CVP2

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Show that |z_{1}z_{2}|=|z_{1}||z_{2}|\,.

|z_{1}z_{2}|=|(x_{1}+iy_{1})(x_{2}+iy_{2})|=|x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+y_{1}x_{2})|\,

={\sqrt  {(x_{1}x_{2}-y_{1}y_{2})^{2}+(x_{1}y_{2}+y_{1}x_{2})^{2}}}={\sqrt  {(x_{1}^{2}+y_{1}^{2})(x_{2}^{2}+y_{2}^{2})}}\,

={\sqrt  {x_{1}^{2}+y_{1}^{2}}}{\sqrt  {x_{2}^{2}+y_{2}^{2}}}=|z_{1}||z_{2}|\,

Alternate Proof

|z_{1}z_{2}|^{2}=(z_{1}z_{2})(\overline {z_{1}z_{2}})=z_{1}z_{2}\overline {z_{1}}\overline {z_{2}}=(z_{1}\overline {z_{1}})(z_{2}\overline {z_{2}})=|z_{1}|^{2}|z_{2}|^{2}\,

\implies |z_{1}z_{2}|=|z_{1}||z_{2}|\,

Complex Variables

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