Skip to main content

A University-Level Understanding of Number with an eye towards elementary education

Shannon Ezzat

Section 6.3 Terminating and Repeating Decimals

In this section, our goal is to understand the relationship between rational numbers and their decimal representations. Specifically, we aim to show that a rational number is equivalent to a terminating decimal if and only if it is equivalent to a decimal fraction. Conversely, a rational number is equivalent to a repeating decimal if and only if it is not equivalent to a decimal fraction. This understanding is crucial for elementary school mathematics teachers as it forms the basis for many mathematical operations and concepts.
🔗

Subsection 6.3.1 Terminating Decimals

A terminating decimal is a decimal number that has a finite number of digits. In other words, it is a decimal that ends or "terminates". It can be expressed as a fraction where the denominator is a power of 10.
🔗

Example 6.3.1. Examples of Terminating Decimals.

Here are some examples of fractions that are equivalent to terminating decimals:
  1. 1/2 = 0.5: Here, we can say that 1/2 is equivalent to the fraction 5/10.
    🔗
  2. 3/4 = 0.75: When we divide 3 by 4, we get 0.75, which is a terminating decimal.
    🔗
  3. 7/8 = 0.875: Dividing 7 by 8 gives us 0.875, another terminating decimal.
    🔗
  4. 3/15 = 0.2: Even though 15 is not a power of 10, it is equivalent to a fraction that is; the fraction 3/15 simplifies to 1/5, and this is equivalent to the decimal fraction 20/100, and thus a terminating decimal.
    🔗
🔗
And here are examples of how to write a terminating decimal as a fraction:
  1. 0.5 = 5/10 = 1/2: Here, we express 0.5 as the fraction 5/10, which simplifies to 1/2.
    🔗
  2. 0.75 = 75/100 = 3/4: We can express 0.75 as the fraction 75/100, which simplifies to 3/4.
    🔗
  3. 0.875 = 875/1000 = 7/8: The decimal 0.875 can be expressed as the fraction 875/1000, which simplifies to 7/8.
    🔗
  4. 0.2 = 2/10 = 1/5: The decimal 0.2 can be expressed as the fraction 2/10, which simplifies to 1/5.
    🔗
🔗
🔗
Now, let’s prove that a rational number is equivalent to a terminating decimal* if and only if it is equivalent to a decimal fraction.
🔗
*Note: this is not quite correct, as we’ll see later that decimals that have repeating 9s to the right are equivalent to terminating decimals. We will discuss this in more depth later.
🔗

Proof.

Since this is an "if and only if" statement in the form \(P \leftrightarrow Q\text{,}\) we have to prove two results, \(P \rightarrow Q\) and \(Q \rightarrow P\text{.}\) Let’s denote a rational number as a/b, where a and b are integers and b ≠ 0. We want to show that a/b is equivalent to a terminating decimal if and only if it is equivalent to a decimal fraction.
🔗
Suppose a/b is equivalent to a terminating decimal. Then, by definition of a terminating decimal, a/b can be written as a decimal with a finite number of digits. This means that a/b can be expressed as a decimal fraction, because a decimal fraction is a fraction where the denominator is a power of 10, and any terminating decimal can be expressed in this form.
🔗
Conversely, suppose a/b is equivalent to a decimal fraction. Then a/b can be expressed as a decimal with a finite number of digits, because a decimal fraction is a fraction where the denominator is a power of 10, and any fraction with a denominator that is a power of 10 can be expressed as a terminating decimal. Therefore, a/b is equivalent to a terminating decimal.
🔗
Putting both results together, a rational number is equivalent to a terminating decimal if and only if it is equivalent to a decimal fraction.
🔗
🔗
Let’s look at a non-example; 7/15. Since \(15=3 \times 5\) and 3 and 7 share no common factors, there is no way of "cancelling" the factor of 3 in the denominator. Thus there is nothing we can multiply both the numerator and denominator by so that both are still integers and the denominator does not have a factor of 3, and thus it isn’t equivalent to a decimal fraction, which has denominators of the form \(10^k=2^k5^k\text{.}\)
🔗
🔗

