The University of Sydney
School of Mathematics and Statistics
Board Tutorial 1
MATH1021: CALCULUS OF ONE VARIABLE
REAL AND COMPLEX NUMBERS
1. Let X = { n ∈ Z | n2 ≤ 5 }.
(a) Rewrite X as an explicit set of numbers.
(b) Decide which of the following statements are true and which are false:
(i) X ⊆ Z
(ii) X ⊇ Z
(iii) 5 ∈ X
(iv) −2 6∈ X
2. If z = 2 − i and w = −4 + 3i, find
(a) z + w
(b) z − w
(c) |z|
(d) w
(e) zw
(f)
z
w
3. Locate each of the following sets on the real number line and then express each as
an interval or as a union of intervals:
(a) x ∈ R | 2 ≤ x ≤ 4
(c) [2, 5] ∩ (3, 6]
(b) x ∈ R | −1 < x ≤ 1 or x ≥ 5
(d) x ∈ R | |x − 1| > 2
4. Express the following complex numbers in Cartesian form:
3i
1
−
i
1−i
(d) i123 − 4i9 − 4i
1+i
1−i
(b) (2 + 3i)(5 − 6i)
(c)
(a)
5. Solve the following equations over C:
(a) 3z 2 − 4z + 4 = 0
(b) z 4 = 1.
6. Sketch the following regions in the complex plane:
(a) { z ∈ C | |z| ≤ 3 }
(c) { z ∈ C | |z − i| ≤ |z − 1| }
(b) { z ∈ C | Im(z) ≥ −1 }
7. In each case decide whether or not the statement is true.
swer.
(a) The square of an imaginary number is always real.
Explain your an-
(b) It does not make sense to write |z| > |w| when z and w are complex numbers
because the complex numbers are not ordered.
(c) Real numbers cannot be graphed on the complex plane.
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(d) When a real number is divided by a complex number the answer can never
be real.
8. Use a Venn diagram with three sets A, B and C to show the following:
(a) A ∪ B ∪ C
(b) (A ∪ B) ∩ C
Brief answers to selected exercises:
1. (a) X = {−2, −1, 0, 1, 2}
(b) (i)
(ii)
(iii)
(iv)
True
False
False
False
2. (a) −2 + 2i
(b) 6 − 4i
√
5
(c)
(d) −4 − 3i
(e) −5 + 10i
2
11
(f) − − i
25 25
4. (a) i
(b) 28 + 3i
3 5
(c)
− i
2 2
(d) −9i
√
2 2 2
5. (a)
±
i
3
3
(b) {1, −1, i, −i}
7. (a) True
(b) False
(c) False
(d) False
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