1. What Determines Terminating vs Repeating Decimals?

A fraction in lowest terms has a terminating decimal if and only if the prime factorization of the denominator has no prime factors other than 2 or 5. Otherwise, the decimal representation will eventually become periodic (repeating).

2. How the Calculator Works

The calculator checks the denominator's prime factors:

Examples:

• Denominator 8 (2×2×2) → Terminating
• Denominator 20 (2×2×5) → Terminating
• Denominator 3 (3) → Repeating
• Denominator 7 (7) → Repeating

3. Importance of Knowing Decimal Type

Details: Understanding whether a fraction terminates or repeats helps in mathematical computations, computer representations, and understanding number theory concepts.

4. Using the Calculator

Tips: Enter any positive integer denominator. The calculator assumes the fraction is in lowest terms (numerator and denominator have no common factors other than 1).

5. Frequently Asked Questions (FAQ)

Q1: What about fractions not in lowest terms?
A: Reduce the fraction first. For example, 3/6 reduces to 1/2 which has a terminating decimal.

Q2: Does numerator affect whether decimal terminates?
A: No, only the denominator matters (assuming fraction is in lowest terms).

Q3: What's the longest possible repeating sequence?
A: For denominator d, the maximum repeating length is d-1 digits.

Q4: Are there denominators that produce purely repeating decimals?
A: Yes, denominators with no 2 or 5 factors (like 3, 7, 9) produce purely repeating decimals.

Q5: What about denominators with both 2/5 and other primes?
A: These produce mixed decimals - non-repeating followed by repeating parts.