1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 11

1.2 REAL AND IMAGINARY COMPLEX NUMBERS.

THE COMPLEX UNITS

Complex numbers of the special form (a, 0) behave as the real numbers and

are called real complex numbers. More precisely, the set of real complex

numbers is isomorphic to the set R of the real numbers under the one-to-one

mapping g : (a, 0) --4 a. In fact, we have

1. (a, 0) = (a', 0) iff a = a^1


2. (a,O) + (a',O) =(a+ a',O) --4 a+ a'


3. (a,O)(a',O) = (aa',O) --4 aa'
   so that equality, as well as the operations of addition and multiplication,
   are preserved under this mapping.
   Because of such isomorphism the system R of the real numbers can be
   embedded in the system C of the complex numbers simply by replacing each
   real number a by the corresponding complex number (a, 0). Conversely,
   the complex number (a, 0) may be replaced by the real number a whenever
   convenient. For simplicity of notation we usually do so. For instance, we
   write 0 to denote the zero complex number (0, 0), and 1 instead of (1, 0).
   In accordance with this convention, by the sum r+(a, b) of a real number
   r and a complex number (a, b ), the sum ( r, 0) + (a, b) is understood, i.e.,

r + (a,b) = (r,O) + (a,b) = (r + a,b) (1.2-1)

Similarly, we have

r(a, b) = (r, O)(a, b) = (ra, rb) (1.2-2)

Complex numbers (a, b ), with b i= 0, are called imaginary complex num-

bers t (also non real complex numbers). In particular, complex numbers of

the form (0, b), with b i= 0, are said to be pure imaginary numbers.

The complex numbers

U1 = (1, 0) and Uz=(0,1)

are of special importance and are called complex units.

Since

(a, b) =(a, 0) + (0, b) = a(l, 0) + b(O, 1) = au1 + bu2 (1.2-3)

tThis unfortunate terminology survives as a result of a historical misconception
about complex numbers. For an account of the history of complex numbers, we
refer the reader to [22]. Numbers in brackets refer to the bibliography at the
end of the chapter.