Repeating Decimal

What Is a Repeating Decimal?

A repeating decimal is a decimal number that has a digit or a group of digits that repeat forever in a specific pattern. Unlike terminating decimals, which have a finite number of digits (like $0.25$), repeating decimals continue infinitely. These numbers are also sometimes called recurring decimals.

For example, if you convert the fraction $1/3$ to a decimal, you get $0.33333...$, where the digit $3$ repeats forever. Similarly, the fraction $4/11$ becomes $0.363636...$, where the block of digits $36$ repeats endlessly.

To show that a decimal repeats, we use a special notation called bar notation or a vinculum. We draw a horizontal line or bar over the digit or group of digits that repeats. This repeating part is officially called the repetend.

• For $0.33333...$, we write $0.\bar{3}$.
• For $0.363636...$, we write $0.\overline{36}$.
• For $0.83333...$, we write $0.8\bar{3}$ because only the $3$ repeats.

This simple bar saves us from writing an infinite number of digits and clearly shows the repeating pattern.

How Do You Find a Repeating Decimal?

Repeating decimals don't just appear out of nowhere; they are the result of converting certain fractions into decimals. The key tool for this conversion is long division. Every fraction is a division problem in disguise. The fraction $a/b$ simply means $a÷b$.

When you perform long division, you keep track of the remainders at each step. There are two possible outcomes:

1. The remainder becomes $0$. If this happens, the division ends, and you have a terminating decimal. For example, $1÷4=0.25$.
2. The remainder never becomes $0$. If the remainder never becomes zero, you will eventually get a remainder that you have seen before. This is the magic moment! As soon as a remainder repeats, the sequence of digits in your answer (the quotient) will also start to repeat. This is the signal that you have found a repeating decimal.

For any fraction, the number of possible remainders is limited. When dividing by a number $n$, the only possible remainders are $1,2,3,...,n-1$. Since you can't have an infinite number of different remainders, one of them is bound to show up again, guaranteeing that the decimal will either terminate or repeat.

Step-by-Step: Converting Fractions to Repeating Decimals

Let's walk through the process of converting a fraction to a decimal using long division. The goal is to divide the numerator by the denominator and watch for repeating remainders.

Here are the steps:

1. Set up the long division problem with the numerator inside the division bracket (the dividend) and the denominator outside (the divisor).
2. Place a decimal point after the numerator and add trailing zeros. You can add as many as you need. Place a decimal point in the answer area directly above the one in the dividend.
3. Perform the division as you normally would.
4. After each subtraction step, bring down the next zero.
5. Pay close attention to the remainder after each subtraction. If you get a remainder of $0$, the decimal terminates. If you get a remainder you've already had, the decimal repeats.
6. Once you identify the repeating block of digits, stop dividing and rewrite your answer using bar notation.

Bar Notation: The Secret Code for Repeating Decimals

Using bar notation correctly is crucial for communicating repeating decimals clearly. The rule is simple but strict: the bar goes over the repeating part only. Placing it incorrectly changes the value of the number entirely.

Let's look at the difference. The number $0.1\bar{6}$ means $0.16666...$. The number $0.\overline{16}$ means $0.161616...$. These are two very different numbers! The first one is the decimal for $1/6$, while the second is the decimal for $16/99$.

Here is a table to help you master the placement of the bar.

Always perform enough long division steps to be certain about which digits are part of the repeating cycle before you place the bar.

What's the Difference Between Pure and Mixed Repeating Decimals?

Repeating decimals can be sorted into two categories: pure and mixed. Understanding the difference can help you recognize patterns related to the fractions they come from.

Pure Repeating Decimals

A pure repeating decimal is one where the repeating part starts immediately after the decimal point. There are no 'in-between' digits.

• $0.\bar{3}$ is a pure repeating decimal.
• $0.\overline{12}$ is a pure repeating decimal.
• $0.\overline{617}$ is a pure repeating decimal.

Pure repeating decimals often come from fractions where the denominator (in simplest form) is not divisible by $2$ or $5$. For example, fractions with denominators like $3$, $7$, $9$, $11$, or $13$ will produce pure repeating decimals.

Mixed Repeating Decimals

A mixed repeating decimal has one or more digits after the decimal point that do not repeat, followed by the part that does repeat. The non-repeating part is sometimes called the pre-period.

• $0.1\bar{6}$ is a mixed repeating decimal. The digit $1$ is the non-repeating part.
• $0.58\bar{3}$ is a mixed repeating decimal. The digits $58$ are the non-repeating part.
• $0.91\bar{6}$ is a mixed repeating decimal. The digits $91$ are the non-repeating part.

Mixed repeating decimals come from fractions where the denominator (in simplest form) has a prime factor of $2$ or $5$ and at least one other prime factor. For example, fractions with denominators like $6=2\times 3$ or $12=2\times 2\times 3$ will produce mixed repeating decimals.

How Do You Compare Repeating Decimals?

Comparing repeating decimals might seem tricky because they go on forever. However, the process is very similar to comparing regular decimals. You just need a systematic approach.

Here’s how to compare two repeating decimals:

1. Write them out: Expand both numbers by writing out the repeating pattern for several decimal places. Aim for at least $5$ or $6$ places to see the pattern clearly.
2. Line them up: Write the two expanded decimals one above the other, making sure to align the decimal points perfectly.
3. Compare from left to right: Start with the digit in the tenths place and compare. If they are the same, move to the hundredths place. Continue moving to the right until you find a place where the digits are different.
4. Make the call: The number with the larger digit in the first place they differ is the larger number.

Common Mistakes to Avoid with Repeating Decimals

Working with repeating decimals is straightforward once you get the hang of it, but there are a few common pitfalls to watch out for.

• Stopping the Division Too Soon: A common error is to stop dividing before a remainder repeats, and mistakenly assume the decimal terminates or misidentify the repeating block. Always continue until you see a remainder for the second time.
• Misplacing the Bar Notation: Be precise! Placing the bar over the wrong digits creates a completely different number. For $1/6=0.1666...$, writing $0.\overline{16}$ is incorrect. The correct answer is $0.1\bar{6}$.
• Rounding the Answer: A repeating decimal is an exact value. Rounding it, for example, writing $1/3$ as $0.33$, turns it into an approximation. Unless you are specifically asked to round, always use the bar notation to represent the exact value.
• Calculator Misinterpretations: Standard calculators have limited screen space and will round the last digit. A calculator might show $2/3$ as $0.66666667$. This can hide the repeating pattern. Rely on long division to be certain.

Repeating Decimals: Quick Summary

Here are the most important takeaways about repeating decimals:

• Definition: A repeating decimal is a decimal number with a digit or group of digits that repeats infinitely.
• Origin: They are formed when converting a rational number (a fraction) into a decimal, and the long division process never results in a zero remainder.
• Key Tool: The primary method to find a repeating decimal from a fraction is long division.
• The Signal: The repeating pattern begins when you encounter a remainder that you have seen before during long division.
• Notation: We use a bar (vinculum) over the repeating part (the repetend) to write the number concisely (e.g., $0.5\overline{23}$).
• Types: They can be pure (repeating part starts right after the decimal point) or mixed (a non-repeating part comes first).