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Using calculator to evaluate $\left( \frac{(3 - i)(i + 5)}{5 + i} \right)^{*}$.

Using calculator to evaluate $\left( \frac{(3 - i)(i + 5)}{5 + i} \right)^{*}$.

Arithmetic with Complex Numbers in Cartesian Form

Complex Conjugates

Complex Numbers

Using calculator to evaluate $\left( \frac{7-3\mathbf{i}}{3+2\mathbf{i}} \right)^{*}$.

Using calculator to evaluate $\left( \frac{7-3\mathbf{i}}{3+2\mathbf{i}} \right)^{*}$.

Using calculator to evaluate $\left( \frac{1+2\mathbf{i}}{3-9\mathbf{i}} \right)^{*}$.

Using calculator to evaluate $\left( \frac{1+2\mathbf{i}}{3-9\mathbf{i}} \right)^{*}$.

Let&nbsp;$z=x+y\text{i}$. The conjugate of ${{z}^{*}}$, is defined by ${{z}^{*}}=x-yi$. Example:
If&nbsp;$z=6+7\mathbf{i}$, then&nbsp;${{z}^{*}}=6-7\mathbf{i}$ If&nbsp;$z=-4-\mathbf{i}$, then&nbsp;${{z}^{*}}=\text{}\hspace{0.5em}$ If&nbsp;$z=-\mathbf{i}$, then&nbsp;${{z}^{*}}=\text{}\hspace{0.5em}$ If&nbsp;$z=4$, then&nbsp;${{z}^{*}}=\text{}\hspace{0.5em}$

For $z=x+yi$, the conjugate is $z^*=x-yi$. Thus $z$ and $z^*$ have equal real parts and opposite imaginary parts.

Multiply numerator and denominator by the conjugate of the nonzero divisor $z_2=x_2+y_2i$.

\[\frac{z_1}{z_2}=\frac{z_1z_2^*}{z_2z_2^*}=\frac{(x_1x_2+y_1y_2)+(x_2y_1-x_1y_2)i}{x_2^2+y_2^2},\qquad z_2\ne0.\]

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