4 ms·

No the headline was not wrong. One of the answers explained Horner's method and its relation to using abacus (which has a small register file, in modern lingo).

No the headline was not wrong. One of the answers explained Horner's method and its relation to using abacus (which has a small register file, in modern lingo). However that is not the extent of the book mentioned. You can find the book here: https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9%89%B4 https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9... The equations dealt with have been transcribed into modern algebraic expressions. They were definitely solved, not merely computed. The more complete answer should have been "The higher order polynomial (as high as 14) came from solving multivariate simultaneous equations (with degrees as high as 4) by elimination of variables."

wbhart 8y ago
I'm not seeing that. I see a degree 10 equation, which oddly has no rational roots (though my guess is this is what is meant by "solving"). I've never seen any mention of the Chinese solving degree 14 equations in any books on the history of mathematics. I'm very dubious of the claim, in any form, other than finding rational roots of polynomials (which the Greeks were doing more than a millennium earlier). Edit: no they were finding rational approximations to roots, which is actually pretty clever.
- fspeech 8y ago
  "Edit: no they were finding rational approximations to roots, which is actually pretty clever." Right they provided a general numerical method that can solve univariate polynomials of any degree. See https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7%AE%97%E6%B3%95 https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7... "南宋数学家秦九韶将贾宪的增乘开方术推广，以求解任意高次方程的实数根的数值解"
leereeves 8y ago
I wasn't referring to the reddit page, but the Wikipedia page it's based on: > Si-yüan yü-jian (《四元玉鑒》), or Jade Mirror of the Four Unknowns, was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations.[48] If there's more to the story you might want to correct Wikipedia.
- fspeech 8y ago
  The page I linked to is the actual page for Si-yüan yü-jian (《四元玉鑒》). The Wikipedia article you referred to may be slightly confusing but is not wrong. If you tried to solve a polynomial numerically, Horner's method would be handy. However we know there is no general solution per Abel and Galois.
  - betterunix2 8y ago
    More precisely, we know there is no general formula using basic arithmetic and radicals. Numerical methods count as a general solution as far as computer scientists are concerned (algebraic numbers are a subset of computable numbers).
  - leereeves 8y ago
    Which part are you're referring to when you say "The equations dealt with have been transcribed into modern algebraic expressions. They were definitely solved, not merely computed"?
    - fspeech 8y ago
      The page I linked to listed the problems posed and solved. The Chinese version is accompanied by modern notations using x,y,z... By the way the numerical solution to high order one variable polynomial was already known and given by another mathematician in an earlier (1247 AD) book: https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7%AB%A0 https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7...
      - leereeves 8y ago
        I thought you were referring to a specific solution for a 14th degree polynomial. If you're just referring to the notation, that's a modern translation using modern notation. What was in the original?
        fspeech 8y ago

wbhart 8y ago

I'm not seeing that. I see a degree 10 equation, which oddly has no rational roots (though my guess is this is what is meant by "solving"). I've never seen any mention of the Chinese solving degree 14 equations in any books on the history of mathematics. I'm very dubious of the claim, in any form, other than finding rational roots of polynomials (which the Greeks were doing more than a millennium earlier). Edit: no they were finding rational approximations to roots, which is actually pretty clever.

fspeech 8y ago
"Edit: no they were finding rational approximations to roots, which is actually pretty clever." Right they provided a general numerical method that can solve univariate polynomials of any degree. See https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7%AE%97%E6%B3%95 https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7... "南宋数学家秦九韶将贾宪的增乘开方术推广，以求解任意高次方程的实数根的数值解"

leereeves 8y ago

I wasn't referring to the reddit page, but the Wikipedia page it's based on: > Si-yüan yü-jian (《四元玉鑒》), or Jade Mirror of the Four Unknowns, was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations.[48] If there's more to the story you might want to correct Wikipedia.

fspeech 8y ago
The page I linked to is the actual page for Si-yüan yü-jian (《四元玉鑒》). The Wikipedia article you referred to may be slightly confusing but is not wrong. If you tried to solve a polynomial numerically, Horner's method would be handy. However we know there is no general solution per Abel and Galois.
- betterunix2 8y ago
  More precisely, we know there is no general formula using basic arithmetic and radicals. Numerical methods count as a general solution as far as computer scientists are concerned (algebraic numbers are a subset of computable numbers).
- leereeves 8y ago
  Which part are you're referring to when you say "The equations dealt with have been transcribed into modern algebraic expressions. They were definitely solved, not merely computed"?
  - fspeech 8y ago
    The page I linked to listed the problems posed and solved. The Chinese version is accompanied by modern notations using x,y,z... By the way the numerical solution to high order one variable polynomial was already known and given by another mathematician in an earlier (1247 AD) book: https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7%AB%A0 https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7...
    - leereeves 8y ago
      I thought you were referring to a specific solution for a 14th degree polynomial. If you're just referring to the notation, that's a modern translation using modern notation. What was in the original?
      - fspeech 8y ago