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Repeating Decimal Rule:
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A repeating decimal is a decimal number that after some point continues infinitely with a repeating pattern of digits. For example, 1/3 = 0.333... or 1/7 = 0.142857142857...
A fraction n/d has a terminating decimal if and only if the denominator d has no prime factors other than 2 or 5. If the denominator has any other prime factors, the decimal representation will repeat.
Why this works: In base 10, the decimal system is based on factors of 2 and 5 (since 10 = 2×5). A fraction terminates if the denominator's prime factors are a subset of the base's prime factors.
Examples:
Instructions: Enter any numerator and denominator (positive integers). The calculator will determine if the fraction would produce a repeating decimal and show the decimal approximation.
Q1: What's the difference between terminating and repeating decimals?
A: Terminating decimals end after a finite number of digits, while repeating decimals continue infinitely with a repeating pattern.
Q2: Are all fractions either terminating or repeating?
A: Yes, in base 10, all rational numbers (fractions) have decimal representations that either terminate or repeat.
Q3: What about denominators with both 2/5 and other factors?
A: If the denominator has any prime factors other than 2 or 5, the decimal will repeat (though it may have some non-repeating digits first).
Q4: How can I predict the length of the repeating part?
A: The length of the repeating part is related to the smallest number k such that 10^k ≡ 1 mod d (after removing factors of 2 and 5).
Q5: Does this work in other number bases?
A: Yes, in base b, a fraction terminates if the denominator's prime factors are a subset of b's prime factors.