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Oulun Lyseon lukio / Galois club 2010-2011 / Complex numbers part one / TL
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1. Real numbers
Given a straight line and two fixed points O and A on this line, we define real numbers as distances along
the line from the point O (origin or zero) using the distance OA as the measurement unit 1. Distances to the
points on the right or left side of O are called respectively positive or negative real numbers. It follows that
there is a one-to-one correspondence between the points of the line and real numbers. Each point P on the
number line corresponds to one and only one real number 𝑥 which is the distance from O to P.
O
A
P
0
1
x
FIGURE 1.1. The number line. The point P corresponds to the real number x.
It is easy to see that there are at least two kinds of real numbers
(i) integers n (the distance OP is an exact multiple of the unit distance OA)
(ii) rational numbers r (the distance OP is an exact multiple of a fraction
1
𝑛
of the unit distance OA)
In fact, the second set contains the first as a subset (case 𝑛 = 1).
According to a legend, it came as an unpleasant surprise to Pythagoras1 that there are distances (for
example the distance between the opposite vertices of the unit square) which are not exact multiples of
any fraction of the unit distance. Such distances and the corresponding real numbers are called irrational.
B
1
A
1
FIGURE 1: Unit square. The diagonal distance AB is irrational (√2 in the modern notation).
We denote the set of all real numbers by ℝ.
1
Pythagoras (ca. 600 BC) had developed, possibly inspired by his studies of musical harmony, a philosophical doctrine
that all relations in the universe can be expressed as ratios of integers (i.e. as rational numbers). The discovery of
irrational relations shattered the foundations of this philosophy.
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Oulun Lyseon lukio / Galois club 2010-2011 / Complex numbers part one / TL
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2. Extending the real line into the complex plane
The basic arithmetic operations (addition and multiplication as well as, consequently, subtraction and
division) of real numbers 𝑥 and 𝑦 can be naturally interpreted geometrically as certain operations of vectors
(arrows from the zero point to the points 𝑥 and 𝑦 on the number line). The point 𝑥 + 𝑦 is obtained by
placing the vectors 𝑥 and 𝑦 one after another and the point 𝑥𝑦 is obtained multiplying the lengths (absolute
values) of the vectors and adding their direction angles (which provides the correct sign). Such vector
operations can be naturally extended to the whole plane which thereby provides definitions for the sum
𝑧 + 𝑤 and product 𝑧𝑤 for all points 𝑧 and 𝑤 of the coordinate plane. It is fairly easy to show that these
operations follow exactly the same laws (e.g. commutativity, associativity and distributivity) as in ℝ. Hence
all usual mathematical machinery extends fully to these new numbers, complex numbers. The set of all
complex numbers, also called the complex plane, is usually denoted by ℂ. We have ℝ ⊂ ℂ .
Every complex number 𝑧 can be represented in the form 𝑧 = 𝑥 + 𝑦𝑖 where 𝑖 - the imaginary unit – is just a
name or symbol for the point (0,1) of the complex plane or, equivalently, for the vector from the origin to
that point. In the same way 1 is the name for the point (1,0) on the 𝑥-axis. This representation is very
useful because it allows us to do arithmetic in the old way just remembering that 𝑖 ⋅ 𝑖 = 𝑖 2 = −1 which fact
follows immediately from the vector definition of multiplication.
imaginary axis
𝑧 = 𝑥 + 𝑦𝑖
real axis
FIGURE 2: The complex plane
Exercise 1. Calculate a) the power 𝑖 𝑛 for 𝑛 = 3, 4, 15, 101 b) (1 + 𝑖)4 c) (1 + 𝑖)8 .
Exercise 2. The conjugate 𝑧̅ of the complex number 𝑧 = 𝑥 + 𝑦𝑖 is defined by 𝑧̅ = 𝑥 − 𝑦𝑖. a) Explain how a
given complex number and its conjugate are located in the complex plane. b) Show that 𝑧 + 𝑧̅ and 𝑧𝑧̅ are
1
1
+ 2 𝑖).
2
√
√
always real numbers. c) Use the fact that 𝑧𝑧̅ is always real to calculate the quotient 𝑖 ∶ (
d) Show
that |𝑧| = √𝑧𝑧̅. Here |𝑧| denotes the absolute value of 𝑧, that is the distance of 𝑧 from the origin.
Exercise 3. For two given angles 𝛼 and 𝛽 let 𝑧1 = cos 𝛼 + 𝑖 sin 𝛼 and 𝑧2 = cos 𝛽 + 𝑖 sin 𝛽. Calculate a) the
absolute values of 𝑧1 and 𝑧2 . b) the product 𝑧1 𝑧2 c) the quotient 𝑧1 /𝑧2 .
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