Chinese 3AD Arithmetics 东晋. 刘徽 《九章算术》更相减损术

Please explain the Number Theory behind this trick :\boxed{\frac {a } {b}= \frac {\frac {a}{b-a}}{\frac {b}{b-a}}}

Example: 246 - 205 = 41

\boxed {\frac {205} {246}= \frac {\frac {205}{41}}{\frac {246}{41}}=\frac{5}{6}}

Example:

27759 – 10227 = 17532 = 2 x 8766 = 2 x (2 x 4383) = 2 x 2 x (3 x 1461) = 2 x 2 x 3 x (3 x 487 )

\boxed {\frac {10227} {27759}= \frac {\frac {10227}{1461}}{\frac {27759}{1461}}=\frac{7}{19}}

Explanation:This method is from《九章算术》295AD 刘徽(曹魏/东晋),he invented the “Limit” 割圆法 method with 95-polygons to get the world’s best pi = 3.1416

https://zhidao.baidu.com/question/109475024.html

更相减损术证明

Bézout’s Theorem :

For a, b CO-PRIME, ie gcd (a, b) = 1
There exist integers x and y such that ax + by = 1

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