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Consider the graph of the equation z^2 + (\overline{z})^2 = 2.

For each complex number in the following list,

1      0    1+i      2 - i sqrt{3}     2-i    -1      i.

figure out whether each one is on the graph.

 May 8, 2019
 #1
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Consider the graph of the equation \(z^2 + (\overline{z})^2 = 2\).

For each complex number in the following list,

1      0    1+i     \( 2 - i \sqrt{3}\)     2-i    -1      i.

figure out whether each one is on the graph.

 

\(\text{Let $z=a+ib$} \\ \text{Let $\overline{z}=a-ib$}\)

\(\begin{array}{|rcll|} \hline \mathbf{z^2 + (\overline{z})^2} &=& \mathbf{2} \\\\ (a+ib)^2 +(a-ib)^2 &=& 2 \\ a^2+2abi-b^2 +a^2-2abi-b^2 &=& 2 \\ 2a^2-2b^2 &=& 2 \quad | \quad : 2 \\ \mathbf{a^2-b^2} &=& \mathbf{1} \\ \hline \end{array} \)

 

\(\begin{array}{|r||r|r|l|} \hline \text{list} && & & \\ z=a+ib && a & b & \mathbf{a^2-b^2=1}\ ? \\ \hline \color{red}1 && 1 & 0 & 1^2 -0^2= 1\ \checkmark \\ \hline 0 && 0 & 0 & 0^2 -0^2\ne 1 \\ \hline 1+i && 1 & 1 & 1^2 -1^2\ne 1 \\ \hline \color{red}2-i\sqrt{3} && 2 & \sqrt{3} & 2^2 - \left(\sqrt{3}\right)^2= 1\ \checkmark \\ \hline 2-i && 2 & -1 & 2^2 -\left(-1\right)^2\ne 1 \\ \hline \color{red}-1 && -1 & 0 & \left(-1\right)^2 -0^2= 1\ \checkmark \\ \hline i && 0 & 1 & 0^2 - 1^2\ne 1 \\ \hline \end{array} \)

 

laugh

 May 8, 2019

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