01 as real numbers


The real numbers 0 and 1 are distinct.

There are four relatively common ways of constructing the real numbers. One can start with the natural numbersMathworldPlanetmath and augment it by adding solutions to particular classes of equations, ultimately considering either equivalence classesMathworldPlanetmathPlanetmath of Cauchy sequencesMathworldPlanetmathPlanetmath of rational numbers or Dedekind cuts of rational numbers. One can instead define the real numbers to be the unique (up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) ordered field with the least upper bound property. Finally, one can characterise the real numbers as equivalence classes of possibly infiniteMathworldPlanetmath strings over the alphabet{0,1,2,3,4,5,6,7,8,9,.}  satisfying certain conditions. We offer a proof for each characterisation.

Cauchy sequences.

This construction proceeds by starting with a standard model of Peano arithmeticMathworldPlanetmathPlanetmath, the natural numbers , extending to by adding additive inverses, extending to by taking the field of fractions of , and finally defining to be the set of equivalence classes of Cauchy sequences in for an appropriately defined equivalence relation.

There is a natural embedding  i:  defined by sending a given number x to the equivalence class of the constant sequenceMathworldPlanetmath (x,x,).  Since i is injectivePlanetmathPlanetmath and 0 and 1 are elements of , to prove that  01  in we need only show that  01  in .

The name 1 is a label for the successorMathworldPlanetmathPlanetmath S0 of 0 in . One of the axioms of Peano arithmetic states that 0 is not the successor of any number. Therefore  0S0  in , and so  01  in . ∎

Dedekind cuts.

This construction agrees with the previous one up to constructing the rationals . Then is defined to be the set of all Dedekind cuts on . Letting x represent the name of an element of and x represent the name of an element of , we define

0 ={x|x<0}
1 ={x|x<1}

The proof that 01 is similar to the previous proof. Observe that 0<1. Since no number is less than itself, it follows that 00 but 01. Thus these Dedekind cuts are not equal. ∎

Ordered field with least upper bound property.

Here the fact that 01 is a consequence of the field axiom requiring 0 and 1 to be distinct. ∎

Decimal strings.

If one defines

0 =(0,0,0,0,)¯
1 =(1,0,0,0,)¯

then since neither defining string ends with a tail of 9s and the strings differ in one position, their equivalence classes are distinct. ∎

Title 01 as real numbers
Canonical name 0ne1AsRealNumbers
Date of creation 2013-03-22 15:23:15
Last modified on 2013-03-22 15:23:15
Owner mps (409)
Last modified by mps (409)
Numerical id 11
Author mps (409)
Entry type Theorem
Classification msc 54C30
Classification msc 26-00
Classification msc 12D99
Related topic DecimalExpansion