Repeating Decimals & Conversion

Repeating decimals possess digits that continue infinitely. These are vital in mathematics, especially when converting between fractions and decimals. Developing a comprehensive understanding of these patterns ensures precision in calculations.

Understanding Rational Numbers

Rational Numbers: Rational numbers can be represented as fractions $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \neq 0$.

• Decimal Representation:
  • Rational numbers manifest as either terminating or repeating decimals.
  • Examples:
    • Terminating Decimal: $\frac{1}{4} = 0.25$
    • Repeating Decimal: $\frac{1}{3} = 0.333\ldots$

Repeating Decimals and Their Notation

Repetend: The sequence in a decimal that repeats.

Overbar Notation: Used to denote the repeating sequence in a decimal. For instance, $0.333...$ is written as $0.\overline{3}$.

Recognising repeating patterns is crucial:

Decimal Number	Overbar Notation	Repetend
$0.333...$	$0.\overline{3}$	3
$0.666...$	$0.\overline{6}$	6
$0.142857...$	$0.\overline{142857}$	142857

Converting Repeating Decimals to Fractions

Converting a repeating decimal into a fraction simplifies mathematical tasks.

Step-by-Step Algebraic Method:

1. Identify the Repeating Decimal: Use overbar notation, e.g., $0.\overline{3}$, $0.\overline{6}$.
2. Formulate Equations:
   • Let $x = 0.\overline{3}$.
   • Multiply to align the decimal point: $10x = 3.\overline{3}$.
3. Subtract and Solve:
   • $10x - x = 3.\overline{3} - 0.\overline{3}$ results in $9x = 3$.
   • Solve $x = \frac{1}{3}$.

Worked Examples

Example 1: Convert $0.\overline{5}$ to a fraction

1. Let $x = 0.\overline{5}$
2. Multiply by 10: $10x = 5.\overline{5}$
3. Subtract: $10x - x = 5.\overline{5} - 0.\overline{5}$
4. Simplify: $9x = 5$
5. Therefore: $x = \frac{5}{9}$

Example 2: Convert $0.1\overline{23}$ to a fraction

1. Let $x = 0.1\overline{23}$
2. Since the repeating part starts after one decimal place, multiply by 100: $100x = 12.\overline{3}$
3. Multiply by 10 again to align the repeating portion: $1000x = 123.\overline{23}$
4. Subtract: $1000x - 100x = 123.\overline{23} - 12.\overline{23}$
5. Simplify: $900x = 111$
6. Therefore: $x = \frac{111}{900} = \frac{37}{300}$

Key Misconceptions

Misconception 1: Terminating vs. Repeating

• Terminating: Decimals that end, e.g., 0.5 ($\frac{1}{2}$).
• Repeating: Decimals with an infinite pattern, e.g., 0.333... ($\frac{1}{3}$).

Misconception 2: Misidentifying Repetends

• Definition: The repeating section, such as "6" in 0.666...

Correct identification is critical for notation and conversion:

• Strategy: Recognise and verify recurring blocks for symmetry.

Practical Applications

Financial and Real-Life Contexts

Finance:

• Double-check fractional conversions in financial statements.
• Convert interest rates accurately (e.g., 6.666...% to $6\frac{2}{3}$%).

Everyday Scenarios:

• Fairly splitting costs, like a restaurant bill where each share is £33.333...
• Cooking conversions, such as converting 1.666... cups of sugar to $1\frac{2}{3}$ cups.