Subsection 6.3.2 Expressing a Fraction as a Repeating Decimal Using Long Division

If a fraction is not equivalent to a decimal fraction, it can be expressed as a repeating decimal. One way to do this is by using long division. During this process, we may notice that we "run out" of non-zero remainders before the remainders must repeat, indicating a repeating decimal.
🔗
Why is this the case? Well, imagine you are dividing \(a\) by some non-zero integer \(b\text{.}\) There are only finitely many possible remainders: \(\{0,1,2, \ldots, b-2, b-1 \}\text{.}\)
🔗
Now, we can convert a fraction a/b to its decimal representation by long division. If we do get a reminder of 0, then we know that the decimal is a terminating decimal. So imagine that a/b cannot be expressed as a terminating decimal. Then we will never see the remainder 0, and thus we only have \(b-1\) possible remainders when performing each partial division in the long division algorithm. Note that we can add as many zeros to the right (after the decimal and after all non-zero decimal digits); e.g. 20.201 = 20.2010000... Let’s call these the trailing zeroes.
🔗
Now when we get to using the place values of the trailing zeros, every time we "bring down" a digit from a new place value, that digit will be 0. If, at this point, we keep track of remainders, we note that we have at most \(b-1\) unique remainders before one must repeat. At the point of repetition, our "previous work" will repeat over and over again, since we will only end up in situations we have seen before and thus the digits of decimal representation of our fraction between the repeated remainders will repeat over and over again to the right. Thus, we know that if fraction is not equivalent to a terminating decimal, it must be a repeating decimal.
🔗

Example 6.3.2. Example: Expressing 2/7 as a Repeating Decimal Using Long Division.

Let’s express the fraction 2/7 as a repeating decimal using long division.
  1. Divide 2 by 7. Since 2 is less than 7, we add a decimal point and a zero to the right of 2, making it 20.
    🔗
  2. Now, divide 20 by 7. The result is 2 with a remainder of 6.
    🔗
  3. Add another zero to the right of the remainder, making it 60.
    🔗
  4. Divide 60 by 7. The result is 8 with a remainder of 4.
    🔗
  5. Repeat this process. Each time, we add a zero to the right of the remainder and divide by 7.
    🔗
  6. Eventually, we notice that the remainders start to repeat (and must repeat, since there are only a finite number of remainders), indicating a repeating decimal.
    🔗
So, 2/7 = 0.\(\overline{285714}\text{.}\)
🔗
🔗

Example 6.3.3. Example: Expressing 6/13 as a Repeating Decimal Using Long Division.

Let’s express the fraction 6/13 as a repeating decimal using long division.
  1. Divide 6 by 13. Since 6 is less than 13, we add a decimal point and a zero to the right of 6, making it 60.
    🔗
  2. Now, divide 60 by 13. The result is 4 with a remainder of 8.
    🔗
  3. Add another zero to the right of the remainder, making it 80.
    🔗
  4. Divide 80 by 13. The result is 6 with a remainder of 2.
    🔗
  5. Repeat this process. Each time, we add a zero to the right of the remainder and divide by 13.
    🔗
  6. Eventually, we notice that the remainders start to repeat, indicating a repeating decimal.
    🔗
So, 6/13 = 0.\(\overline{461538}\text{.}\)
🔗
🔗
🔗

Subsection 6.3.3 Representing All Repeating Decimals as Rational Numbers

In this subsection, we will show that all repeating decimals can be represented by rational numbers. We will start by demonstrating that 0.9999... = 1 using decimal and fraction notation. Then, we will define the repetend and period of a repeated decimal and provide examples.
🔗

Subsubsection 6.3.3.1 Can Repeating Decimals Equal 1?

