Math 2.0

Field theory and polynomials Theorem

The reals form an ordered field

Builds on The reals form a fieldOrder on the realsOrdered field

Statement

The real numbers ℝ\mathbb{R}, with their addition, multiplication and order ≤\le, form an ordered field. That is, ≤\le is a total order with two properties for all real numbers aa, bb and cc. If a≤ba \le b, then a+c≤b+ca + c \le b + c. If 0≤a0 \le a and 0≤b0 \le b, then 0≤ab0 \le a b.

Remarks

The ordered field ℝ\mathbb{R} contains ℚ\mathbb{Q} as an ordered subfield. The order of ℝ\mathbb{R} is the only order that makes ℝ\mathbb{R} an ordered field. The nonnegative reals are exactly the squares.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectRing of integers, statement, no evidence levelRing of integersstatementThe reals form a field, statement, no evidence levelThe reals form a fieldstatementThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementOrder on the reals, definition, no evidence levelOrder on the realsdefinitionThe reals form an ordered field, statement, no evidence levelThe reals form an ord…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Integers object
  3. Ordered field definition
  4. Rational numbers object
  5. Real numbers object
  6. Ring of integers statement
  7. The reals form a field statement
  8. The rationals form a field statement
  9. The rationals form an ordered field statement
  10. Order on the reals definition
  11. The reals form an ordered field statement

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordered_field, Examples of ordered fields
About this page
Tag0QSPNASE50
URIm2://0QSPNASE50
Kindstatement, theorem
WritingW0 stub
MSC12J15
StatusThis copy of the site has no build data. Run m2 status 0QSPNASE50.
UsesThe reals form a field, Order on the reals, Ordered field

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