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# The Arithmetic of Complex Numbers

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What is the product of (1+3i) and (3+2i)?

Aug 27, 2015

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$$\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{i}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{i}\right) = {\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{9}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}\left[{{i}}^{{\mathtt{2}}}\right] = {\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}{i}{\mathtt{\,-\,}}{\mathtt{6}} = {\mathtt{\,-\,}}{\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}{i}$$

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Aug 27, 2015

$$\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{i}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{i}\right) = {\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{9}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}\left[{{i}}^{{\mathtt{2}}}\right] = {\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}{i}{\mathtt{\,-\,}}{\mathtt{6}} = {\mathtt{\,-\,}}{\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}{i}$$