Let’s consider the sum of 1/3 + 1/3 + 1/3 in decimal notation and fraction notation.
🔗
In decimal notation, 1/3 is equivalent to 0.333..., so the sum is 0.333... + 0.333... + 0.333... = 0.999.... ADD PIC OF THIS ADDITION
🔗
In fraction notation, the sum is 1/3 + 1/3 + 1/3 = 1.
🔗
Therefore, 0.999... = 1.
🔗
Another way to look at this is to ask yourself "if \(0.\overline{9} < 1\text{,}\) what would I add to \(0.\overline{9}\) to make it equal 1? Any positive number that you think of adding would be too big!
🔗
🔗

Subsubsection 6.3.3.2 Repetend and Period of a Repeated Decimal

The repetend of a repeating decimal is the first instance of the shortest group of digits that repeat indefinitely. It’s important to note that the repetend is the smallest repeating unit in the decimal expansion. The period of a repeating decimal is the number of digits in the repetend.
🔗
Example 6.3.4. Examples of Non-Repetends.
Here are some examples of non-repetends in repeating decimals:
  1. In the repeating decimal 0.\(\overline{142857}\text{,}\) the number 142857 is the repetend. The number 142 is not a repetend because it is not the shortest group of repeating digits.
    🔗
    🔗
  2. In the repeating decimal 0.\(\overline{27}=0.272727...\text{,}\) the number 27 is the repetend. The number 2727 is not the repetend because it is not the shortest group of repeating digits. The number 72 is not the repetend since it is not the first instance of repeating digits. So we would not write \(0.\overline{2727}\) or \(0.2\overline{72}.\)
    🔗
    🔗
🔗
🔗
Example 6.3.5. Examples of Repetend and Period.
Here are two examples of identifying the repetend and period of a repeating decimal:
  1. In the repeating decimal 0.\(\overline{3}\text{,}\) the repetend is 3 and the period is 1.
    🔗
    🔗
  2. In the repeating decimal 0.00\(\overline{142857}\text{,}\) the repetend starts in the thousandths place and is 142857, and the period is 6.
    🔗
    🔗
>
🔗
🔗
🔗

