Table 3 List of operators implemented for each type of n-dimensional vector, quaternions, octonions, sedenions, and complex numbers.

u + v

\((u_1+v_1, u_2+v_2, \ldots , u_n+v_n)\)

u - v

\((u_1-v_1, u_2-v_2, \ldots , u_n-v_n)\)

u * v

(Hamilton multiplication)

\(\dagger \)

v * k

\((v_1\,k,v_2\,k, \ldots , v_n\,k)\)

k * v

\((k\,v_1,k\,v_2, \ldots , k\,v_n)\)

u / v

(Hamilton division)

\(\dagger \)

v / k

\((v_1/k, v_2/k, \ldots , v_n/k )\)

u > v

\(u_1^2 + u_2^2 + \ldots + u_n^2 > v_1^2 + v_2^2 + \ldots + v_n^2 \)

u < v

\(u_1^2 + u_2^2 + \ldots + u_n^2 < v_1^2 + v_2^2 + \ldots + v_n^2 \)

u >= v

\(u_1^2 + u_2^2 + \ldots + u_n^2 \ge v_1^2 + v_2^2 + \ldots + v_n^2 \)

u <= v

\(u_1^2 + u_2^2 + \ldots + u_n^2 \le v_1^2 + v_2^2 + \ldots + v_n^2 \)

u == v

\(u_1=v_1 \text { and } u_2=v_2 \; \ldots \text { and } u_n=v_n\)

u != v

\(u_1 \ne v_1 \text { or } u_2 \ne v_2 \;\ldots \text { or } u_n \ne v_n\)