Explain how you know for sure the process will repeat forever.<\/li>\n
                            2. Use \u201cDots & Boxes\u201d division to compute the decimal representation of \"\frac 1{12}\".\u00a0 Explain how you know for sure the process will repeat forever.<\/li>\n
                            3. What are the possible remainders<\/i> you can get when you use division to compute the fraction \"\frac 1 7\"?\u00a0 How can you be sure the process will eventually repeat?<\/li>\n
                            4. What are the possible remainders<\/i> you can get when you use division to compute the fraction \"\frac 1 9\"?\u00a0 How can you be sure the process will eventually repeat?<\/li>\n<\/ol>\n

                               <\/p>\n

                              \n

                              Problem 13<\/h3>\n

                              Suppose that\u00a0\"n\"\u00a0is a whole number, and it has some prime factors besides 2\u2019s and 5\u2019s.\u00a0 Write a convincing argument that:<\/p>\n

                                \n
                              1. The decimal representation of\u00a0\"\frac 1 n\"\u00a0will go on forever (it will not terminate).<\/li>\n
                              2. The decimal representation of\u00a0\"\frac 1 n\"\u00a0will be an infinite repeating<\/i> decimal.<\/li>\n
                              3. The period of the decimal representation\u00a0 of\u00a0\"\frac 1 n\" will be less than \"n\".<\/li>\n<\/ol>\n<\/div>\n

                                 <\/p>\n

                                \n

                                Problem 14<\/h3>\n
                                  \n
                                1. Find the \u201cdecimal\u201d expansion for\u00a0\"\frac 1 2\"\u00a0in the following bases.\u00a0 Be sure to show your work:\n

                                    <\/span>   <\/span>\"\[ \text{two}, \quad \text{three}, \quad \text{four},\quad \text{five}, \quad \text{six}, \quad \text{seven}.\]\"<\/p>\n<\/li>\n

                                2. Make a conjecture: If I write the decimal expansion of\u00a0\"\frac 1 2\"\u00a0in base\u00a0\"b\", when will that expansion be finite and when will it be an infinite repeating decimal expansion?<\/li>\n
                                3. Can you prove your conjecture is true?<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":21,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1153","chapter","type-chapter","status-publish","hentry"],"part":1068,"_links":{"self":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1153","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":27,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1153\/revisions"}],"predecessor-version":[{"id":2449,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1153\/revisions\/2449"}],"part":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/parts\/1068"}],"metadata":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1153\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/media?parent=1153"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapter-type?post=1153"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/contributor?post=1153"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/license?post=1153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}