【Study Notes】Supercomplex Numbers

This is a study note that will be continuously supplemented and improved.

hypercomplex number

Hypercomplex numbers are a generalized complex number . Hamilton generalized complex numbers and introduced quaternions . This generalized complex number is a hypercomplex number .

Combined with ordinary plurals, it is easy to understand the following.

1 definition

$A=a_0+a_{1}i+a_{2}j+a_{3}k$ is a multiple number.
$a=|A|=(a_0^2+a_1^2+a_2^2+a_3^2{)}^{1/2}$ is called hyperparameterAAThe modulus of$A.$
$i,j,k$ is called hyperparameterAAThe imaginary units,second, and third dimensions of$A.$

2 Related properties

2.1 Hypercomplex equality

Two supermultiples $A=a_0+a_{1}i+a_{2}j+a_{3}k$ sum$B=b_0+b_{1}i+b_{2}j+b_{3}k$。
(1)$A=B\iff a_n=b_n(n=0,1,2,3)$
(2)$m(a_0+a_{1}i+a_{2}j+a_{3}k)=m{a}_0+m{a}_{1}i+m{a}_{2}j+m{a}_{3}k$，$m$ is a real number.

2.2 Conjugate hypercomplex numbers

Supermultiple $A=a_0+a_{1}i+a_{2}j+a_{3}k$，$\bar{A}=a_0-a_{1}i-a_{2}j-a_{3}k$ isAATheconjugate hypercomplex numbersof$A.$

$$|A|=|\bar{A}|,\bar{A}=\bar{B}\iff a_n=b_n(n=0,1,2,3)$$

2.3 Multiplication rule

Two supermultiples $A=a_0+a_{1}i+a_{2}j+a_{3}k$ sum$B=b_0+b_{1}i+b_{2}j+b_3$product of$k$

$$\begin{array}{rl}AB& =[a_0+a_{1}i+a_{2}j+a_{3}k][b_0+b_{1}i+b_{2}j+b_{3}k]\\ & =\cdots \\ & =a_0{b}_0-a_1{b}_1+(a_0{b}_1+b_0{a}_1)i\end{array}$$
That is, the regular product of complex numbers.

3 Trigonometric expressions

Supermultiple $A=a_0+a_{1}i+a_{2}j+a_{3}k$ names$u,v,w$ forAAThe first, second and third corners of$A.$
$a_0=a\cos u\cos v\cos w$，$a_1=a\cos u\cos v\sin w$，$a_2=a\cos u\sin v$，$a_3=a\sin u$。

$$\begin{array}{rl}A& =a_0+a_{1}i+a_{2}j+a_{3}k\\ & =(a,u,v,w)\\ & =a(\cos u\cos v\cos w+i\cos u\cos v\sin w+j\cos u\sin v+k\sin u)\end{array}$$

some conclusions

Set-up supermultiple $A=(a,u_1,v_1,w_1),B=(b,u_2,v_2,w_2)$ , real number$m$，则
(1)$m(a,u_1,v_1,w_1)=(ma,m{u}_1,m{v}_1,m{w}_1)$
(2)$(a,u_1,v_1,w_1)(b,u_2,v_2,w_2)=(ab,u_1+u_2,v_1+v_2,w_1+w_2)$
(3) $(a,u_1,v_1,w_1)/(b,u_2,v_2,w_2)=(a/b,u_1-u_2,v_1-v_2,w_1-w_2)$
(4)$(a,u_1,v_1,w_1{)}^n=(a^n,n{u}_1,n{v}_1,n{w}_1)$
(5) Supermultiple$A=(a,u,v,w)$ conjugate hypercomplex number$\bar{A}=(a,-u,-v,-w)$ , 则
$(\bar{A}{)}^n$ is$A^{}$The conjugate hypercomplex numbersof${}^n$ .
(6) Let hypercomplex number$A=(a,u,v,w)$，$f(A)=c_0+c_{1}A+\cdots +c_n{A}^n$，$c_j(j=0,1,\cdots n)$ is a real number, then$f(\bar{A})$isf ( A ) f(A)Conjugate hypercomplex numbers of$f$$($$A$$)\; .$

For (6), there is a simple way to understand: we can put $\stackrel{ˉBring}{A}$ into$f(A)$中，可以得到：
$$\begin{array}{rl}f(A)& =c_0+c_{1}A+\cdots +c_n{A}^n\\ & =c_0+c_1(a,u,v,w)+\cdots +c_n(a^n,no,\_nv,nw)\end{array}$$
$$\begin{array}{rl}f(\bar{A})& =c_0+c_1\bar{A}+\cdots +c_n(\bar{A}{)}^n\\ & =c_0+c_1(a,-u,-v,-w)+\cdots +c_n(a^n,-nu,-nv,-nw)\end{array}$$

It is not difficult to see that the absolute value of the quaternion is completely symmetrical, and the three angles are mutually opposite numbers. After spatial addition, $f(A)$、$f(\bar{A})$and$A$、$\stackrel{ˉSimilarity}{A}$ is the conjugate relationship.