1. Let n be a positive number. a. When we divide a decimal number, n, by 10, how are the numeral and the quotient related?

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1 Black Converting between Fractions and Decimals Unit Number Patterns and Fractions. Let n be a positive number. When we divide a decimal number, n, by 0, how are the numeral and the quotient related?. Let x be a number with a decimal expansion that terminates m digits after the decimal point. Find the smallest power of 0 that we can multiply by x to get an integer product.. For these problems, n represents a natural number. Write as a decimal for each of n =,,, and. n How many digits past the decimal point do we need to express n as a decimal?. For a given natural number n, how many digits past the decimal point do we need to express as a decimal? n 5 5. Find the number of digits past the decimal point needed to express each of the following. 8 e f c. 0 g. 000 d. 50 h c. Find the first three digits after the decimal point in the decimal expansion of / How do we know that the decimal expansion of / repeats endlessly? Find the repeating decimal expansions of the fractions /, /, and 0/. What is the decimal expansion for /? How can we be sure it repeats endlessly? 7. Find the first 9 digits after the decimal point in the decimal expansion of /7 8. Find the decimal expansion of.

2 9. Convert each of the following fractions into decimal numerals. Unit Number Patterns and Fractions 9 e. i. 8 f. 9 j. 7 c. 5 g. 90 k. 08 d. 5 h. 90 l Find the th digit past the decimal point in the decimal expansion of 5. Converting Decimals to Fraction We can convert fractions involving integers into their decimal forms using long division, but the problem of converting repeating decimals to fractions is a bit more difficult. In this section we use the repetition process to convert repeating decimals into fractions.. Notice the similarity between the repeating decimals 0. and 0 0. =.. c. Find the difference between. and 0. Set up an algebraic equation, using x = 0. and 0x =., to solve for 9x Now express x as a fraction in simplest form.. Express.09 as a fraction in reduced form. Express 0.09 as a fraction in reduced form.. Express 0.07 as a fraction in reduced form. Express as a fraction in reduced form.. What s the difference between and 0.9?

3 5. Express each of the following as a fraction in lowest terms. Unit Number Patterns and Fractions 0.5 e f c..7 g d. 0.7 h Solutions. Let s take a look at a few examples of what happens when we divide numbers by 0: Dividing a number by 0 seems to cause the decimal point to shift one digit to the left in the quotient. This makes sense as 0 is the number base in which are working.. The number x has m digits to the right of the decimal, so lox has m - digits to the right of the decimal point. Multiplying by 0 again, 0 x has m - digits to the right of the decimal point. Each time we multiply by 0, we reduce the number of digits to the right of the decimal point by. in order to get rid of all the digits to the right of the decimal point, we multiply x by 0 m to get an integer, 0 m x.. We first take a look at a few examples to help us establish what happens to decimals when we divide them by powers of : n = : 0.5 = n = = digit n = : 0.5 = n = = digits n = : 0.5 = n = 8 = digits n = : = n = 6 = digits

4 Unit Number Patterns and Fractions It appears that we need another digit past the decimal point for each power of by which we divide. To try to confirm this observation, we take a look at the process of dividing by in decimal form. Dividing by means multiplying by 0.5 = Let's take a look at this same process of multiplication in decimal form: n = : n = : n = : n = : 5 = = 5 (0.) n = i digit = = 5 (0.) n = i digits = = 5 (0.) n = i digits = = 65 (0.) n = i digits We already know that multiplying a number by O. (which is the same as dividing it by 0) causes the decimal point to shift one digit to the left. Multiplying an odd number by 5 results in an odd number, so multiplying by 5 in these cases will not result in a new 0 digit. The decimal point still shifts one digit to the left. The result is that for any natural number n, the fraction has exactly n digits after the decimal point. n Note also that the decimal point shifts one digit to the left any time we divide an odd number by : n = :.5 = n = = digit n = : 0.75 = n = = digits n = : 0.75 = n = 8 = digits n = : = n = 6 = digits Make sure you see why this works.. We use the same method as we did in Problem, but reversing the roles of and 5: n = : n = : n = : n = : = = = (0.) n = i digit = = = (0.) n = i digits = = = (0.) n = i digits = = = (0.) n = i digits

