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 and where for . Then, . Therefore, on the complex plane, adding to to obtain corresponds to translating the point by the vector to obtain the point . Thus, the four points, , and comprise a parallelogram (Figure 2).
Figure 2. Adding two complex numbers.
Multiplication by a real number is scaling
Let and with . Then, , which corresponds to the point on the complex plane. Meanwhile, we have , so the modulus is scaled by . If , and are in the same direction from the origin; if , 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 with is rotation around the origin by .
Figure 4. Multiplication is rotation.
Theorem
Let with . If we multiply by , then is rotated around the origin by an angle of on the complex plane (Figure 4).
Proof.
(Step 1) Note
(Step 2) Now consider . . Suppose . If we multiply by , we have . Note that and . On the complex plane, the squared distance between and is given by . Therefore we have
By the Pythagorean theorem, the angle made by , 0 (the origin), and is the right angle, . Furthermore, this rotation is counter-clockwise. Hence multiplying by rotates around the origin by . However, . In other words, .
(Step 3) Let . . Thus multiplying by preserves distance.
(Step 4) Let . Consider the triple of distances . These are the three distances of from and , respectively.
It is noted that this is the only point on the complex plane which has this particular triple of distances from and . Now, since multiplying by preserves distances. Note that the triangle with vertices is obtained by rotating the triangle with by an angle . If a point has a distance from equal to , then that point must also be obtained by rotating by an angle .
Thus, we conclude that multiplication by has the effect of rotating points in the complex plane by . ■
Next, consider multiplying
If we define
Comments
Post a Comment