Math 2.0

statement theorem

The rationals form an ordered field

Statement

Define an order on the rational numbers as follows. For integers a,ca, c and positive integers b,db, d, let ab≤cd\frac{a}{b} \le \frac{c}{d} when ad≤bca d \le b c in Z\mathbb{Z}. This relation is well defined, and with it Q\mathbb{Q} is an ordered field.

Remarks

The order on Q\mathbb{Q} extends the order on the integers. This is the only order on Q\mathbb{Q} that makes it an ordered field. The ordered field Q\mathbb{Q} is Archimedean, but it is not complete: the set {q∈Q:q2<2}\{q \in \mathbb{Q} : q^2 < 2\} has no least upper bound in Q\mathbb{Q}.

Explanations

L1 human

To compare two fractions, write them with positive denominators and cross-multiply. For example, 2/3<3/42/3 < 3/4 because 2⋅4=8<9=3⋅32 \cdot 4 = 8 < 9 = 3 \cdot 3.

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Integers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectRing of integers, statement, no evidence levelRing of integersstatementThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statement
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Sources

  1. Web https://en.wikipedia.org/wiki/Ordered_field, Examples of ordered fields
  2. Web https://en.wikipedia.org/wiki/Rational_number, Ordering

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