Adding [latex](3-4i)+(2+5i)[/latex], we add the like terms by combining the real parts and the imaginary parts.

[latex]3+2-4i+5i[/latex]

[latex]5+i[/latex]

The initial point is [latex]3-4i[/latex]. When we add [latex]-1+3i[/latex], we add [latex]-1[/latex] to the real part, moving the point [latex]1[/latex] units to the left, and we add [latex]5[/latex] to the imaginary part, moving the point [latex]5[/latex] units vertically. This shifts the point [latex]3-4i[/latex] to [latex]2+1i[/latex].

[latex]\begin{array}{l}\left(2+5i\right)\left(4+i\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Expand.}\\=8+20i+2i+5i^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Since }i=\sqrt{-1},i^{2}=-1\\=8+20i+2i+5\left(-1\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Group like terms.}\\=8-5+20i+2i\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify.}\\=3+22i\end{array}[/latex]

Multiplying we’d get:

[latex]\begin{align}&2\cdot1+2\cdot2i\\&=2+4i\\\end{align}[/latex]

Notice both the real and imaginary parts have been scaled by [latex]2[/latex]. Visually, this will stretch the point outwards, away from the origin.

Multiplying [latex]1+2i[/latex] by [latex]1+i[/latex],

[latex]-4+2i[/latex]

[latex]\begin{align}&(1+2i)(1+i)\\&=1+i+2i+2{{i}^{2}}\\&=1+3i+2(-1)\\&=-1+3i\\\end{align}[/latex]

Multiplying by [latex]1+i[/latex] again,

[latex]\begin{align}&(-1+3i)(1+i)\\&=-1-i+3i+3{{i}^{2}}\\&=-1+2i+3(-1)\\&=-4+2i\\\end{align}[/latex]

If we multiplied by [latex]1+i[/latex] again, we’d get [latex]–6–2i[/latex]. Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by [latex]45°[/latex].