Octonions
Let $x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7$ be real numbers, the octonion $x$ is an expression in the form \[ x = x_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7 \] where $1,e_1,\ldots,e_7$ are the unit octonions and $e_i^2=-1$ for $i=1,\ldots,7$. Their multiplication are shown in the table below \[ \begin{array}{|c|rrrrrrrr|} \hline e_ie_j & 1 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ \hline 1 & 1 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ e_1 & e_1 & -1 & e_3 & -e_2 & e_5 & -e_4 & -e_7 & e_6 \\ e_2 & e_2 & -e_3 & -1 & e_1 & e_6 & e_7 & -e_4 & -e_5 \\ e_3 & e_3 & e_2 & -e_1 & -1 & e_7 & -e_6 & e_5 & -e_4 \\ e_4 & e_4 & -e_5 & -e_6 & -e_7 & -1 & e_1 & e_2 & e_3 \\ e_5 & e_5 & e_4 & -e_7 & e_6 & -e_1 & -1 & -e_3 & e_2 \\ e_6 & e_6 & e_7 & e_4 & -e_5 & -e_2 & e_3 & -1 & -e_1 \\ e_7 & e_7 & -e_6 & e_5 & e_4 & -e_3 & -e_2 & e_1 & -1 \\ \hline \end{array} \] The multiplication of octonions is not only non-commutative but also non-associative, i.e. there are octonions $x_1$, $x_2$, $x_3$ such that $x_1(x_2x_3)$ and $(x_1x_2)x_3$ are different.
In addition to four octonion basic operations: addition, subtraction, multiplication, and division, this calculator also supports the following 10 functions.
| function | description/example |
|---|---|
| $\mathrm{abs}(x)$ | If $x=x_0+\sum_{k=1}^7x_ke_k$, this function return $\sqrt{x_0^2+\cdots+x_7^2}$ |
| $\mathrm{conj}(x)$ | If $x=x_0+\sum_{k=1}^7x_ke_k$, this function return $x=x_0-\sum_{k=1}^7x_ke_k$ |
| $\mathrm{pow}(x,n)$ or $x\hat{\;} n$ | Calculate $x^n$, where $x$ is an octonion and $n$ is an integer. Undefined if $n$ is not an integer. |
| $\exp(x)$ | The base-e exponential function of octonion $x$ |
| $\sin(x)$ | The sine function of octonion $x$ |
| $\cos(x)$ | The cosine function of octonion $x$ |
| $\tan(x)$ | The tangent function of octonion $x$ |
| $\sinh(x)$ | The hyperbolic sine function of octonion $x$ |
| $\cosh(x)$ | The hyperbolic cosine function of octonion $x$ |
| $\tanh(x)$ | The hyperbolic tangent function of octonion $x$ |
Input Data
This calculator uses the same input format as used in Quaternion Calculator except that in this case the input matrix dimension is $n\times 8$ for $n$ octonions.