Download Complex Numbers Worksheet

Math 285
Complex Number System
June 27, 2011
1. If z1 = 1 + 2i and z2 = 3 + 4i are complex numbers, compute:
(a) z1 + z2
(b) z1 − z2
(c) z2 − z1
(d) z1 z2
1
(e)
z1
z2
(f)
z1
2. Show each of the complex numbers from the previous problem in the complex
plane.
3. Suppose z is the conjugate of z, show that
(a) z1 ± z2 = z1 ± z2
(b) z1 z2 = z1 z2
(c) zz = |z|2
4. Let a0 , a1 , a2 , . . . , an be real constants. Show that if z0 is a root of the polynomial
equation an z n + an−1 z n−1 + · · · + a0 = 0, then so is z0 .
5. Application of Euler’s formula: Prove the following identities.
(a) cos(A + B) = cos A cos B − sin A sin B and
sin(A + B) = sin A cos B + cos A sin B
(Hint: consider z1 = Ai and z2 = Bi where A and B > 0.)
(b) cos(A − B) = cos A cos B + sin A sin B and
sin(A − B) = sin A cos B − cos A sin B
(Hint: The cosine and sine functions are even and odd, respectively.)
(c) State a single expression for both cos(A ± B) and sin(A ± B)
Math 285
Complex Number System
June 27, 2011
6. Use the previous identities to derive the following identities:
(a) cos(A + B) + cos(A − B) = 2 cos A cos B
(b) sin(A + B) + sin(A − B) = 2 sin A cos B
(c) sin(2θ) = 2 sin θ cos θ
(d) cos(2θ) = cos2 θ − sin2 θ
7. Show cos x =
eix + e−ix
, and derive a similar relationship for sin x.
2
8. Write each of the following complex numbers in polar form.
(a) −3 + 3i
(b) −πi
√
(c) −2 3 − 2i
9. Suppose |ẑ| = 1, interpret geometrically what multiplying by ẑ does to any
other complex number z.