NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 8•7 Lesson 10: Converting Repeating Decimals to Fractions Student Outcomes . Students develop a convincing argument establishing that every real number with a repeating decimal is a rational number. Classwork Discussion (4 minutes) . In Lesson 8, we say that every rational number, that is, every fraction, has a decimal expansion that falls into a repeating pattern. A natural question now is the converse: If a real number has an infinitely long decimal expansion with a repeating pattern, must it be a rational number? . We begin by observing the effect of multiplying decimals by powers of 10. Consider, for example, the finite decimal 1.2345678. If we multiply it by 105, we get the following: 2 3 4 5 6 7 8 105 × 1.2345678 = 105 × (1 + + + + + + + ) 10 100 1000 10000 100000 1000000 10000000 7 8 = 100000 + 20000 + 3000 + 400 + 50 + 6 + + 10 100 = 123 456.78 This example illustrates how to think through such products. Example 1 (10 minutes) Example 1 There is a fraction with an infinite decimal expansion of ퟎ. ퟖퟏ̅̅̅̅. Find the fraction. We want to find the fraction that is equal to the infinite decimal 0. 81̅̅̅̅. Why might we want to write an infinite decimal as a fraction? Maybe we want to use 0. 81̅̅̅̅ in some calculation. It is unclear how to do arithmetic with infinitely long decimals. But if we recognize the decimal as a fraction, then we can do the arithmetic with the fraction. Let’s start by giving the decimal a name. Let 푥 = 0. 81̅̅̅̅ = 0.8181818181…. Any thoughts on what we might do to this number 푥? (Of course, our previous discussion was probably a hint!) Allow students time to work in pairs or small groups to attempt to find the fraction equal to 0. 81̅̅̅̅. Students should MP.1 guess that multiplying 푥 by some powers of 10 might yield something informative. Lesson 10: Converting Repeating Decimals to Fractions 125 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 8•7 . Let’s try multiplying 푥 = 0. 81̅̅̅̅ by some powers of 10. 푥 = 0.8181818181… 10푥 = 8.1818181… 100푥 = 81.81818181… 1000푥 = 818.1818181… 8 1 8 1 (Perhaps have students write 푥 as + + + + ⋯ to help with this process.) 10 100 1000 10000 Ask students to pause over the expression 100푥. Can they observe anything interesting about it? . We see MP.7 100푥 = 81.81818181… = 81 + 0.818181… = 81 + 푥. This now gives an equation for 푥 students can solve. 100푥 = 81 + 푥 100푥 − 푥 = 81 + 푥 − 푥 (100 − 1)푥 = 81 99푥 = 81 99푥 81 = 99 99 81 푥 = 99 9 푥 = 11 9 . Therefore, the repeating decimal 0. 