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Section 21.2 Ordered fields
Objectives
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Explain how axioms provide a precise foundation for number systems.
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Use the field axioms to justify basic arithmetic properties.
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Identify when a field can be ordered and what the order axioms require.
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Motivation: We need to start somewhere. Discuss the role of axioms in building a logical system. Alternatively: what the heck
are numbers?
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Field axioms:
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Commutativity of addition and multiplication
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Distributivity of multiplication over addition
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Associativity of addition and multiplication
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Existence of additive and multiplicative identities
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Existence of additive and multiplicative inverses
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There is a nonempty subset \(P\subseteq F\text{,}\) called the positive numbers, such that:
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If
\(a,b\in P\text{,}\) then
\(a+b\in P\) and
\(a\cdot b\in P\text{.}\)
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If
\(a\in F\) and
\(a\neq 0\text{,}\) then
\(a\in P\) or
\(-a\in P\text{,}\) but not both.
A field that satisfies this axiom is called an ordered field.
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Real numbers are ordered. Complex numbers are not.
