Additive Group of Complex Numbers
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Contents |
Theorem
Let $\C$ be the set of complex numbers.
The structure $\left({\C, +}\right)$ is an infinite abelian group.
Proof
Taking the group axioms in turn:
G0: Closure
G1: Associativity
Complex Addition is Associative.
G2: Identity
From Complex Addition Identity is Zero, we have that the identity element of $\left({\C, +}\right)$ is the complex number $0 + 0 i$:
- $\left({x + i y}\right) + \left({0 + 0 i}\right) = \left({x + 0}\right) + i \left({y + 0}\right) = x + i y$
and similarly for $\left({0 + 0 i}\right) + \left({x + i y}\right)$.
G3: Inverses
From Inverses for Complex Addition, the inverse of $x + i y \in \left({\C, +}\right)$ is $-x - i y$.
C: Commutativity
Complex Addition is Commutative.
Infinite
Complex Numbers are Uncountable.
$\blacksquare$
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.1$
- Ian D. Macdonald: The Theory of Groups (1968): $\S 1$: Example $1.04$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (1)$
- John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.4$