81̅̅̅̅ = . 11 Have students use calculators to verify that this is correct. Exercises 1–2 (5 minutes) Students complete Exercises 1–2 in pairs. Allow them to use calculators to check their work. Exercises 1–2 1. There is a fraction with an infinite decimal expansion of ퟎ. ퟏퟐퟑ̅̅̅̅̅̅. Let 풙 = ퟎ. ퟏퟐퟑ̅̅̅̅̅̅. a. Explain why looking at ퟏퟎퟎퟎ풙 helps us find the fractional representation of 풙. We have 풙 = ퟎ. ퟏퟐퟑퟏퟐퟑퟏퟐퟑ…, and we see that ퟏퟎퟎퟎ풙 = ퟏퟐퟑ. ퟏퟐퟑퟏퟐퟑퟏퟐퟑ…. This is the same as ퟏퟐퟑ + ퟎ. ퟏퟐퟑퟏퟐퟑퟏퟐퟑ…, which is ퟏퟐퟑ + 풙. So we have the equation ퟏퟎퟎퟎ풙 = ퟏퟐퟑ + 풙, which we can use to solve for 풙. Lesson 10: Converting Repeating Decimals to Fractions 126 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 8•7 b. What is 풙 as a fraction? ퟏퟎퟎퟎ풙 − 풙 = ퟏퟐퟑ + 풙 − 풙 ퟗퟗퟗ풙 = ퟏퟐퟑ ퟗퟗퟗ풙 ퟏퟐퟑ = ퟗퟗퟗ ퟗퟗퟗ ퟏퟐퟑ 풙 = ퟗퟗퟗ ퟒퟏ 풙 = ퟑퟑퟑ c. Is your answer reasonable? Check your answer using a calculator. Yes, my answer is reasonable and correct. It is reasonable because the denominator cannot be expressed as a product of ퟐ’s and ퟓ’s; therefore, I know that the fraction must represent an infinite decimal. It is also ퟒퟏ reasonable because the decimal value is closer to ퟎ than to ퟎ. ퟓ, and the fraction is also closer to ퟎ than ퟑퟑퟑ ퟏ ퟒퟏ to . It is correct because the division of using a calculator is ퟎ. ퟏퟐퟑퟏퟐퟑ…. ퟐ ퟑퟑퟑ 2. There is a fraction with a decimal expansion of ퟎ. ퟒ̅. Find the fraction, and check your answer using a calculator. Let 풙 = ퟎ. ퟒ̅ 풙 = ퟎ. ퟒ̅ ퟏퟎ풙 = (ퟏퟎ)ퟎ. ퟒ̅ ퟏퟎ풙 = ퟒ. ퟒ̅ ퟏퟎ풙 = ퟒ + 풙 ퟏퟎ풙 − 풙 = ퟒ + 풙 − 풙 ퟗ풙 = ퟒ ퟗ풙 ퟒ = ퟗ ퟗ ퟒ 풙 = ퟗ Example 2 (10 minutes) Example 2 Could it be that ퟐ. ퟏퟑퟖ̅ is also a fraction? . We want to see if there is a fraction that is equal to the infinite decimal 2.138̅. Notice that this time there is just one digit that repeats, but it is three places to the right of the decimal point. Let’s multiply 푥 = 2.138̅ by various powers of 10 and see if any of the results seem helpful. 푥 = 2.138888… 10푥 = 21.38888… 100푥 = 213.8888… 1000푥 = 2138.888… . Do any of these seem helpful? Students might not have any direct thoughts in response to this. Lesson 10: Converting Repeating Decimals to Fractions 127 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 8•7 . What if I asked as a separate question: Is 0.8888… the decimal expansion of a fraction? If knowing that 0.888… is a fraction, would any one of the equations we have then be of use to us? 푎 푎 If we know that 0.888… = , then we would see that 100푥 = 213 + 0.888… = 213 + . We could 푏 푏 work out what 푥 is from that. Okay. As a side problem: Is 0.8888… the decimal expansion of some fraction? Let 푦 = 0. 