8 1. The Complex Numbers

Exercise Set 1.1

1. Express (3 + i) + (2 − 7i) and (3 + i)(2 − 7i) in the standard form x + yi.

2. Express

1

3 + 5i

in the standard form x + yi.

3. Express (1 +

2i)2

and (1 +

2i)−2

in the standard form x + yi.

4. Express

2 − 3i

3 + 2i

in the standard form x + yi.

5. Find a square root for i.

6. If z = x + iy, express

z3

in standard form.

7. By direct substitution, prove that the two solutions to the quadratic equation

given in Example 1.1.3 really do satisfy equation (1.1.1).

8. Prove Part (e) of Theorem 1.1.4.

9. Prove Parts (a), (b) and (c) of Theorem 1.1.7.

10. Prove Parts (d), (e) and (f) of Theorem 1.1.7.

11. For z ∈ C and a ∈ R prove the following:

(a) Re(az) = a Re(z) and Im(az) = a Im(z);

(b) Re(iz) = − Im(z) and Im(iz) = Re(z).

12. Prove Theorem 1.1.8 – that is, prove that if z = x + iy, then

max{|x|, |y|} ≤ |z| ≤ |x| + |y|.

13. Graph the set of points z ∈ C which satisfy the equation |z − i| = 1.

14. Graph the set of points z ∈ C which satisfy the equation

z2

+

z2

= 2. Hint: If

z = x + iy, rewrite this equation as an equation in x and y.

15. Prove that if z is a non-zero complex number, then 1/z = 1/z and |1/z| = 1/|z|.

16. If z is any non-zero complex number, prove that z/z has modulus one.

17. Prove that every complex number of modulus 1 has the form cos θ + i sin θ for

some angle θ.

18. Prove that every line or circle in C is the solution set of an equation of the form

a|z|2

+ wz + wz + b = 0,

where a and b are real numbers and w is a complex number. Conversely, show

that every equation of this form has a line, circle, point, or the empty set as its

solution set.

1.2. Convergence in C

We assume the reader is familiar with the basics concerning convergent sequences

and series of real numbers – particularly those results which follow from the com-

pleteness of the real number system, such as the fact that bounded monotone

sequences converge and the various convergence tests for series. We also assume

a familiarity with the basics of power series in a real variable. The purpose of