Multiplying Complex Numbers

This graph shows how we can interpret the multiplication of complex numbers geometrically.[br]Given two complex numbers:[br][math]w=a+bi[/math], [math]z=c+di[/math] where [math]a,b,c,d\in\mathbb{R}[/math][br][br]Consider their product[br][math]wz=\left(a+bi\right)z=az+biz[/math][br][table][tr][td]1[/td][td][math]az[/math][/td][td]Dilate [math]z[/math] by a scale factor of [math]a[/math][/td][/tr][tr][td]2[/td][td][math]iz[/math][/td][td]Rotate [math]z[/math] by [math]90^\circ[/math] about [math]0[/math][/td][/tr][tr][td]3[/td][td][math]biz[/math][/td][td]Dilate [math]iz[/math] by a scale factor of [math]b[/math][/td][/tr][tr][td]4[/td][td][math]az+biz[/math][/td][td]Translate [math]az[/math] by [math]biz[/math][/td][/tr][/table][br]Focus on the two right triangles in the diagram:[br][list=1][*]The right triangle formed by [math]w[/math], [math]0[/math] and the positive real axis.[/*][*]The right triangle formed by [math]az[/math], [math]biz[/math] and [math]0[/math][br][br][/*][/list]The first right triangle has sides of length: [math]a[/math], [math]b[/math], [math]\left|w\right|[/math].[br]The second right triangle has sides of length [math]a\left|z\right|[/math], [math]b\left|z\right|[/math], and [math]\left|wz\right|[/math].[br][br]Since we have the proportion: [math]\frac{a\left|z\right|}{a}=\frac{b\left|z\right|}{b}[/math], we can conclude the triangles are similar since two pairs of corresponding sides are proportional and their included angles (the right angles) are congruent.[br][br]This has two implications:[br][list=1][*]The ratio of similitude is [math]\left|z\right|[/math], which means that [math]\left|wz\right|=\left|w\right|\left|z\right|[/math] (this is an alternative to the algebraic proof you did for homework)[/*][*]The angle formed by [math]wz[/math], [math]0[/math] and [math]az[/math] is congruent to [math]Arg\left(w\right)[/math], since they are corresponding angles of similar triangles[/*][/list][br]This leads us to our other conclusion, that [math]Arg\left(wz\right)=Arg\left(z\right)+Arg\left(w\right)[/math][br][br]Key results:[br][math]\left|wz\right|=\left|w\right|\left|z\right|[/math][br][math]Arg\left(wz\right)=Arg\left(w\right)+Arg\left(z\right)[/math]

Information: Multiplying Complex Numbers