8̅. 푦 = 0. 8̅ 10푦 = 8. 8̅ 10푦 = 8 + 0. 8̅ 10푦 = 8 + 푦 10푦 − 푦 = 8 + 푦 − 푦 9푦 = 8 9푦 8 = 9 9 8 푦 = 9 8 . Now that we know that 0. 8̅ = , we will go back to our original problem. 9 100푥 = 213 + 0. 8̅ 8 100푥 = 213 + 9 213 ⋅ 9 8 100푥 = + 9 9 213 ⋅ 9 + 8 100푥 = 9 1925 100푥 = 9 1 1 1925 (100푥) = ( ) 100 100 9 1925 푥 = 900 77 푥 = 36 . We can see that this technique applies to any infinite repeating decimal, even if there is a delay before the repeat begins, to show that every real number that has a repeating decimal expansion is, for sure, a rational number, that is, can be expressed as a fraction. And, conversely, we saw in Lesson 8 that every rational number has a repeating decimal expansion. So we have proven that the set of real numbers with repeating decimal expansions precisely matches the set of all rational numbers. Any number that has an infinitely long decimal expansion with no repeating pattern cannot be rational; that is, it must be an irrational number. Lesson 10: Converting Repeating Decimals to Fractions 128 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 8•7 Exercises 3–4 (6 minutes) Students complete Exercises 3–4 independently or in pairs. Allow students to use calculators to check their work. Exercises 3–4 3. Find the fraction equal to ퟏ. ퟔퟐퟑ̅̅̅̅. Check your answer using a calculator. Let 풙 = ퟏ. ퟔퟐퟑ̅̅̅̅ Let 풚 = ퟎ. ퟐퟑ̅̅̅̅ ퟏퟎ풙 = ퟏퟔ. ퟐퟑ̅̅̅̅ ퟏퟎ풙 = ퟏퟔ + 풚 풙 = ퟏ. ퟔퟐퟑ̅̅̅̅ 풚 = ퟎ. ퟐퟑ̅̅̅̅ ퟐퟑ ̅̅̅̅ ퟐ ퟐ ̅̅̅̅ ퟏퟎ풙 = ퟏퟔ + ퟏퟎ풙 = ퟏퟎ(ퟏ. ퟔퟐퟑ) ퟏퟎ 풚 = ퟏퟎ (ퟎ. ퟐퟑ) ퟗퟗ ퟏퟎ풙 = ퟏퟔ. ퟐퟑ̅̅̅̅ ퟏퟎퟎ풚 = ퟐퟑ. ퟐퟑ̅̅̅̅ ퟏퟔ ⋅ ퟗퟗ ퟐퟑ ퟏퟎ풙 = + ퟏퟎퟎ풚 = ퟐퟑ + 풚 ퟗퟗ ퟗퟗ ퟏퟔ ⋅ ퟗퟗ + ퟐퟑ ퟏퟎퟎ풚 − 풚 = ퟐퟑ + 풚 − 풚 ퟏퟎ풙 = ퟗퟗ ퟗퟗ풚 = ퟐퟑ ퟏퟔퟎퟕ ퟗퟗ풚 ퟐퟑ ퟏퟎ풙 = = ퟗퟗ ퟗퟗ ퟗퟗ ퟏ ퟏ ퟏퟔퟎퟕ (ퟏퟎ풙) = ( ) ퟐퟑ ퟏퟎ ퟏퟎ ퟗퟗ 풚 = ퟗퟗ ퟏퟔퟎퟕ 풙 = ퟗퟗퟎ ퟏퟔퟎퟕ ퟏ. ퟔퟐퟑ̅̅̅̅ = ퟗퟗퟎ 4. Find the fraction equal to ퟐ. ퟗퟔퟎ̅̅̅̅. Check your answer using a calculator. Let 풙 = ퟐ. ퟗퟔퟎ̅̅̅̅ Let 풚 = ퟎ. ퟔퟎ̅̅̅̅ ퟏퟎ풙 = ퟐퟗ. ퟔퟎ̅̅̅̅ 풙 = ퟐ. ퟗퟔퟎ̅̅̅̅ ퟏퟎ풙 = ퟐퟗ + 풚 풚 = ퟎ. ퟔퟎ̅̅̅̅ ̅̅̅̅ ퟐퟎ ퟏퟎ풙 = ퟏퟎ(ퟐ. ퟗퟔퟎ) ퟐ ퟐ ̅̅̅̅ ퟏퟎ풙 = ퟐퟗ + ퟏퟎ 풚 = ퟏퟎ (ퟎ. ퟔퟎ) ퟑퟑ ퟏퟎ풙 = ퟐퟗ. ퟔퟎ̅̅̅̅ ퟏퟎퟎ풚 = ퟔퟎ. ퟔퟎ̅̅̅̅ ퟐퟗ ⋅ ퟑퟑ ퟐퟎ ퟏퟎ풙 = + ퟏퟎퟎ풚 = ퟔퟎ + 풚 ퟑퟑ ퟑퟑ ퟐퟗ ⋅ ퟑퟑ + ퟐퟎ ퟏퟎퟎ풚 − 풚 = ퟔퟎ + 풚 − 풚 ퟏퟎ풙 = ퟑퟑ ퟗퟗ풚 = ퟔퟎ ퟗퟕퟕ ퟏퟎ풙 = ퟗퟗ풚 ퟔퟎ ퟑퟑ = ퟗퟗ ퟗퟗ ퟏ ퟏ ퟗퟕퟕ ퟔퟎ (ퟏퟎ풙) = ( ) 풚 = ퟏퟎ ퟏퟎ ퟑퟑ ퟗퟗ ퟗퟕퟕ 풙 = ퟐퟎ ퟑퟑퟎ 풚 = ퟑퟑ ퟗퟕퟕ ퟐ. ퟗퟔퟎ̅̅̅̅ = ퟑퟑퟎ Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson. We know that every decimal that has a repeating pattern is a rational number. We know how to find the fraction that has a given repeating decimal expansion.