Introduction

Centuries later, Descartes showed how it is systematically possible to reduce questions of geometry to algebra. This brought up the issue of irrational numbers again -- if one is going to reformulate everything in terms of algebra, then one cannot have recourse to defining magnitudes geometrically, but have to find some sort of number which can adequately represent things like the hypotenuse of a square with rational sides. At first, such problems of logical consistency were swept under the rug, but eventually mathematicians realized that their subject needed to be put on a firm logical foundation. In particular, Dedekind solved this difficulty by defining the real numbers as a certain type of partition of the set of rational numbers which he termed a cut and defining operations on these numbers, such as addition, subtraction, multiplication, and division in terms of operations on these partitions.

Index of entries on real numbers

1. Rational numbers
2. Axiomatic definition of the real numbers.
3. Constructions of real numbers (advanced):
   1. Dedekind cuts
   2. Cauchy sequences
   3. Characterization of real numbers
   4. Reals not isomorphic to $p$ -adic numbers
4. commensurable numbers
5. positive
6. Inequalities for real numbers
7. index of inequalities
8. rational numbers are real numbers
9. interval
10. nested interval theorem
11. Real numbers are uncountable
12. Archimedean property
13. Operations for real numbers
    1. infimum and supremum for real numbers
    2. minimal and maximal number
    3. absolute value
    4. square root
    5. fraction power
14. Topic entry on algebraic and transcendental numbers
    1. Irrational number
    2. Transcendental number
    3. Algebraic number
15. Particular real numbers
    1. Natural log base
    2. pi
    3. Mascheroni constant

Generalizations