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MAT067 University of California, Davis Winter 2007
Homework Set 1: Exercises on Complex Numbers
Directions: You are assigned the Calculational Problems 1(a, b, c), 2(b), 3(a, b), 4(b,
c), 5(a, b), and the Proof-Writing Problems 8 and 11.
Please submit your solutions to the Calculational and Proof-Writing Problems separately
at the beginning of lecture on Friday January 12, 2007. The two sets will be graded by
different persons.
- Express the following complex numbers in the form x + yi for x, y โ R:
(a) (2 + 3i) + (4 + i)
(b) (2 + 3i)
2 (4 + i)
(c)
2 + 3i
4 + i
(d)
i
1 + i
(e) (โi)
โ 1
- Compute the real and imaginary parts of the following expressions, where z is the
complex number x + yi and x, y โ R:
(a)
z^2
(b)
3 z + 2
(c)
z + 1
2 z โ 5
(d) z
3
- Solve the following equations for z a complex number:
(a) z
5 โ 2 = 0
(b) z
4
(c) z
6
(d) z
3 โ 4 i = 0
- Calculate the
(a) complex conjugate of the fraction (3 + 8i)
4 /(1 + i)
10 .
(b) complex conjugate of the fraction (8 โ 2 i)
10 /(4 + 6i)
5 .
(c) complex modulus of the fraction i(2 + 3i)(5 โ 2 i)/(โ 2 โ i).
(d) complex modulus of the fraction (2 โ 3 i)
2 /(8 + 6i)
2 .
- Compute the real and imaginary parts:
(a) e
2+i
(b) sin(1 + i)
(c) e
3 โi
(d) cos(2 + 3i)
- Compute the real and imaginary part of e
ez for z โ C.
- Let a โ R and z, w โ C. Prove that
(a) Re(az) = aRe(z) and Im(az) = aIm(z).
(b) Re(z + w) = Re(z) + Re(w) and Im(z + w) = Im(z) + Im(w).
- Let z โ C. Prove that Im(z) = 0 if and only if Re(z) = z.
- Let p be a polynomial with real coefficients and z โ C.
Show that p(z) = 0 if and only p(z) = 0.
- Let z, w โ C. Prove the parallelogram law |z โ w|
2
2 = 2(|z|
2
2 ).