5 Unit Number Patterns and Fractions Dividing by 5 means multiplying by 0., moving the decimal point one place over unless the integer we are dividing has at least one power of 5 in its prime factorization. This means that, has exactly n digits to the right of the decimal point. n 5 We extend the method from Problem. For an integer m that is not divisible by 5, m the decimal expansion of 5 n point. For instance, when m = 7 we have n = : n = : n = : also has exactly n digits to the right of its decimal 7 7 = = 7 i = 7 i i (0.) =. n digit = = 7 i = 7 i i (0.) = 0.8 n digits = = 7 i = 7 i i (0.) = n digits When the denominator of a fraction expressed in lowest terms has a prime factorization in the form a 5 b, the number of digits past the decimal point needed to express that fraction is the greater of a and Each answer represents the exponent of the smallest power of 0 that is a multiple of each denominator. c. d. e. f. g. h. The denominator is = 5 0, so this number needs digits to the right of the decimal. The denominator is 5 = 0 5, so this number needs digits to the right of the decimal. The denominator is 0 = 5, so this number needs digits to the right of the decimal. The denominator is 50 = 5, so this number needs digits to the right of the decimal. The denominator is 5 = 0 5, so this number needs digits to the right of the decimal. The denominator is 500 = 5, so this number needs digits to the right of the decimal. The denominator is 000 = 5, so this number needs digits to the right of the decimal. The denominator is 600 = 6 5, so this number needs 6 digits to the right of the decimal. 5

6 Unit Number Patterns and Fractions 6. We begin dividing by and see that the decimal expansion does not terminate one, two, or even three digits past the decimal point. In fact at each step in the division process, we find ourselves dividing into 0. The quotient is, which becomes a new digit of the decimal of /. The remainder is, to which we append a 0 and continue the long division. Since this process never changes, the decimal representation of / repeals endlessly. Follow the division yourself and make sure you understand the repetition of division. 7. Dividing 7 into results in 6 digits before is left as a remainder. Continuing the division process, dividing 7 into again looks exactly as it does for the first 6 digits. This means those first 6 digits repeat endlessly, always leaving a remainder of to restart the pattern. We can also determine the decimal forms of other fractions with a denominator of 7. Here we show a few with the first two blocks of their repeating digits: = = = There is a way for writing repeating decimals that is a little easier than writing out the entire block of repeating digits two or three times to be certain that the repeating block is understood. We draw a line over the entire repeating block of digits to show that those digits repeat: 0. = = 0.6 = = This convention helps us to more easily write repeating decimals within mathematical statements: = Note also that some repeating decimals have non-repeating digits before their repeating blocks of decimals. Here are a couple of examples in which we do not write the bar over the non-repeating decimals: 0.6 =.857. = 6

7 Unit Number Patterns and Fractions 8. We can find the decimal expansion through long division as we do at left. This involves a process of multiplying each remainder by 0 and dividing again. However, multiplying a fraction by powers of 0 can help us convert fractions to decimals more easily. In particular, it helps its in cases where the fraction repeats and the denominator contains factors of or 5. Let's take a look at this second possible method in action. First, we multiply / by powers of 0 until there are no factors of or 5 left in the denominator: Next, we convert the product of 00 and / to decimal form. This should be easy at this point as you should know the fraction to. decimal conversion of / quite well: 00 i = 5 = 8 =8. To finish, divide the decimal expansion by 00 (since you multiplied it by 00 earlier). All you have to do for this is to move the decimal places to the left. Hence the final answer is In some of the following solutions, we multiply in powers of or 5 in order to pair all factors of and 5 into powers of 0 in the denominator in order to make computation simpler. This method can save a great deal of valuable time 7

