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Turning Repeating Decimals Into Fractions:

Algorithm and Reasoning

W. Blaine Dowler

June 5, 2010

Abstract

This document details exactly how to convert any repeating decimal

into its fractional form, and explains exactly why the method works.

1 The Algorithm

First, some notation. The overline notation will be used to represent the re-

peating portion of a decimal. For example,

12.343434343434 . . . = 12.34

Now, the algorithm for turning these repeating decimals into fractions is

fairly straightforward. The explanation for it requires math generally left for

High School, which will come later. Elementary school students can learn how

to convert, even if they can't immediately be shown why it works.

When the repeating digits start right at the decimal, such as with 0.3, then

we start by counting the number of digits that repeat. In 0.3, only one digit

(the 3) repeats. In 0.34, two digits repeat (the 3 and 4.) We write the fractions

by writing the repeating digit(s) over a matching number of 9s, remembering to

reduce the fraction into lowest terms afterwards. So, for example,

0.3 = 3 = 1

0.34 = 3

93

0.123456789 = 1

4

99

23456789 = 13717421

999999999 111111111

12 = 4

0.012 = 999 333

Notice in the fourth example that we have three 9s in the denominator

instead of just two. The 0 that repeats counts.

Things get trickier when the repeating digits don't start repeating immedi-

ately after the decimal. For example, if you have 0.166666666 . . . = 0.16, then

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the 1 that appears after the decimal is not involved in the repetitions. We need

to have all repeating digits start repeating right after the decimal for the above

trick to work. If the decimal is followed by one digit that doesn't repeat, then

10 times the number will start repeating immediately:

10 ? 0.16 = 1.6

If you have four non-repeating digits after the decimal, than 10,000 times that

number will start repeating right after the decimal:

10, 000 ? 0.11116 = 1111.6

If we multiply our decimal to turn the number into a decimal we can work

with, convert it into a fraction, and then divide by the multiple of 10 we origi-

nally multiplied by (10 or 10,000 above) then we can convert our original number

into a fraction. We will need to remember to convert any mixed numerals into

improper fractions before we divide.

10 ? 0.16 = 1.6

6

10 ? 0.16 = 1 9

2=5

10 ? 0.16 = 1 3 3

0.16 = 5 ? 10

3

5

0.16 = 30

0.16 = 1

6

Similarly,

10, 000 ? 0.11116 = 1111.6

10, 000 ? 0.11116 = 1111 6

9

10, 000 ? 0.11116 = 1111 2 = 3335

0.11116 = 3335

3

33

3 ? 10, 000

335

0.11116 = 30, 000

667

0.11116 = 6000

With these two skills, we can now convert any repeating decimal into a

fraction.

2

2 The Explanation

We use standard notation for sums. For example,

10

21 + 22 + 23 + . . . + 210 = 2i

i=1

The symbol (a capital sigma from the Greek alphabet) means we will

take the sum of a series of similar terms. Each term can be written in the form

2i, where i ranges from 1 (below the ) to 10 (above the ). The notation

always means taking the number below the symbol to the number above the

symbol in steps of one number at a time. The variables, starting points, and

ending points may change, which is why they are included in the notation.

We start by deriving a general result for one particular type of sum. In most

high school curricula, student learn about geometric sequences and series. A

sequence is a series of numbers that have some sort of connection or relationship.

A geometric sequence is one in which the relationship is defined by a common

ratio r. So, if the sequence is 1, 2, 4, 8, 16, ..., then the common ratio is 2 because

2 ? 1 = 2, 4 ? 2 = 2, 8 ? 4 = 2, 16 ? 8 = 2, .... If the common ratio is less than 1

(such as 1 ,) then the numbers in the sequence get smaller each time.

A seri

2

es is the sum of the terms in a sequence. So, if the first four terms in

a sequence are 1, 2, 4 and 8, then the first four terms in the series would be 1,

3, 7 and 15 because 1 = 1, 1 + 2 = 3, 1 + 2 + 4 = 7, and 1 + 2 + 4 + 8 = 15. We

are almost at the stage to connect these ideas to repeating decimals.

High school students also see a formula that gives the value of any term in

the series if the underlying sequence is understood. Just as some people prefer

to cook with a microwave, others prefer to take the time cook from scratch.

If you're a "cook from the microwave" kind of mathematician, jump ahead to

the finished (and numbered) equation. If not, keep reading and we'll derive the

thing from scratch.

Let's start by finding a way to describe every term in a sequence. We'll need

some notation: the nth term in a sequence is denoted tn, so the first term is

t1, the second term is t2, and so forth. The first term can also be represented

by the letter a, and usually is. The common ratio between consecutive terms

is r as above, which leaves n to denote the number of terms we are into the

sequence. For example, if our sequence is 1, 2, 4, 8, ... as above, then a = 1 and

r = 2 for every term in the sequence. If n = 1, we have tn = 1. If n = 2, then

tn = 2. If n = 3, then tn = 4, and so forth. We need to construct a formula

that gives the correct tn when given a, r, and n.

We start by recognizing that the difference between consecutive terms is a

multiplied factor of r. We need to make sure that tn = a when n = 1, and that

each increment of n by 1 introduces another factor of r so that t2 = ar, t3 = r2,

and so forth. If we recall that r0 = 1 for any value of r, then we can build our

formula by pure logic:

tn = arn-1

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How do you turn decimals into a fraction? Convert Decimals to Fractions. Step 1: Write down the decimal divided by 1, like this: decimal 1. Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.) Step 3: Simplify (or reduce) the fraction.

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