Complex Numbers

• Benjamin Fine
• Gerhard Rosenberger
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Recall that a field F is a set with two binary operations, addition, denoted by +, and multiplication denoted by. or just by juxtaposition, defined on it satisfying the following nine axioms:
1. (1)

Addition is commutative: a +b = b + a for each pair a, b in F.

2. (2)

Addition is associative: a + (b + c) = (a + b) + c for a, b, c ∈ F.

3. (3)

There exists an additive identity, denoted by 0, such that a + 0 = a for each a ∈ F.

4. (4)

For each a E F there exists an additive inverse denoted —a, such that a + (-a) = 0.

5. (5)

Multiplication is associative: a(bc) = (ab)c for a, b, c ∈ F.

6. (6)

Multiplication is distributive over addition: a(b + c) = ab + ac for a, b, c ∈ F.

7. (7)

Multiplication is commutative: ab = ba for each pair a, b in F.

8. (8)

There exists an multiplicative identity denoted by 1 (not equal to 0) such that al = a for each a in F.

9. (9)

For each a ∈ F, with a ≠ 0 there exists a multiplicative inverse denoted by a-1, such that aa-1 = 1.

Keywords

Complex Number Arithmetic Operation Cauchy Sequence Polar Form Multiplicative Identity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.