8 Unit Number Patterns and Fractions 0. First, we find the repeating decimal expansion of 5/: = i = = i = (.578)(0.) = The th digit after the decimal point is the th digit in the 6-digit repeating block Since 6 leaves a remainder of 5, our answer is the 5 th digit in the 6-digit black, which is.. It is at first difficult to grasp how we might convert numbers with repeating decimals into fractions. Focusing on the repeating decimal part gives us a clue. Moving the decimal point over a digit allows us to examine 0. in relation to another number with the same repeating decimal part: a) We can now create a terminating decimal by subtracting these numbers to rid ourselves of the repeating decimal part:. 0. =. b) Motivated by this decimal similarity, we apply algebra to this problem by using a variable to represent 0.: Thus, 0x x = x = c) After dividing by 9 we see that x = /9, so 0. =. 9 Concept: Algebraic methods are useful in developing techniques to solve some arithmetic problems. In Problem we used algebra to take advantage of the self-similarity of a repeating decimal. We assigned a variable to represent the repeating decimal and found a way to express the variable as a fraction, thereby giving us a way to convert a repeating decimal to a fraction. 8

9 Unit Number Patterns and Fractions. Once again we focus on the self-similarity of repeating decimals. First, we let x =.09. Since there are digits in x's repeating block, we compare x to 0 x. We subtract these numbers to get rid of the repeating decimal: 00x x = x = 07 x = # =!!!! Note that.09 = 0 i This 0 to ratio allows us to divide our answer from part (a) by 0 to get our answer:. Apply the ideas from problems and.!!!!! =!!! #. The number 0.9 presents difficulties for many students of mathematics. Hopefully, the following methods will convince you that the only difference between and 0.9 is the way in which we chose to write them. They are in fact equal. First, notice that 0.9 = i 0. = i =. Multiplying integers and fractions makes evaluation of 0.9 easier. Next, we find the value of 0.9 using algebr Let x = 0.9, so x = 0.9 0x = 9.9 Subtracting the first equation from the second we see that 0x x = = 9. Solving 9x = 9, we get x =, so 0.9 =. 9

10 Unit Number Patterns and Fractions If you re still not convinced, try subtracting 0.9 from, one step at a time: As we subtract out more and more of the decimal expansion of 0.9 from, we find that the difference between and 0.9 gets continually closer to 0. This process never ends and the difference diminishes to a number smaller than any positive number you can think of, so it must be 0. Think about it this way: if there is no number between and 0.9, then they must be the same number! Finally, we write 0.9 as an infinite geometric series and find its sum: 5. Think about each of these arguments until you are convinced that 0.9 = (a) Let! = 0. 5 So 0!! = And 9! = 5 Thus! =!! (b) Let! = So 0!! = And 9! = 0.05 (which is the same as! " Thus! =!! =! # (c) Let! =. 7 So 00!! = And 99! = 6 Thus! = #!!!! ) 0

11 Unit Number Patterns and Fractions (d) Note that.7 = 0 i 0.7. This 0 to ratio allows us to divide our answer from (c) by 0 to get our answer: 0.7 =!. So 0.7 =!! Thus! =! (e) Let! =. 05 So 000!! = And 999! = 050 Thus! = #"!!!!! (f) Note that. 05 is 0 times bigger than This 0 to ratio allows us to divide our answer from (e) by 0 to get our answer: =!.# So = # Thus! = (g) Let! = So 00!! = And 99! = 76 Thus! =!! (h) We can use the result from part (g) to help us solve this problem. Let! = 0.76 We know 0.76 = This also means Thus! = # #! +!. =!! #!! =!! = #!

12 Unit Number Patterns and Fractions Bibliography Information Teachers attempted to cite the sources for the problems included in this problem set. In some cases, sources may not have been known. Problems Bibliography Information -5 Crawford, Matthew. The Art of Problem Solving: Introduction to Number Theory