Recall that a real number is a point on the real number line. Just as a real number should be
considered as a point on the line, a complex number is considered a point in the plane which can be
identified in the usual way using the Cartesian coordinates of the point. Thus
(a,b)
identifies a
point whose x coordinate is a and whose y coordinate is b. In dealing with complex numbers,
such a point is written as a + ib. For example, in the following picture, I have graphed the
point 3 + 2i. You see it corresponds to the point in the plane whose coordinates are
(3,2)
.
PICT
Multiplication and addition are defined in the most obvious way subject to the convention that i^{2} = −1.
Thus,
(a+ ib) + (c +id) = (a +c)+ i(b+ d)
and
(a + ib)(c+ id) = ac+ iad + ibc +i2bd
= (ac − bd) +i(bc+ ad).
Every non zero complex number a + ib, with a^{2} + b^{2}≠0, has a unique multiplicative inverse.
Theorem 1.5.1The complex numbers with multiplication and addition defined as above form afield satisfying all the field axioms.These are the following list of properties. In this list, F is thesymbol for a field.
x + y = y + x, (commutative law for addition)
There exists 0 such that x + 0 = x for all x, (additive identity).
For each x ∈ F, there exists −x ∈ F such that x +
(− x)
= 0, (existence of additive inverse).
(x +y)
+ z = x +
(y+ z)
,(associative law for addition).
xy = yx,(commutative law for multiplication). You could write this as x × y = y × x.
(xy)
z = x
(yz)
,(associative law for multiplication).
There exists 1 such that 1x = x for all x,(multiplicative identity).
For each x≠0, there exists x^{−1}such that xx^{−1} = 1.(existence of multiplicative inverse).
x
(y +z)
= xy + xz.(distributive law).
When you have a field F some things follow right away from the above axioms.
Theorem 1.5.2Let F be a field. This means it satisfies the axioms of the above theorem. Then thefollowing hold.
0 is unique.
−x is unique
0x = 0
(− 1)
x = −x
x^{−1}is unique
Proof: Consider the first claim. Suppose
ˆ0
is another additive identity. Then
ˆ0 = ˆ0+ 0 = 0
and so sure enough, there is only one such additive identity. Consider uniqueness of −x next. Suppose y is
also an additive inverse. Then
− x = − x+ 0 = − x +(x + y) = (− x + x)+ y = 0 + y = y
so the additive inverse is unique also.
0x = (0+ 0)x = 0x+ 0x
Now add −0x to both sides to conclude that 0 = 0x. Next
0 = (1+ − 1)x = x +(− 1)x
and by uniqueness of −x, this implies
(− 1)
x = −x as claimed. Finally, if x≠0 and y is a multiplicative
inverse,
( )
x −1 = 1x−1 = (yx)x−1 = y xx−1 = y1 = y
so y = x^{−1}. ■
Something which satisfies these axioms is called a field. Linear algebra is all about fields, although in
this book, the field of most interest will be the field of complex numbers or the field of real numbers. You
have seen in earlier courses that the real numbers also satisfy the above axioms. The field of complex
numbers is denoted as ℂ and the field of real numbers is denoted as ℝ. An important construction
regarding complex numbers is the complex conjugate denoted by a horizontal line above the number. It is
defined as follows.
-----
a+ ib ≡ a − ib.
What it does is reflect a given complex number across the x axis. Algebraically, the following formula is
easy to obtain.
(a+-ib)(a+ ib) = (a − ib)(a+ ib)
2 2 2 2
= a + b − i(ab − ab) = a + b .
Definition 1.5.3Define the absolutevalue of a complex number as follows.
|a + ib| ≡ ∘a2-+-b2.
Thus, denoting by z the complex number z = a + ib,
- 1∕2
|z| = (zz) .
Also from the definition, if z = x + iy and w = u + iv are two complex numbers, then