Decimal Representation of Rational Numbers

Every real number can be represented in decimal form. For example, 1/3 = .3333 . . . and 1/2 = .5. The decimal representation of many numbers, such as π, have no pattern to the sequence of digits. Rational numbers, those that can be written as the quotient of two whole numbers, either have a terminating decimal representation or have a repeating representation. We will investigate this fact and see how to convert to and from the decimal representation in this assignment. Recall that to denote a series of repeating digits, we draw a bar over the repeating digits. Therefore, .333 . . . = 3 and 3.14242 . . . = 3.142

If you have a rational number written as a fraction, you get the decimal representation by long division. To convert from terminating decimals to fractions is quick if you remember what decimals represent. For example, .7 means 7 tenths, so .7 = 7/10. Also, .63 means 63 hundredths, so .63 = 63/100. Alternatively, if x = .63, then 100x = 63, so x = 63/100.

For repeating decimals, the process is a little more involved, but basically uses the same ideas. Here is an example. Suppose we want to convert .232323 . . . to a fraction. Call this number x. If we multiply x by 100, we get 100x = 23.23 = 23 + 23, so 100x = 23 + x. This gives us an equation we can solve to find x. Subtracting x from both sides gives 99x = 23, so x = 23/99. The reason we multiplied x by 100 is because we had two repeating digits. If we had three repeating digits, we would multiply by 1000.

Problem 1. Convert the following fractions to decimals:

  1. 15/8
  2. 7/11
  3. 2/7

In the following problems it is not necessary to reduce the fractions to lowest terms.

Problem 2. Convert the following decimals to fractions:

  1. .35
  2. 2.461
  3. 3.13432

Problem 3. Convert the following repeating decimals to fractions:

  1. .7
  2. .32
  3. .432
Do you see a pattern in the numerators and denominators of these fractions? Based on the pattern you find, make an educated guess for how to immediately write down a fraction that is equal to a given repeating decimal, when the first repeating digit is immediately to the right of the decimal place. Test your guess by using it to convert .13 and using our procedure to see if you get the same value.

Problem 4. Convert the following repeating decimals to fractions:

  1. 1.5
  2. .23
  3. -3.2178