Complex Numbers

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


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.



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.


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Benjamin Fine
    • 1
  • Gerhard Rosenberger
    • 2
  1. 1.Department of MathematicsFairfield UniversityFairfieldUSA
  2. 2.Department of MathematicsUniversity of DortmundDortmundGermany

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