Math 2.0

Real functions Definition

Ceiling function

Builds on Floor functionIntegers

Definition

Let xx be a real number. The ceiling of xx, written ⌈x⌉\lceil x \rceil, is the least integer nn with x≤nx \le n. The ceiling function is the function ℝ→ℤ\mathbb{R} \to \mathbb{Z}, x↦⌈x⌉x \mapsto \lceil x \rceil.

Remarks

The ceiling and the floor are related by ⌈x⌉=−⌊−x⌋\lceil x \rceil = -\lfloor -x \rfloor. For each real xx, x≤⌈x⌉<x+1x \le \lceil x \rceil < x + 1. If xx is not an integer, then ⌈x⌉=⌊x⌋+1\lceil x \rceil = \lfloor x \rfloor + 1. If xx is an integer, then ⌈x⌉=⌊x⌋=x\lceil x \rceil = \lfloor x \rfloor = x.

Reading path

To understand this page from the ground up, read these 18 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…statementFloor function, definition, no evidence levelFloor functiondefinitionCeiling function, definition, no evidence levelCeiling functiondefinition
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. Floor function definition
  18. Ceiling function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Floor_and_ceiling_functions, Definition and properties
About this page
Tag0ASM4EPMB2
URIm2://0ASM4EPMB2
Kinddefinition
WritingW1 stated
MSC26A09
DefinesCeiling function
UsesFloor function, Integers

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.