Rod Calculus - Fractions

Fractions

If there is a remainder in a place value decimal rod calculus division, both the remainder and the divisor must be left in place with one on top of another. In Liu Hui's notes to Jiuzhang suanshu (2nd century BCE), the number on top is called "shi实", while the one at bottom is called "fa法". In Sun Tzu's Calculation Classic, the number on top is called "zi子" or "fenzi" (lit., son of fraction), and the one on the bottom is called "mu母" or "fenmu" (lit., mother of fraction). Fenzi and Fenmu are also the modern Chinese name for numerator and denominator, respectively. As shown on the right, 1 is the numerator remainder, 7 is the denominator divisor, formed a fraction . The quotient of the division 309/7 is 44 + . Liu Hui's used a lot of calculations with fraction in The Sea Island Mathematical Manual.

This form of fraction with numerator on top and denominator at bottom without a horizontal bar in between, was transmitted to Arabic country in a 825AD book by al Khwarizmi via India, and in use by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithematic Key".