Distributive Law For Complex Numbers
Let Z be a vector with length s and angle q. Then Z == (s , q) = s * ( 1, q) = sR whenever s is a length > 0. Suppose B and C are vectors in the plane. Multiplication by Z has the same result as rotation R through the angle q followed by a scalar multiplication by s. Therefore
This suggests the distributive law
for vector multiplication over vector addition. The assumption that the addition of angles > 0 and the multiplication of lengths > 0 commutes implies the order of multiplication with the polar coordinate rule Add the angles, Multiply the lengths, does not matter. Therefore we can write
as well whenever B, C and Z are points or arrows in the plane. Multiplication with Components (optional)Suppose Z and W are arrows in the plane with tails or initial points at the origin. The product ZW has been defined by means of the Polar coordinate multiplication rule: add their angles, multiply their lengths. Before showing or deriving how to multiply these factors together using the rectangular coordinates of their head locations, we will show how to compute the product using horizontal and vertical components. To this end
Now observe ZW = (A+B)W = AW + BW by the distributive law. AW = A(C+D) = AC+ AD by the distributive law. BW = B(C+D) = BC + BD by the distributive law. Now the polar multiplication rule implies a horizontal * horizontal and vertical* vertical gives a horizontal result, while horizontal*vertical and vertical*horizontal gives a vertical result. Therefore, the component addition method implies the horizontal component of the product ZW is AC + BD while the vertical component of ZW is BC + AD. Therefore
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The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers. Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision . D1 to D6 after provide a review of vectors. More on Complex Numbers:
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