1 Complex Numbers
1.1 The Complex Number System
Definitions 1.1 Let R = {a, b, c, ... } be the set of real numbers, and let
R^2 =IR; x IR;= {(a,b),(a,c),(b,c), ... } be the set of all ordered pairs of
real numbers. The complex number system C is a field whose elements are
those of R^2 , with equality, addition, and multiplication defined as follows:
- Equality: (a, b) = ( c, d) iff a = c and b = d.
2. Addition: (a,b)+(c,d) = (a+c,b+d).
- Multiplication: (a, b) · ( c, d) = (a, b )( c, d) = ( ac - bd, ad+ be).
Briefly, C = {~^2 , =, +, · }. Thus C is obtained from R^2 by imposing
on ~^2 a certain algebraic structure. By subjecting R^2 to other algebraic
structures other algebraic systems will result (see Sections 1.5 and 1.18).
Each pair (a, b) E C is called a complex number. The real numbers a
and bare called the components of the complex number, a being the first
component, b the second. For convenience, a complex number is usually
denoted by a single letter. Thus we may use z for the complex number
(x,y) and write z = (x,y).
That the system C is a field means that the following properties are
satisfied:
(a) Equality is a determinative relation that is reflexive, symmetric, and
transitive: i.e., given any two complex numbers z1 and z2, either z1 =7