Math 2.0

Real functions Theorem

Density of the irrationals

Builds on Density of the rationalsIrrational numberIrrationality of the square root of 2

Statement

Let xx and yy be real numbers with x<yx < y. Then there is an irrational number zz with x<z<yx < z < y.

Remarks

Proof: by the density of the rationals, there is a rational qq with x/2<q<y/2x / \sqrt{2} < q < y / \sqrt{2} and q≠0q \neq 0. Then z=q2z = q \sqrt{2} is irrational and x<z<yx < z < y. Thus both ℚ\mathbb{Q} and ℝ∖ℚ\mathbb{R} \setminus \mathbb{Q} are dense in ℝ\mathbb{R}.

Reading path

To understand this page from the ground up, read these 22 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…definitionIrrational number, definition, no evidence levelIrrational numberdefinitionThe 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 axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementSquare root, definition, no evidence levelSquare rootdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionIrrationality of the square root of 2, statement, no evidence levelIrrationality of the …statementThe 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…statementDensity of the irrationals, statement, no evidence levelDensity of the irrati…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. Irrational number definition
  10. The reals form a field statement
  11. Supremum definition
  12. The rationals form a field statement
  13. Completeness axiom statement
  14. Existence of square roots of nonnegative reals statement
  15. The rationals form an ordered field statement
  16. Square root definition
  17. Order on the reals definition
  18. Irrationality of the square root of 2 statement
  19. The reals form an ordered field statement
  20. The reals are Archimedean statement
  21. Density of the rationals statement
  22. Density of the irrationals statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Irrational_number, Set of all irrationals
About this page
Tag04J6G4H78J
URIm2://04J6G4H78J
Kindstatement, theorem
WritingW0 stub
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 04J6G4H78J.
UsesDensity of the rationals, Irrational number, Irrationality of the square root of 2

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