Chapter 1

Construction of set C of complex numbers

Def: an orderly pair z= (a, b) of real numbers it is called COMPLEX NUMBER, and the set C=R2= {(a, b) | a, bR} it is called the SET of COMPLEX NUMBERS.

If (a, b), (a’, b’) C, then (a, b) = (a’, b’) if and only if a=a’ and b=b’.

z+ z’= (a, b) + (a’, b’) = (a+ a’, b+ b’)

MULTIPLICATION

z× z’= (aa’- bb’, ab’+ a’b)

The set C which has the 2 properties presented above has also the following properties:

1.0 For every z1, z2C, z1+ z2C(consistence of the operation of the multiplication)

1.1 For every z1, z2, z3C (z1+ z2) + z3= z1+ (z2+ z3) (associatively)

1.2 There is in C an element with “neutral effect” for addition (neutral element);

0= (0, 0), which means that for every zC, z+ 0= 0+ z= z;

1.3 For every element zC is another (-z) C, with the property that

z + (-z) = (-z) + z=0; in this case (-z) is called the opposite of z, and we can notice that if z=(a, b) then –z=(-a, -b);

Def: In general, a set G which has an operation “*” (for every pair of distinct or the same elements form G, taken in a particular order, it is associated after a law, a third element, well determined, from the same set G: x, yG → x× yG) which satisfies the axioms 1.0-1.3, it is called group and it has the symbol (G, *).

1.4 For every z1, z2C, z1+ z2 = z2+ z1 (commutatively). In particular we can say that (C, +) is commutative group (additive).

2.0 For every z1, z2C, z1× z2C, (consistence of multiplication)

2.1 For every z1, z2, z3C (z1 ×z2) × z3= z1 × (z2× z3) (associatively)

2.2 There is in C an element with “neutral effect” for addition (neutral element);

1= (1, 0), which means that for every zC, z×1= 1×z= z;

2.3 For every element zC*=C\ {0} is another (z) C, with the property that

z × (z)  = (z) ×z=1; in this case (z) is called the inverse of z, and we can notice that if z=(a, b) then z=;

Def: In general, a set M with an operation “*” which satisfies the axioms

2.0-2.2 (similar to 1.0-1.2) it is called monomial and it has the symbol (M, *);

2.4 For every z1, z2C, (z1∙z2) = (z2∙z1) (commutative). In particular we can say that (C, ∙) is multiplicative.

3.0 For every z1, z2, z3C, we have:

3.1 z1∙ (z2+z3) = (z1∙z2) + (z1∙z3);

3.2 (z2+z3) ∙ z1= (z2∙z1) + (z3∙z1);

Relations which express that the operation of multiplication (the second) is distributive at left and right from the addition (the first!).

In general, a set A with an orderly pair of operations: first (“+”) and the second (“∙”) which satisfies the axioms 1.0-1.4. ((A, +) - commutative group), 2.0-2.2. ((A, ∙) - monomial) and 3.1-3.2, it is called ring and it has the symbol (A, +, ∙). If it also satisfies the axiom 2.3, then the ring (A, +, ∙) it is called body.

In particular we can say that (C, +, ∙) is commutative body!

Def: In general, in a ring (A, +, ∙) the neutral element towards the first operation (addition) has the symbol 0 and it is called the symbol of the ring, and the neutral element towards the second operation (multiplication) has the symbol 1 and it is called the unity of the ring.

We’ll write: i =(0, 1)- imaginary unity; already 1=(1, 0)- real unity; for every xR, x=(x, 0). In particular, 0=0, 1=1 we notice that, for every z=(x, y) C there is the relation z=x+ y∙i.

Def: z1-z2 = z1+(-z2) it is called difference of complex numbers z1 and z2 (difference is a consistent operation over C,  non- commutative).