topic entry on real numbers

Introduction

The real number system may be conceived as an attempt to fill in thegaps in the rational number system. These gaps first became apparentin connection with the Pythagorean theorem, which requires one to extracta square root in order to find the third side of a right triangle two ofwhose sides are known. Hypossos, a student of Pythagoras, showed that thereis no rational number whose square is exactly 2. In particular, this meansthat there is no rational number which may be used to describe the lengthof the diagonal of a square the length of whose sides is rational. Thisresult ruined the philosophical program of Pythagoras, which was to describeeverything in terms of whole numbers (or ratios of whole numbers) and,according to legend, resulted in Hypossos drowning himself. Eventually,geometers reconciled themselves to the existence of irrational magnitudesand Eudoxos devised his method of exhaustion which allowed one to proveresults about irrational magnitudes by considerations of rational magnitudeswhich are smaller and larger than the the irrational magnitude in question.

Centuries later, Descartes showed how it is systematically possible toreduce questions of geometry to algebra. This brought up the issue ofirrational numbers again — if one is going to reformulate everythingin terms of algebra, then one cannot have recourse to defining magnitudesgeometrically, but have to find some sort of number which can adequatelyrepresent 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 beput on a firm logical foundation. In particular, Dedekind solved thisdifficulty by defining the real numbers as a certain type of partition ofthe set of rational numbers which he termed a cut and defining operationson these numbers, such as addition, subtraction, multiplication, and divisionin terms of operations on these partitions.

Index of entries on real numbers

The below list presents entries on real numbers in an ordersuitable for studying the subject.

1. Rational numbers

2. Axiomatic definition of the real numbers.

3. Constructions of real numbers (advanced):

   1. Dedekind cuts

   2. Cauchy sequences (http://planetmath.org/RealNumber)

   3. Characterization of real numbers (http://planetmath.org/EveryOrderedFieldWithTheLeastUpperBoundPropertyIsIsomorphicToTheRealNumbers)

   4. Reals not isomorphic to $p$-adic numbers (http://planetmath.org/NonIsomorphicCompletionsOfMathbbQ)

4. commensurable numbers

5. positive

6. Inequalities for real numbers (http://planetmath.org/InequalityForRealNumbers)

7. index of inequalities

8. rational numbers are real numbers

9. interval

10. nested interval theorem

11. Real numbers are uncountable (http://planetmath.org/CantorsDiagonalArgument)

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 (http://planetmath.org/TheoryOfAlgebraicNumbers)

    1. Irrational number (http://planetmath.org/Irrational)

    2. Transcendental number

    3. Algebraic number (http://planetmath.org/AlgebraicNumber)

15. Particular real numbers

    1. natural log base

    2. pi

    3. Mascheroni constant

    4. golden ratio

Generalizations

There are many generalizations of real numbers. These includethe complex numbers, quaternions, extended real numbers,hyperreal numbers (http://planetmath.org/Hyperreal),and surreal numbers.  Of course the field $ℝ$ has many other field extensions, e.g. the field $ℝ(x)$ of the rational functions.