${\displaystyle {\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill \end{array}\right)}^2=\left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill -1\hfill \end{array}\right).}$

Arithmetic for the set $R^{*}$ is isomorphic to the arithmetic of real numbers $R$ where the matrix $\left(\begin{array}{cc}\hfill x\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill x\hfill \end{array}\right)$ corresponds to the real number $x$

- both sets are closed under addition and multiplication which are commutative

- addition and multiplication are associative

- the additive identity is $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}\right)$ corresponds to the real number zero

- the multiplicative identity $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\right)$ corresponds to the real number 1

- each element has an additive and multiplicative inverse

- the distributive property holds.

Arithmetic for the set $C^{*}$ is isomorphic to the arithmetic of complex numbers $C$ where the matrix $\left(\begin{array}{cc}\hfill x\hfill & \hfill -y\hfill \\ \multicolumn{1}{c}{y}& \hfill x\hfill \end{array}\right)$ corresponds to the complex number $x+iy$

- both sets are closed under addition and multiplication which are commutative

- addition and multiplication are associative

- the additive identity is $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}\right)$ corresponds to the complex number $0+i0$

- the multiplicative identity $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\right)$ corresponds to the complex number $1+i0$

- the additive inverse of $\left(\begin{array}{cc}\hfill x\hfill & \hfill -y\hfill \\ \multicolumn{1}{c}{y}& \hfill x\hfill \end{array}\right)~x+\mathrm{iy}$ is $\left(\begin{array}{cc}\hfill -x\hfill & \hfill y\hfill \\ \multicolumn{1}{c}{-y}& \hfill -x\hfill \end{array}\right)~-x-\mathrm{iy}$

- the multiplicative inverse of $\left(\begin{array}{cc}\hfill x\hfill & \hfill -y\hfill \\ \multicolumn{1}{c}{y}& \hfill x\hfill \end{array}\right)~x+iy$ is

${\displaystyle \frac{1}{(x^2+y^2)}\left(\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \multicolumn{1}{c}{-y}& \hfill x\hfill \end{array}\right)~\frac{x-\mathrm{iy}}{x^2+y^2}}$

- the distributive property holds.

(2)(a) $i^2=j^2=k^2=\left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill -1\hfill \end{array}\right)$

(b) $ij=k=-ji$, (c) $jk=i=-kj$ (d) $ki=j=-ik$

(3) $ij=k,\hspace{0.5em}ijk=k^2=-1,\hspace{0.5em}ijki=-i,\hspace{0.5em}ijkij=-ij=-k,\hspace{0.5em}ijkijk=-k^2=1$ so the sequence $i,\hspace{0.5em}ij,\hspace{0.5em}ijk,\hspace{0.5em}ijki,\hspace{0.5em}ijkij,\hspace{0.5em}...$ is a cycle of order 6 repeating the terms $i,k,-1,-i,-k,1$