Page 16 - Linear Algebra Done Right

Chapter 1. Vector Spaces
2
Complex Numbers
You should already be familiar with the basic properties of the set R
of real numbers. Complex numbers were invented so that we can take
square roots of negative numbers. The key idea is to assume we have
The symbol i was first a square root of −1, denoted i, and manipulate it using the usual rules
√
used to denote −1 by of arithmetic. Formally, a complex number is an ordered pair (a, b),
the Swiss where a, b ∈ R, but we will write this as a + bi. The set of all complex
mathematician numbers is denoted by C:
Leonhard Euler in 1777.
C ={a + bi : a, b ∈ R}.
If a ∈ R, we identify a + 0i with the real number a. Thus we can think
of R as a subset of C.
Addition and multiplication on C are defined by

(a + bi) + (c + di) = (a + c) + (b + d)i,
(a + bi)(c + di) = (ac − bd) + (ad + bc)i;
here a, b, c, d ∈ R. Using multiplication as defined above, you should
verify that i 2 =−1. Do not memorize the formula for the product
of two complex numbers; you can always rederive it by recalling that
2
i =−1 and then using the usual rules of arithmetic.
You should verify, using the familiar properties of the real num-
bers, that addition and multiplication on C satisfy the following prop-
erties:

commutativity
w + z = z + w and wz = zw for all w, z ∈ C;

associativity
(z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ) and (z 1 z 2 )z 3 = z 1 (z 2 z 3 ) for all
z 1 ,z 2 ,z 3 ∈ C;

identities
z + 0 = z and z1 = z for all z ∈ C;

additive inverse
for every z ∈ C, there exists a unique w ∈ C such that z + w = 0;

multiplicative inverse
for every z ∈ C with z = 0, there exists a unique w ∈ C such that
zw = 1;