1. What Are Terminating and Non-Terminating Decimals?

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. A non-terminating decimal continues infinitely without ending, either repeating a pattern (repeating decimal) or without repetition (irrational number).

2. How Does the Calculator Work?

The calculator checks the denominator of the simplified fraction:

3. Mathematical Explanation

Theorem: A fraction a/b in lowest terms has a terminating decimal expansion if and only if the prime factorization of the denominator b has no prime numbers other than 2 or 5.

Examples:

• 1/2 = 0.5 (terminating)
• 1/3 = 0.333... (non-terminating repeating)
• 1/4 = 0.25 (terminating)
• 1/5 = 0.2 (terminating)
• 1/6 = 0.1666... (non-terminating repeating)

4. Using the Calculator

Instructions: Enter any integer numerator and denominator (≠ 0). The calculator will simplify the fraction and determine if its decimal representation terminates or repeats.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between repeating and irrational decimals?
A: Repeating decimals have an infinitely repeating pattern (like 0.333...), while irrational numbers (like π) have infinite non-repeating decimals.

Q2: Can this calculator identify irrational numbers?
A: No, with integer inputs you'll only get terminating or repeating decimals. Irrational numbers can't be expressed as fractions of integers.

Q3: Why does the denominator's prime factors matter?
A: Our number system is base-10 (2×5). Only fractions with denominators that divide powers of 10 can be expressed as finite decimals.

Q4: What about mixed numbers?
A: The whole number part doesn't affect the decimal type. Only the fractional part matters.

Q5: How can I convert repeating decimals to fractions?
A: Use algebra. For example, x = 0.333..., then 10x = 3.333..., subtract to get 9x = 3 → x = 1/3.