What are Terminating and Recurring Decimals?
When we convert a rational number (p/q) to decimal form by dividing p by q, we get either:
Terminating Decimals: The division ends after a finite number of steps.
- Example: 1/2 = 0.5, 3/4 = 0.75, 7/8 = 0.875
- Occurs when denominator (in simplest form) has only prime factors 2 and 5
Recurring (Repeating) Decimals: The division never ends; a digit or block of digits repeats forever.
- Example: 1/3 = 0.333..., 2/7 = 0.285714285714...
- Represented with a bar: 0.3̅, 0.285714̅
Why Do Recurring Decimals Occur?
When denominator has prime factors other than 2 and 5 (like 3, 7, 11, etc.), the decimal repeats.