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

Math Online Tom Circle

Please explain the Number Theory behind this trick :$latex boxed{frac {a } {b}= frac {frac {a}{b-a}}{frac {b}{b-a}}} &fg=aa0000&s=4$

Example: $latex 246 – 205 = 41$

$latex boxed {frac {205} {246}= frac {frac {205}{41}}{frac {246}{41}}=frac{5}{6}&fg=0000aa&s=3}$

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 )

$latex boxed {frac {10227} {27759}= frac {frac {10227}{1461}}{frac {27759}{1461}}=frac{7}{19}&fg=00aa00&s=3}$

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 integersxandysuch thatax+by= 1

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Author: tomcircle

Math amateur

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