1 / 21

Decimal Expansion of Fractions Brent Murphy

Decimal Expansion of Fractions Brent Murphy. P Q. Problem: Under What conditions will the decimal expansion of p/q terminate? Under what conditions will it repeat? p/q can be investigated as p*(1/q). Terminating.

harriet-raymond
Télécharger la présentation

Decimal Expansion of Fractions Brent Murphy

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Decimal Expansion of FractionsBrent Murphy

  2. PQ Problem: Under What conditions will the decimal expansion of p/q terminate? Under what conditions will it repeat? p/q can be investigated as p*(1/q).

  3. Terminating • When placing 1 over q as a fraction the decimal expansion will terminate if it can be factored down to a prime number < 5, unless q is also a multiple of 3 and a prime number > 5 is not a factor.

  4. Examples of Terminating decimal expansion • ½ and 1/5 terminate. • 1/10 will terminate because 10 can be factored down to 2 and 5 and it is not a multiple of 3. • 1/16 terminates. 16 can be factored down to four 2’s, and is not a multiple of 3. • 1/25 terminates. 25 can be factored down to 5 and 5 and is not a multiple of 3.

  5. Repeating Decimal Expansion • If q is a multiple of 3 it will repeat. • If q can be factored down to a prime number > 5 then it will repeat.

  6. More examples of repeating Decimals • 1/29 repeats. 29 is a prime number > 5. • 1/35 repeats. 35 can be factored to 5 and 7. 7 is a prime number greater than 5, so it must repeat the decimal when placed under a number as a fraction. • 1/14 repeats. It can be factored to a 2 and 7. 7 is prime and > 5 so 1/14 repeats.

  7. Examples of repeating Decimals • 1/3 repeats. • 1/9 repeats. 9 is a multiple of 3. • 1/15 repeats. 15 can be factored to 3 and 5. The fact that it is a factor of 5 (one of the first 3 prime numbers) would cause it to terminate if it were not a multiple of 3. 15 is a multiple of 3 so it repeats.

  8. Problem 2 • Suppose you are given a decimal expansion of a fraction. How can you represent that as p/q? • For Terminating decimals • For Repeating decimals

  9. Terminating Decimals • Terminating decimals are a little easier than repeating decimals. If you make the decimal a whole number by multiply it by 10^x, where x is the number of decimal places needed to move to make the decimal a whole number. Put that number over 10^x and reduce.

  10. Examples of terminating decimals to fractions Example 1: • N= 3.74 • N*10^2 = 374 • 374/10^2= 374/100 = 187/50 Example 2: N = 4.169 N*10^3 = 4,169 N/10^3= 4,169/1000

  11. Repeating Decimals • Repeating decimals are a little harder and require a little more logic. N= a repeating decimal number. Declare a variable (M) that M= N*10^x, where x is the number of decimal places needed to move the decimal to where it begins to repeat. • Note: This step may not be needed if the decimal begins to repeat after immediately. In this case, assume M=N

  12. Repeating decimalsStep 2 • Next, multiple M by 10^y, where y is the number terms in the geometric sequence before it begins to repeat once more. Then, setup an equation where 10^yM = the number M multiplied by 10^y.

  13. Repeating DecimalsStep 3 • Subtract M from both sides leaves the right side of the equation as a whole number. Divide that whole number by 10^yM – M and it will create a fraction. Remember, that M = N*10^y. You are looking for N, not M so you must setup and equation for N*10^y and then divide the fraction found for M by N*10^y and it will give you the fraction for N. Reduce and Enjoy!!!

  14. Examples of repeating decimals • N = 3.135135 • N = M • M*10^3 = 3,135.135135 • 1000M – M = 3,132 • 999M = 3,132 • M = 3,132/999 • M = 116/37 M= N so N = 116/37

  15. Repeating Decimal Example • N = 5.163333 • M = 516.3333 = 100N • 10M = 5,163.3333 • 10M – M = 4,647 • 9M = 4,647 • M = 4647/9 • M = 1549/3 • 100N = 1549/3 • N = 1549/300

  16. Problem 3 • Express as rationals: • A) 13.201… • B) .27… • C) .23… • D)4.163333… • Show that .99…. Represents 1.0

  17. A • N = 13.201… • 1000N = 13,201.201 • 1000N – N = 13,201.201 – N • 999N = 13,188 • N = 13,188/999 • N = 4,396/333

  18. B • N = .27… • 100N = 27.27 • 100N – N = 27 • 99N = 27 • N = 27/99 • N = 3/11

  19. C • N = .23 • 100N = 23.23 • 100N – N = 23 • 99N = 23 • 23/99 = N

  20. D • N = 4.16333… • M= 100N • M = 416.33 • 10M = 4,163.33 • 10M – M = 3,747 • 9M = 3,747 • M = 3,747/9 = 1249/3 • 100N = 1249/3 • N = 1249/300

  21. Show that .9999 represents 1.0 • N = .999… • 10N = 9.999 • 10N – N = 9 • 9N = 9 • N = 1.0

More Related