Download Math A1a (Calculus) Complex numbers Dr. Mohamed Khaled 1

Math A1a (Calculus)
Complex numbers
Dr. Mohamed Khaled
1. Suppose z is any complex number. Is it always true that ez is positive? Is it always true that
ez is a real number?
2. Verify directly from the definition that
1
eit
= e−it for any t ∈ R.
3. The imaginary part of a complex number is known to be twice its real part. The absolute
value of this number is 4. Which number is this?
4. The midpoint of an equilateral triangle is the origin. One of its vertices is the complex number
z1 = 1 + i. Determine the other two vertices z2 and z3 .
5. Compute the following complex numbers (by hand) in the algebraic form, then draw them in
the Argand plane.
• i2 ; i3 ; i4 ; 1i .
• (1 + 2i)(2 − i).
• (1 + i)(1 + 2i)(1 + 3i).
√
√
• ( 21 2 + 2i 2)2 .
√
• ( 12 + 2i 3)3 .
•
1
5
1+i ; 2−i .
6. Express the following complex numbers in the trigonometric form, then draw the in the
complex plane.
√ √
(a) z = (1 + i)(1 + i 3)( 3 − i).
(b) z =
√
(1+i)5 (1−i 3)5
√
.
( 3+i)4
7. Find formulas for cos 6θ, sin 4θ, cos 5θ and sin 6θ in terms of sin θ and cos θ using de
Moivre’s theorem.
8. Find and draw (in the complex plane) all real and complex solutions of:
(a) z 2 + 6z + 10 = 0.
(b) z 3 + 8 = 0.
(c) z 3 + 125 = 0.
(d) 2z 2 + 4z + 4 = 0.
(e) z 4 + 2z 2 − 3 = 0.
(f) 3z 6 = z 3 + 2.
(g) z 5 − 16z = 0.
9.
√
• Find the square roots of 1 + i 3.
• Find the fourth roots of i.
• Find the cube roots of −8i.
• Find the fourth roots of 2 − 2i.
10. Describe the set of points determined by z satisfy the condition:
(a) | z |= 1.
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Math A1a (Calculus)
Complex numbers
Dr. Mohamed Khaled
(b) | z − i |< 2.
(c) Re(z) = 1.
(d)
π
4
< arg(z) <
π
2
and 1 <| z |≤ 2.
(e) | z |≤| z + 2i |.
11. Factorize the polynomial z 5 − 1 as a product of real linear and quadratic factors.
12. Give a complex number for which 1 − i is a fourth root, then give the other fourth roots.
13. Prove that if zw = 0, then at least one of the factors (i.e., either z or w) is zero.
14. Prove the following for any complex numbers z and w.
• | z + w |≤| z | + | w |.
• | z − w |≥|| z | − | w ||.
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