1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 9

We note that

(e, d) - (a, b) = (e, d) +(-a, -b) (1.1-4)

so the difference between ( e, d) and (a, b) may be obtained by adding

(-a, -b) to (e, d).

The complex number (-a, -b) whose components are the negatives of

those of (a, b) is called the opposite or additive inverse of (a, b ). Clearly,

(a, b) +(-a, -b) = (0, 0)

The complex number (0,0) is the neutral element or zero with respect

to addition; that is, it is the only complex number satisfying the equation

(a, b) + (x, y) =(a, b)

The number z = (u,v) satisfying equation (1.1-2), namely,

(a, b)(u, v) = (e, d), (a, b) =fa (0, 0)

is called the quotient of Z2 = ( e, d) by z1 = (a, b ), and can also be
determined easily. By the definition of multiplication we have

(au - bv, bu+ av)= (e, d)

and by definition of equality, we obtain

au - bv = e

bu+ av= d

(1.1-5)

which is a •linear system of two equations in two unknowns. This system
has a unique solution provided that

l

a b -b1 a = a2 + b2 =fa 0

i.e., provided that (a, b) =fa (0, 0). Solving (1.1-5), we find that

ae+ bd

U= V=---,-,-

a2 + b2 ' a2 + b2

ad-be

(1.1-6)

Hence if we represent by f ~·.~~ = ( e, d) / (a, b) the quotient of ( e, d) by (a, b ),
we have

( e, d) _ ( ac + bd ad - be )

(a, b) - a^2 + b^2 ' a^2 + b2

(1.1-7)

The quotient of two complex numbers is not defined when (a, b) = (0, 0).
Certain standard names that are employed in operations with real
numbers are kept in use for the complex number system. For instance,
in (1.1-3), ( e, d) is called the minuend and (a, b) the subtrahend, while