Transformations on the Argand Diagram
Transformations on the Argand Diagram
TGM Original Questions
Conjugation
\(z_1\mapsto z_1^*\): reflect in the real axis.
Negation
\(z_1\mapsto-z_1\): rotate anticlockwise by \(\pi\), or reflect in the origin.
Addition and subtraction
Adding \(z_2=x_2+y_2i\) translates by \((x_2,y_2)\); subtracting \(z_2\) translates by \((-x_2,-y_2)\).
Multiplication by i
\(z_1=x_1+y_1i\mapsto iz_1=-y_1+x_1i\): rotate anticlockwise by \(\pi/2\).
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Answer:\(\text{Conjugation:}\hspace{0.5em}x+y\mathbf{i}\mapsto x-y\mathbf{i}\) \(\text{Negation:}\hspace{0.5em}z\mapsto-z\) \(\text{Addition/subtraction:}\hspace{0.5em}z_1\mapsto z_1\pm z_2\) \(\text{Multiplication by}\hspace{0.5em}\mathbf{i}\text{: rotate anticlockwise by}\hspace{0.5em}\dfrac{\pi}{2}\)
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