# 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.

Dinates

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