Subsubsection 6.3.3.3 Converting a Repeating Decimal to a Fraction

To convert a repeating decimal to a fraction, we can use the following algorithm:
  1. Let x be the number in decimal form.
    🔗
  2. Let p be the period of the repetend.
    🔗
  3. Calculate (10^p)x - x in repeated decimal form.
    🔗
  4. Solve for x.
    🔗
  5. Convert any numerators and denominators to integers by multiplying both by the appropriate power of 10 to clear decimal places.
    🔗
  6. Simplify by converting the fraction to lowest terms.
    🔗
🔗
Example 6.3.6. Example: Converting a Repeating Decimal to a Fraction.
Let’s convert the repeating decimal 0.\(\overline{3}\) to a fraction using the above algorithm:
  1. Let x = 0.\(\overline{3}\text{.}\)
    🔗
  2. The period of the repetend is 1, so p = 1.
    🔗
  3. Calculate (10^p)x - x = 10x - x = 9x. In decimal form, this is 3.333... - 0.333... = 3.
    🔗
  4. Solve for x: x = 3/9. This is the equation we get when we equate the decimal form and the fraction form.
    🔗
  5. Convert the numerator and denominator to integers by multiplying both by 1 (no need to clear decimal places in this case).
    🔗
  6. Simplify by converting the fraction to lowest terms: x = 1/3.
    🔗
So, the repeating decimal 0.\(\overline{3}\) is equal to the fraction 1/3.
🔗
Let’s convert another repeating decimal, 0.\(\overline{27}\text{,}\) to a fraction:
  1. Let x = 0.\(\overline{27}\text{.}\)
    🔗
  2. The period of the repetend is 2, so p = 2.
    🔗
  3. Calculate (10^p)x - x = 100x - x = 99x. In decimal form, this is 27.2727... - 0.2727... = 27.
    🔗
  4. Solve for x: x = 27/99.
    🔗
  5. Convert the numerator and denominator to integers by multiplying both by 1 (no need to clear decimal places in this case).
    🔗
  6. Simplify by converting the fraction to lowest terms: x = 3/11.
    🔗
So, the repeating decimal 0.\(\overline{27}\) is equal to the fraction 3/11.
🔗
🔗
Example 6.3.7. More Examples of Converting Repeating Decimals to Fractions.
Let’s convert more repeating decimals to fractions using the algorithm we discussed earlier:
🔗
  1. Convert 0.5 to a fraction:
    1. Let x = 0.5. Notice that this is a terminating decimal, not a repeating decimal. However, we can still use our method to convert it to a fraction.
      🔗
    2. Since there’s no repetend, we can say p = 0.
      🔗
    3. Calculate (10^p)x - x = x - x = 0. In decimal form, this is 0.5 - 0.5 = 0.
      🔗
    4. Solve for x: x = 0/1.
      🔗
    5. Convert the numerator and denominator to integers by multiplying both by 1 (no need to clear decimal places in this case).
      🔗
    6. Simplify by converting the fraction to lowest terms: x = 0.
      🔗
    So, the decimal 0.5 is equal to the fraction 1/2.
    🔗
    🔗
  2. Convert 0.01\(\overline{4}\) to a fraction:
    1. Let x = 0.01\(\overline{4}\text{.}\)
      🔗
    2. The period of the repetend is 1, so p = 1.
      🔗
    3. Calculate (10^p)x - x = 10x - x = 9x. In decimal form, this is 0.014444... - 0.0144 = 0.0004.
      🔗
    4. Solve for x: x = 0.0004/9.
      🔗
    5. Convert the numerator and denominator to integers by multiplying both by 10000 (to clear four decimal places): x = 4/90000.
      🔗
    6. Simplify by converting the fraction to lowest terms: x = 1/22500.
      🔗
    So, the repeating decimal 0.01\(\overline{4}\) is equal to the fraction 1/22500.
    🔗
    🔗
  3. Convert 1.\(\overline{6}\) to a fraction:
    1. Let x = 1.\(\overline{6}\text{.}\) This is an improper fraction because the decimal is greater than 1.
      🔗
    2. The period of the repetend is 1, so p = 1.
      🔗
    3. Calculate (10^p)x - x = 10x - x = 9x. In decimal form, this is 16.666... - 1.666... = 15.
      🔗
    4. Solve for x: x = 15/9.
      🔗
    5. Convert the numerator and denominator to integers by multiplying both by 1 (no need to clear decimal places in this case).
      🔗
    6. Simplify by converting the fraction to lowest terms: x = 5/3.
      🔗
    So, the repeating decimal 1.\(\overline{6}\) is equal to the fraction 5/3.
    🔗
    🔗
  4. Convert -0.\(\overline{8}\) to a fraction:
    1. Let x = -0.\(\overline{8}\text{.}\) This is a negative number.
      🔗
    2. The period of the repetend is 1, so p = 1.
      🔗
    3. Calculate (10^p)x - x = 10x - x = 9x. In decimal form, this is -8.888... - (-0.888...) = -8.
      🔗
    4. Solve for x: x = -8/9.
      🔗
    5. Convert the numerator and denominator to integers by multiplying both by 1 (no need to clear decimal places in this case).
      🔗
    6. Simplify by converting the fraction to lowest terms: x = -8/9.
      🔗
    So, the repeating decimal -0.\(\overline{8}\) is equal to the fraction -8/9.
    🔗
    🔗
🔗
🔗
A natural question is if a number has two or more different ways of expressing it using decimals. Let’s explore this in the next example.
🔗
Example 6.3.8.
  1. Consider the decimal number \(x=0.24\overline{9}\text{.}\) Using the procedure above, determine the fraction represented by this decimal. Do you know of another way of expressing this fraction using decimals?
    🔗
    🔗
  2. Using the example above as a guide, conjecture a second decimal expression for the fraction \(\frac{3}{10}\text{.}\) Then check that this is correct.
    🔗
    🔗
  3. Using the ideas above, make a conjecture about when two decimal expressions represent the same fraction. Do your best to prove/justify why this is the case.
    🔗
    🔗
🔗
🔗
🔗
🔗
🔗
PrevTopNext