Terminating Decimals Calculator
Determine if fractions have terminating or repeating decimal representations
Analyze Decimal Representation
Top number of the fraction
Bottom number of the fraction (cannot be zero)
Analysis Results
Prime Factorization Analysis
Reduced denominator: 2
Prime factors: 2
Unique factors: 2
Long Division Steps
Example Calculations
Terminating Examples
1/2 = 0.5 (factors: 2)
3/4 = 0.75 (factors: 2²)
7/8 = 0.875 (factors: 2³)
3/5 = 0.6 (factors: 5)
1/10 = 0.1 (factors: 2×5)
Repeating Examples
1/3 = 0.333... (factors: 3)
1/7 = 0.142857... (factors: 7)
5/6 = 0.8333... (factors: 2×3)
1/9 = 0.111... (factors: 3²)
2/11 = 0.181818... (factors: 11)
Quick Guide
Enter Fraction
Input numerator and denominator
Choose Method
Select analysis approach
View Results
See decimal type and steps
Mathematical Rules
A fraction in lowest terms has a terminating decimal if and only if the denominator has no prime factors other than 2 and 5
Terminating decimals end after a finite number of digits
Repeating decimals have a pattern that repeats infinitely
The length of the repeating cycle is at most (denominator - 1)
Understanding Terminating and Repeating Decimals
What are Terminating Decimals?
A terminating decimal is a decimal number that ends after a finite number of digits. These decimals can be exactly represented without any repeating patterns.
Recognition Rule
A fraction in lowest terms will produce a terminating decimal if and only if the denominator contains no prime factors other than 2 and 5. This is because our decimal system is base 10, and 10 = 2 × 5.
Examples
- •1/4 = 0.25 (denominator = 2²)
- •3/8 = 0.375 (denominator = 2³)
- •7/20 = 0.35 (denominator = 2² × 5)
What are Repeating Decimals?
A repeating decimal has a digit or group of digits that repeats infinitely. The repeating part is often indicated with a bar over the repeating digits or with ellipsis (...).
Mathematical Properties
- •Every rational number has either a terminating or repeating decimal representation
- •The period length is at most (denominator - 1) digits
- •Irrational numbers have non-repeating, non-terminating decimals
Long Division Method
Use long division to find the decimal representation. When a remainder repeats, you've found the start of the repeating cycle. If the remainder becomes 0, the decimal terminates.