Recurring Decimals

Recurring Decimals are decimals that never end but instead repeat a specific digit or group of digits forever. This repetition is periodic, meaning the same sequence appears over and over in the same order.

Every recurring decimal is a rational number, which means it can always be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. This distinguishes them from irrational numbers, like $\pi$ or $\sqrt{2}$, which have non-repeating, infinite decimal expansions.

A terminating decimal is the counterpart to a recurring decimal; it is a decimal that has a finite number of digits (e.g., $0.25$). Both terminating and recurring decimals represent rational numbers.

To avoid writing an infinite string of digits, mathematicians use dot notation to indicate the repeating part. A single dot placed directly above a digit indicates that only that specific digit repeats infinitely (e.g., $0.\dot{5} = 0.555...$).

When a sequence of multiple digits repeats, dots are placed over the first and last digits of the repeating cycle. For example, $0.\dot{1}2\dot{3}$ indicates that the sequence '123' repeats ($0.123123...$), while $0.4\dot{5}\dot{6}$ indicates only the '56' repeats ($0.4565656...$).

In some regions, a bar (vinculum) is used instead of dots. A horizontal line is drawn over the entire repeating sequence (e.g., $0.\overline{12}$), which serves the same purpose as the dot notation.

To convert a recurring decimal to a fraction, the goal is to create two equations with the exact same repeating decimal part so they can be subtracted to eliminate the infinite tail.

Step 1: Assign a Variable. Let $x$ equal the recurring decimal. Write out a few repetitions to visualize the pattern (e.g., $x = 0.151515...$).

Step 2: Multiply to Shift. Multiply $x$ by a power of 10 ($10, 100, 1000, etc.$) corresponding to the length of the repeating cycle. If two digits repeat, multiply by $100$ (e.g., $100x = 15.151515...$).

Step 3: Subtract Equations. Subtract the original equation from the new one ($100x - x = 15.1515... - 0.1515...$). This results in an integer ($99x = 15$).

Step 4: Solve and Simplify. Divide to isolate $x$ ($x = \frac{15}{99}$) and simplify the resulting fraction to its lowest terms (e.g., $\frac{5}{33}$).

You can predict if a fraction will result in a recurring or terminating decimal by looking at the prime factors of the denominator once the fraction is in its simplest form.

A fraction will terminate if and only if the prime factors of the denominator consist solely of $2$s, $5$s, or both (e.g., $\frac{1}{20}$ where $20 = 2^2 \times 5$).

If the denominator contains any prime factor other than 2 or 5 (such as $3, 7, 11, etc.$), the decimal expansion will be recurring (e.g., $\frac{1}{6}$ where $6 = 2 \times 3$).

Check the Multiplier: Always ensure the power of 10 you multiply by matches the number of digits in the repeating cycle. If you have $0.1\dot{2}\dot{3}$, the cycle is two digits long, so you must multiply by $100$ to align the decimals.

Verification: On a calculator, you can verify your fraction by performing the division. If your fraction is $\frac{7}{9}$, entering $7 \div 9$ should result in $0.7777...$ on your screen.

Simplification: Marks are often lost for not simplifying the final fraction. Always check if the numerator and denominator share common factors like $3$ or $9$, which are very common in these problems.

Show All Steps: In non-calculator exams, the algebraic method is usually required to be shown in full. Simply writing the answer without the 'Let $x = ...$' steps may result in zero marks.