Definition:Zero (Algebra)
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Definition
The number zero is defined as being the cardinal of the empty set.
Establish the equivalence of this definition and that for the number fields as introduced below.
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Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.
Then from NO 1, $\left({S, \circ, \preceq}\right)$ has a smallest element.
This smallest element of $\left({S, \circ, \preceq}\right)$ is called zero and has the symbol $0$.
That is:
- $\forall n \in S: 0 \preceq n$
Natural Numbers
Integers
Rational Numbers
Real Numbers
Complex Numbers
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Still to be done: Define this in the context of the standard number fields.
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