1550251515-Classical_Complex_Analysis__Gonzalez_

8 Chapter^1

z2 or z1 -/:-z2; also, z1 = z1 (reflexive property), z1 = z2 implies that

z2 = z1 (symmetric property), and z1 = z2, z2 = za implies that

z1 = za (transitive property).

(b) Addition and multiplication are uniquely defined, and both the sum

z 1 + z 2 and the product z 1 z 2 belong to C (closure property).

( c) Addition and multiplication are commutative: i.e.,

and

(cl) Addition and multiplication are associative: i.e.,

and

( e) Multiplication is distributive with respect to addition: i.e.,

(z1 + z2)za = z1za + z2za

(f) Given any two complex numbers z1 and. z2, there exists just one

complex number z such that

(1.1-1)

(g) Given any two complex numbers z 1 and z 2 , with z 1 -/:- (0, 0), there

exists just one complex number z such that

(1.1-2)

The validity of the preceding properties for complex numbers can easily

be verified by applying the definitions of equality, addition, and/or multi-

plication, together with the fact that the real components of the complex

numbers do satisfy the same laws, since IR itself is a field. We leave this

straightforward verification to the reader.

The number z satisfying equation (1.1~1), called the difference between

z 2 and z 1 , can be determined as follows: Let z 1 = (a, b ), z 2 = ( c, d), and

z = (x, y). By the definition of addition, equation (1.1-1) gives

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

and the definition of equality implies that

a+x = c,

Hence there is just one solution given by

x = c-a, y=d-b

Thus z = ( c - a, d - b). That this number is the difference between z 2 and

z 1 is denoted by writing z 2 - z 1 = z, or equivalently,

( c, d) -(a, b) = ( c - a, d - b) (1.1-3)