Geometric interpretations of complex arithmetic

The addition and multiplication of complex numbers have interesting geometric interpretations in terms of translation, rotation, and scaling on the complex plane.



Addition is translation

Let z1=a1+ib1 and z2=a2+ib2 where ai,biR for i=1,2. Then, z1+z2=(a1+a2)+i(b1+b2). Therefore, on the complex plane, adding z1 to z2 to obtain z1+z2 corresponds to translating the point (a1,b1) by the vector (a2,b2) to obtain the point (a1+a2,b1+b2). Thus, the four points, 0,z1,z1+z2, and z2 comprise a parallelogram (Figure 2).
Figure 2. Adding two complex numbers.




Multiplication by a real number is scaling

Let cR and z=a+ibC with a,bR. Then, cz=ca+i(cb), which corresponds to the point (ca,cb) on the complex plane. Meanwhile, we have |cz|=|c||z|, so the modulus is scaled by |c|. If c>0, cz and z are in the same direction from the origin; if c<0, then they are in the opposite directions from the origin (Figure 3).



Figure 3. Multiplying by a real number is scaling.



Multiplication by a complex number u with |u|=1 is rotation around the origin by argu.

Figure 4. Multiplication is rotation.





Theorem

Let z,uC with |u|=1. If we multiply z by u, then z is rotated around the origin by an angle of Arg(u) on the complex plane (Figure 4).
Proof.

(Step 1) Note 1C and |1|=1. If we multiply 1 by u, we have u1=u so Arg(u1)=Arg(u). Thus, 1 is rotated around the origin by Arg(u).


(Step 2) Now consider iC. |i|=1. Suppose z=a+ib. If we multiply z by i, we have zi=(a+ib)i=b+ia. Note that |z|2=a2+b2 and |zi|2=|z|2|i|2=a2+b2. On the complex plane, the squared distance between z and zi is given by |zzi|2=|(a+b)+i(ba)|2=(a+b)2+(ba)2=2a2+2b2. Therefore we have
|z|2+|zi|2=(a2+b2)+(a2+b2)=2a2+2b2=|zzi|2.
 By the Pythagorean theorem, the angle made by z, 0 (the origin), and zi is the right angle, π/2. Furthermore, this rotation is counter-clockwise. Hence multiplying by i rotates z around the origin by π/2. However, Arg(i)=π/2. In other words, Arg(zi)=Arg(z)+Arg(i).

(Step 3) Let z,wC. |u(zw)|=|u||zw|=1|zw|=|zw|. Thus multiplying by u preserves distance.

(Step 4) Let zC. Consider the triple of distances (|z1|,|z0|,|zi|). These are the three distances of z from 1,0 and i, respectively.
  It is noted that this z is the only point on the complex plane which has this particular triple of distances from 1,0 and i. Now, (|uzu|,|uz0|,|uzui|)=(|z1|,|z0|,|zi|) since multiplying by u preserves distances. Note that the triangle with vertices u,0,ui is obtained by rotating the triangle with 1,0,i by an angle Arg(u). If a point has a distance from u,0,ui equal to |z1|,|z|,|zi|, then that point must also be obtained by rotating z by an angle Arg(u)
Thus, we conclude that multiplication by u has the effect of rotating points in the complex plane by Arg(u). ■

Next, consider multiplying z by an arbitrary non-zero complex number α. We can decompose α as

α=|α|α|α|.

If we define u=α|α|, then |u|=1. Of course, |α| is a positive real number. Thus, multiplication by α can be decomposed into two steps: (1) rotating by argu, then (2) scaling by |α|:

zrotateuzscale|α|(uz)=αz.


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