Jade Mirror of the Four Unknowns

Summarize Timeline Top Qs Fact Check

The four quantities are x, y, z, w can be presented with the following diagram

x

y 太w

z

The square of which is:

This section deals with Tian yuan shu or problems of one unknown.

Question: Given the product of huangfan and zhi ji equals to 24 paces, and the sum of vertical and hypotenuse equals to 9 paces, what is the value of the base?

Answer: 3 paces

Set up unitary tian as the base( that is let the base be the unknown quantity x)

Since the product of huangfang and zhi ji = 24

in which

huangfan is defined as：${\displaystyle (a+b-c)}$^[4]

zhi ji：${\displaystyle ab}$

therefore ${\displaystyle (a+b-c)ab=24}$

Further, the sum of vertical and hypotenuse is

${\displaystyle b+c=9}$

Set up the unknown unitary tian as the vertical

${\displaystyle x=a}$

We obtain the following equation

（${\displaystyle x^{5}-9x^{4}-81x^{3}+729x^{2}=3888}$）

太

Solve it and obtain x=3

太 Unitary

equation: ${\displaystyle -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0}$;

from the given

太

equation: ${\displaystyle 2y^{2}-xy^{2}+2xy+x^{3}=0}$;

we get:

太

${\displaystyle 8x+4x^{2}=0}$

and

太

${\displaystyle 2x^{2}+x^{3}=0}$

by method of elimination, we obtain a quadratic equation

${\displaystyle x^{2}-2x-8=0}$

solution: ${\displaystyle x=4}$.

Template for solution of problem of three unknowns

Zhu Shijie explained the method of elimination in detail. His example has been quoted frequently in scientific literature.^[5][6][7]

Set up three equations as follows

太

${\displaystyle -y-z-y^{2}x-x+xyz=0}$ .... I

${\displaystyle -y-z+x-x^{2}+xz=0}$.....II

太

${\displaystyle y^{2}-z^{2}+x^{2}=0;}$....III

Elimination of unknown between II and III

by manipulation of exchange of variables

We obtain

太

${\displaystyle -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y}$ ...IV

and

太

${\displaystyle -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}}$.... V

Elimination of unknown between IV and V we obtain a 3rd order equation

${\displaystyle x^{4}-6x^{3}+4x^{2}+6x-5=0}$

Solve to this 3rd order equation to obtain ${\displaystyle x=5}$;

Change back the variables

We obtain the hypothenus =5 paces

This section deals with simultaneous equations of four unknowns.

${\displaystyle {\begin{cases}-2y+x+z=0\\-y^{2}x+4y+2x-x^{2}+4z+xz=0\\x^{2}+y^{2}-z^{2}=0\\2y-w+2x=0\end{cases}}}$

Successive elimination of unknowns to get

${\displaystyle 4x^{2}-7x-686=0}$

Solve this and obtain 14 paces

There are 18 problems in this section.

Problem 18

Obtain a tenth order polynomial equation:

${\displaystyle 16x^{10}-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-4x-177162=0}$

The root of which is x = 3, multiply by 4, getting 12. That is the final answer.

There are 18 problems in this section

There are 9 problems in this section

There are 6 problems in this section

There are 7 problems in this section

There are 13 problems in this section