Math 2.0

Real functions Theorem

Density of the rationals

Builds on The reals are ArchimedeanRational numbers

Statement

Let xx and yy be real numbers with x<yx < y. Then there is a rational number qq with x<q<yx < q < y.

Remarks

Proof idea: choose nn with 1/n<y−x1/n < y - x by the Archimedean property. Then some fraction m/nm/n with m∈ℤm \in \mathbb{Z} lies between xx and yy. Thus between two real numbers there are infinitely many rational numbers. The irrationals are also dense.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
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 numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionRing of integers, statement, no evidence levelRing of integersstatementArchimedean ordered field, definition, no evidence levelArchimedean ordered f…definitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementSupremum, definition, no evidence levelSupremumdefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementCompleteness axiom, statement, no evidence levelCompleteness axiomstatementThe 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…statementThe reals are Archimedean, statement, no evidence levelThe reals are Archime…statementDensity of the rationals, statement, no evidence levelDensity of the ration…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. Set bounded above definition
  7. Ring of integers statement
  8. Archimedean ordered field definition
  9. The reals form a field statement
  10. Supremum definition
  11. The rationals form a field statement
  12. Completeness axiom statement
  13. The rationals form an ordered field statement
  14. Order on the reals definition
  15. The reals form an ordered field statement
  16. The reals are Archimedean statement
  17. Density of the rationals statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dense_order, Example
  2. Web https://en.wikipedia.org/wiki/Archimedean_property, Equivalent definitions of Archimedean ordered field
About this page
Tag0STX0WRDNG
URIm2://0STX0WRDNG
Kindstatement, theorem
WritingW0 stub
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 0STX0WRDNG.
UsesThe reals are Archimedean, Rational numbers

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