Math 2.0

Real functions Definition

Floor function

Builds on The reals are ArchimedeanIntegers

Definition

Let xx be a real number. The floor of xx, written ⌊x⌋\lfloor x \rfloor, is the greatest integer nn with n≤xn \le x. The floor function is the function ℝ→ℤ\mathbb{R} \to \mathbb{Z}, x↦⌊x⌋x \mapsto \lfloor x \rfloor.

Remarks

The greatest such integer exists because the reals are Archimedean and because of the well-ordering principle. For each integer nn, ⌊n⌋=n\lfloor n \rfloor = n. For example, ⌊π⌋=3\lfloor \pi \rfloor = 3 and ⌊−0.5⌋=−1\lfloor -0.5 \rfloor = -1. The characterization of the floor gives an equivalent definition.

Reading path

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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 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…statementFloor function, definition, no evidence levelFloor functiondefinition
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  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. Floor function definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Floor_and_ceiling_functions, Definition and properties
About this page
Tag0YVRNEDSVG
URIm2://0YVRNEDSVG
Kinddefinition
WritingW1 stated
MSC26A09
DefinesFloor function
UsesThe reals are Archimedean, Integers

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