Bibliography:
- www.quantamagazine.org/rational-or-not-this-basic-math-question-took-decades-to-answer-20250108/ Rational or Not? This Basic Math Question Took Decades to Answer (2025)
A "reduced fraction" is a fraction that has the smallest possible integer numerator and denominator for its value.
For example:is not a reduced fraction, because there is another fraction equal to it but with smaller numerator and denominator:
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A rational number is a quotient of integers with . The numerator can be negative or zero; the denominator can be chosen positive by transferring its sign to . For example, and .
Different fractions describe the same rational number exactly when cross multiplication gives . Dividing the entries by their greatest common divisor produces the unique reduced fraction with positive denominator, including for zero.
The rational numbers form a field: addition, subtraction, multiplication and division by a nonzero rational number remain rational. They are dense in the real line, by density of the rational numbers, although they do not fill it: the square root of two is an irrational number.
A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \) is not equal to zero. In other words, rational numbers include integers, finite decimals, and repeating decimals. For example: - The number \( \frac{1}{2} \) is a rational number.