Measure, Integration & Real Analysis, 2021a

Section 6B Vector Spaces 155
6B
Vector Spaces
Integration of Complex-Valued Functions
Complex numbers were invented so that we can take square roots of negative numbers.
The idea is to assume we have a square root of −1, denoted i, that obeys the usual
rules of arithmetic. Here are the formal definitions:
6.17 Definition complex numbers; C; addition and multiplication in C
• A complex number is an ordered pair (a, b), where a, b ∈ R, but we write
this as a + bi.
• The set of all complex numbers is denoted by C:
C = {a + bi : a, b ∈ R}.
• Addition and multiplication in C are defined by
here a, b, c, d ∈ R.
If a ∈ R, then we identify a + 0i
with a. Thus we think of R as a subset of
C. We also usually write 0 + bi as bi, and
we usually write 0 + 1i as i. You should
verify that i 2 = −1.
(a + bi)+(c + di) =(a + c)+(b + d)i,
(a + bi)(c + di) =(ac − bd)+(ad + bc)i;
The √ symbol i was first used to denote
−1 by Leonhard Euler
(1707–1783) in 1777.
With the definitions as above, C satisfies the usual rules of arithmetic. Specifically,
with addition and multiplication defined as above, C is a field, as you should verify.
Thus subtraction and division of complex numbers are defined as in any field.
The field C cannot be made into an ordered
field. However, the useful concept
of an absolute value can still be defined
on C.
Much of this section may be review
for many readers.
6.18 Definition real part; Re z; imaginary part; Im z; absolute value; limits
Suppose z = a + bi, where a and b are real numbers.
• The real part of z, denoted Re z, is defined by Re z = a.
• The imaginary part of z, denoted Im z, is defined by Im z = b.
• The absolute value of z, denoted |z|, is defined by |z| = √ a 2 + b 2 .
• If z 1 , z 2 ,...∈ C and L ∈ C, then lim
k→∞
z k = L means lim
k→∞
|z k − L| = 0.
Measure, Integration & Real Analysis, by Sheldon Axler