Calculus Tutorial: Real & Complex Numbers - University Level

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 Xas 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) −26∈ X
2. If z= 2 −iand 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
(b) x∈R| −1< x ≤1 or x≥5
(c) [2,5] ∩(3,6]
(d) x∈R| |x−1|>2
4. Express the following complex numbers in Cartesian form:
(a) 1 + i
1−i
(b) (2 + 3i)(5 −6i)
(c) 1
i−3i 1−i
(d) i123 −4i9−4i
5. Solve the following equations over C:
(a) 3z2−4z+ 4 = 0 (b) z4= 1.
6. Sketch the following regions in the complex plane:
(a) {z∈C| |z| ≤ 3}
(b) {z∈C|Im(z)≥ −1}
(c) {z∈C| |z−i|≤|z−1|}
7. In each case decide whether or not the statement is true. Explain your an-
swer.
(a) The square of an imaginary number is always real.
(b) It does not make sense to write |z|>|w|when zand ware 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,Band Cto 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) True
(ii) False
(iii) False
(iv) False
2. (a) −2+2i
(b) 6 −4i
(c) √5
(d) −4−3i
(e) −5 + 10i
(f) −11
25 −2 25i
4. (a) i
(b) 28 + 3i
(c) 3
2−5 2i
(d) −9i
5. (a) 2
3±2√2 3i
(b) {1,−1, i, −i}
7. (a) True
(b) False
(c) False
(d) False
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