# Two concrete new constructions of the real numbers

@article{Knopfmacher1988TwoCN, title={Two concrete new constructions of the real numbers}, author={Arnold Knopfmacher and John Knopfmacher}, journal={Rocky Mountain Journal of Mathematics}, year={1988}, volume={18}, pages={813-824} }

Two new methods are put forward for constructing the complete ordered field of real numbers out of the ordered field of rational numbers. The methods are motivated by some known theorems on so-called Engel and Sylvester series. Amongst advantages of the methods are the facts that they do not require an arbitrary choice of "base", or any equivalence classes or similar constructs. Introduction. By old theorems of Lambert (1770) and Engel (1913) (see Perron [2]), every real number A has a unique…

## 8 Citations

### Two constructions of the real numbers via alternating series

- Mathematics
- 1989

Two further new methods are put forward for constructing the complete
ordered field of real numbers out of the ordered field of rational numbers. The
methods are motivated by some little known…

### A New Set Theory for Analysis

- MathematicsAxioms
- 2019

It is concluded that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

### A new construction of the real numbers by alternating series

- Mathematics
- 2012

We put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J.…

### The reals as rational Cauchy filters

- Mathematics
- 2015

We present a detailed and elementary construction of the real
numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in Q (where…

### The real numbers--A survey of constructions

- Mathematics, Economics
- 2015

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known…

### SEL Series Expansion and Generalized Model Construction for the Real Number System via Series of Rationals

- MathematicsInt. J. Math. Math. Sci.
- 2014

We present an algorithm for constructing infinite series expansion for real numbers, which yields generalized versions of three famous series expansions, namely, Sylvester series, Engel series, and…

### KNOPFMACHER EXPANSIONS IN NUMBER THEORY

- Mathematics
- 2001

Dedicated to the memory of John Knopfmacher (1937–1999) Abstract We survey in this paper J. Knopfmacher results on number theoretical expansions as they arised from our common collaboration.

## References

SHOWING 1-3 OF 3 REFERENCES

### A New Construction of the Real Numbers

- Mathematics
- 1974

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy…