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What are Recurring Decimals?

Dividing a cake equally among three people gives each person exactly one-third, but if you type $1 \div 3$ into a calculator, the screen fills with an endless string of threes! This is an example of a recurring decimal, which is a number where a digit or a block of digits repeats infinitely in a regular pattern. Because they can all be written as an exact fraction (a fraction where both the numerator and denominator are whole numbers), all recurring decimals are rational numbers.

To write these concisely, mathematicians use dot notation. You simply place a dot above the repeating digits:

Single digit: $0.777...$ becomes $0.\dot{7}$ .

A pure recurring decimal has a repeating pattern that begins immediately after the decimal point (like $0.\dot{3}$ ). In contrast, a mixed recurring decimal has one or more non-repeating digits before the repeating pattern starts (like $0.1\dot{6}$ ). The number of repeating digits in the cycle is known as the period of a decimal.

Terminating vs. Recurring Decimals

Not all fractions create recurring decimals; some create a terminating decimal, which has a finite number of digits and does not repeat infinitely. You can predict whether an exact fraction will terminate or recur by looking at the prime factors of its denominator.

First, you must ensure the fraction is in its simplest form. If the prime factors of the simplified denominator are exclusively $2$ and/or $5$ , the decimal will terminate. If the denominator contains any other prime factors (such as $3$ , $7$ , or $11$ ), the decimal will recur.

Converting a Fraction to a Decimal

To convert a fraction into a decimal manually, you must divide the numerator by the denominator using the bus stop method (short division). You confirm the repeating pattern when a remainder reappears during your calculation.

Convert $\frac{4}{11}$ into a recurring decimal.

Step 1: Set up the division.

We use the bus stop method to divide $4.0000...$ by $11$ .

Step 2: Perform the division and track the remainders.

$11$ into $4$ goes $0$ times, remainder $4$ .
$11$ into $40$ goes $3$ times (), remainder .

Step 3: Identify the repeat.

The remainder $4$ has reappeared, meaning the pattern $36$ will repeat infinitely.

Step 4: Write the final answer using dot notation.

$0.3636... = 0.\dot{3}\dot{6}$

Converting a Decimal to a Fraction using Algebraic Proof

To convert a recurring decimal back into an exact fraction, you must use an algebraic proof. This involves setting the decimal as an unknown variable ( $x$ ), multiplying it by a power of $10$ to align the repeating digits, and subtracting to eliminate the infinite tail.

If $1$ digit repeats, multiply by $10$ . If $2$ digits repeat, multiply by $100$ , and so on.

Prove algebraically that $0.\dot{4}\dot{5}$ can be written as the exact fraction $\frac{5}{11}$ .

Step 1: Define $x$ as the recurring decimal.

$x = 0.454545...$

Step 2: Multiply by a power of $10$ based on the period of the decimal.

Because two digits repeat, multiply both sides by $100$ .
$100x = 45.454545...$

Step 3: Subtract the original $x$ equation from the new equation to eliminate the repeating decimals.

$100x - x = 45.4545... - 0.4545...$
$99x = 45$

Step 4: Solve for $x$ and simplify the fraction.

$x = \frac{45}{99}$
Divide the numerator and denominator by $9$ :
$x = \frac{5}{11}$

Handling Mixed Recurring Decimals

When converting a mixed recurring decimal, you need to generate two different equations where the decimal point is positioned just before and just after the first repeating block. Subtracting these two equations will eliminate the repeating tail entirely.

Convert $0.3\dot{8}$ into a fraction in its simplest form.

Step 1: Define $x$ as the recurring decimal.

$x = 0.3888...$

Step 2: Multiply to move the decimal to the start of the repeating sequence.

$10x = 3.888...$

Step 3: Multiply to move the decimal one full cycle past the start of the repeat.

$100x = 38.888...$

Step 4: Subtract the two new equations.

$100x - 10x = 38.888... - 3.888...$
$90x = 35$

Step 5: Solve for $x$ and simplify.

$x = \frac{35}{90}$
Divide by $5$ :
$x = \frac{7}{18}$

Students often place dots over every digit in a long repeating sequence, but OCR mark schemes require dots only on the first and last digits of a repeating block of three or more digits (e.g., write $0.123123...$ as $0.\dot{1}2\dot{3}$ ).

Treating a recurring decimal as a terminating one in calculations, such as mistakenly writing $0.\dot{4}\dot{5}$ as $\frac{45}{100}$ instead of the correct .

In calculator papers, you can use the $S \leftrightarrow D$ button to check your conversion, but for 'Prove algebraically' questions, writing an unsupported answer will score zero marks.

Always simplify your fraction completely before checking its denominator's prime factors to predict if it will terminate or recur.

A decimal number where a digit or a block of digits repeats infinitely in a regular pattern.

A fraction where both the numerator and denominator are integers, representing the precise value of a recurring decimal.

The bottom number in a fraction, representing the total number of equal parts.

The standard mathematical notation used to represent recurring decimals by placing dots over the repeating digits.

A recurring decimal where the repeating pattern begins immediately after the decimal point.

A recurring decimal where there are one or more non-repeating digits before the repeating pattern starts.

The number of digits in the repeating string of a recurring decimal.

A decimal that has a finite number of digits and does not repeat infinitely.

A shorthand method of short division used to divide a multi-digit number by a smaller number.

A formal sequence of steps using variables to demonstrate why a mathematical statement is true.