Complex Arithmetic

Exercises

These complex numbers

C = 1 + 2i D = 2 + 2i

A = 1 + i B = 2 + i

lie on the vertices of a square. Plot them on a piece of graph paper. All the exercises will be based on these numbers.

1. (a) Calculate (1 + i)*A, (1 + i)*B, (1 + i)*C, and (1 + i)*D and plot these points.

(b) Calculate 2i*A, 2i*B, 2i*C, and 2i*D and plot these points.

(d) By examples convince yourself that for any complex numbers u, v, and w, if c lies on the line beteen u and v, then w*c lies on the line between w*u and w*v. Can you prove this in general? Use this information to find the images of the box with corners A, B, C, and D under the multiplications in (a), (b), and (c).

Here is the answer.

2. (a) Calculate (.707 + .707i)*A, (.707 + .707i)*B, (.707 + .707i)*C, and (.707 + .707i)*D and plot these points.

(b) Calculate i*A, i*B, i*C, and i*D and plot these points.

(d) Find the images of the box with corners A, B, C, and D under the multiplications in (a), (b), and (c).

Here is the answer.

3. (a) Calculate (.5 + .5i)*A, (.5 + .5i)*B, (.5 + .5i)*C, and (.5 + .5i)*D and plot these points.

(b) Calculate .5i*A, .5i*B, .5i*C, and .5i*D and plot these points.

(d) Find the images of the box with corners A, B, C, and D under the multiplications in (a), (b), and (c).

Here is the answer.

4. What are the relations between the multipliers in (a), (b), and (c) for each problem? Use this information to interpret the pictures.

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