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,
Every non zero complex number a + ib, with a^{2} + b^{2}≠0, has a unique multiplicative
inverse.
--1-- = a2−-ib2 = -2a--2 − i-2-b-2.
a+ ib a + b a + b a + b
You should prove the following theorem.
Theorem 1.12.1The complex numbers with multiplication and additiondefined as above form a field satisfying all the field axioms.These are the followingproperties.
x + y = y + x, (commutative law for addition)
x + 0 = x, (additive identity).
For each x ∈ ℝ, there exists −x ∈ ℝ such that x +
(− x)
= 0, (existence ofadditive inverse).
(x +y)
+ z = x +
(y+ z)
,(associative law for addition).
xy = yx,(commutative law for multiplication). You could write this asx × y = y × x.
(xy)
z = x
(yz)
,(associative law for multiplication).
1x = x,(multiplicative identity).
For each x≠0, there exists x^{−1}such that xx^{−1} = 1.(existence of multiplicativeinverse).
x
(y +z)
= xy + xz.(distributive law).
Something which satisfies these axioms is called a field. In this book, the field of most
interest will be the field of real numbers. You have seen in earlier courses that the
set of real numbers with the usual operations also satisfies 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 .
Observation 1.12.2The conjugate of a sum of complex numbers equals the sum ofthe complex conjugates and the conjugate of a product of complex numbers equals theproduct of the conjugates. To illustrate, consider the claim about the product.
Showing the claim works for a sum is left for you. Of course this means the conclusionholds for any finite product or finite sum. Indeed, for z_{k}a complex number, theassociative law of multiplication above gives