Math 2.0

Number theory Theorem

Every real number has a decimal expansion

Builds on Decimal expansionFloor functionCompleteness axiom

Statement

Let xx be a real number with x≥0x \ge 0. Then xx has a decimal expansion: there are a natural number a0a_0 and digits dk∈{0,…,9}d_k \in \{0, \dots, 9\} for k≥1k \ge 1 with x=a0+∑k=1∞dk10−kx = a_0 + \sum_{k=1}^{\infty} d_k 10^{-k}.

Remarks

One choice is a0=⌊x⌋a_0 = \lfloor x \rfloor and dk=⌊10kx⌋−10⌊10k−1x⌋d_k = \lfloor 10^k x \rfloor - 10 \lfloor 10^{k-1} x \rfloor, with the floor function. With this choice, the truncation a0.d1…dka_0.d_1 \dots d_k satisfies 0≤x−a0.d1…dk<10−k0 \le x - a_0.d_1 \dots d_k < 10^{-k}. The expansion is unique, except for the numbers with a finite expansion. They also have a second expansion that ends in infinitely many 99.

Reading path

To understand this page from the ground up, read these 19 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…statementDecimal expansion, definition, no evidence levelDecimal expansiondefinitionOrder 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 functiondefinitionEvery real number has a decimal expansion, statement, no evidence levelEvery real number has…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. Decimal expansion definition
  15. Order on the reals definition
  16. The reals form an ordered field statement
  17. The reals are Archimedean statement
  18. Floor function definition
  19. Every real number has a decimal expansion statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Decimal_representation, lead section
  2. Web https://en.wikipedia.org/wiki/Decimal_representation, Finite decimal approximations
About this page
Tag047KEVEZ9Q
URIm2://047KEVEZ9Q
Kindstatement, theorem
WritingW0 stub
MSC11A63
StatusThis copy of the site has no build data. Run m2 status 047KEVEZ9Q.
UsesDecimal expansion, Floor function, Completeness axiom

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (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.