Transformations

Transformations on complex numbers involve applying operations or mappings that change their position in the complex plane. The complex plane represents complex numbers as points, with the real part as the $x$-coordinate and the imaginary part as the $y$-coordinate.

Adding & Subtracting Complex Numbers

If $z_1=2-3i$ and $z_2=1+2i$, then adding $z_2$ to $z_1$ can be viewed as shifting $z_1$ to the right in the real component by $1$ unit and $2$ units in the imaginary axis.

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Scaling

Multiplying a complex number $z_1$ by some scalar constant $k$ expand or compress its magnitude. For example, consider $z_1=2+3i$, then $2z_1$ doubles its magnitude.

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Reflection

The conjugate of a complex number flips or rotates it across the the real axis.

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Rotation

Multiplying a complex number by $i$ rotates it $90 \degree$anticlockwise, while multiplying it by $-i$ rotates it $90 \degree$ clockwise.