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[parent] topic entry on real numbers (Topic)

Introduction

The real number system may be conceived as an attempt to fill in the gaps in the rational number system. These gaps first became apparent in connection with the Pythagorean theorem, which requires one to extract a square root in order to find the third side of a right triangle two of whose sides are known. Hypossos, a student of Pythagoras, showed that there is no rational number whose square is exactly 2. In particular, this means that there is no rational number which may be used to describe the length of the diagonal of a square the length of whose sides is rational. This result ruined the philosophical program of Pythagoras, which was to describe everything 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 magnitudes and Eudoxos devised his method of exhaustion which allowed one to prove results about irrational magnitudes by considerations of rational magnitudes which are smaller and larger than the the irrational magnitude in question.

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

The below list presents entries on real numbers in an order suitable 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
    3. Characterization of real numbers
    4. Reals not isomorphic to $ p$-adic numbers
  4. positive
  5. Inequalities for real numbers
  6. rational numbers are real numbers
  7. interval
  8. nested interval theorem
  9. Real numbers are uncountable
  10. Archimedean property
  11. 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
  12. Topic entry on algebraic and transcendental numbers
    1. Irrational number
    2. Transcendental number
    3. Algebraic number
  13. Particular real numbers
    1. Natural log base
    2. pi
    3. Mascheroni constant

Generalizations

There are many generalizations of real numbers. These include the complex numbers, quaternions, extended real numbers, hyperreal numbers, and surreal numbers. Of course the field $ \mathbb{R}$ has many other field extensions, e.g. the field $ \mathbb{R}(x)$ of the rational functions.



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See Also: Gelfand-Tornheim theorem, positivity in ordered ring


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infimum and supremum for real numbers (Topic) by matte
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Cross-references: rational functions, field extensions, field, surreal numbers, extended real numbers, quaternions, complex numbers, Mascheroni constant, pi, natural log base, transcendental number, fraction power, absolute value, minimal and maximal number, infimum and supremum for real numbers, Archimedean property, nested interval theorem, interval, rational numbers are real numbers, positive, Dedekind cuts, axiomatic definition of the real numbers, division, multiplication, subtraction, addition, operations, cut, rational numbers, partition, type, foundation, hypotenuse, represent, sort, algebra, geometry, method of exhaustion, irrational, eventually, terms, rational, diagonal, length, square, right triangle, side, order, square root, Pythagorean theorem, connection, rational number, real number
There are 3 references to this entry.

This is version 16 of topic entry on real numbers, born on 2006-02-13, modified 2007-12-05.
Object id is 7618, canonical name is TopicEntryOnRealNumbers.
Accessed 6705 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 26-00 (Real functions :: General reference works )
 54C30 (General topology :: Maps and general types of spaces defined by maps :: Real-valued functions)
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jump to page: 1 2 3 >> of 3 (14 items)

function by PARASHAR on 2008-02-19 22:22:40
I am below depicting a definition of a function. It is a simple conditional loop but there is some logic behind it which can be used in algorithms, Can anybody please explain me what is that?

sd(integer a, integer k, paramter){
if(a > k*parameter && "a is not an integer multiple of parameter"){
return true;
}else{
return false;
}
}
[ reply | up ]
February ponder this by PARASHAR on 2008-02-19 00:29:42
This month is very simple problem i think, i have just reached the solution but i need your help. Please visit the link for the problem
http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/February2008.html
I have found the missing values but I am still in search of a perfect answer for the first one, can anybody please help me?
[ reply | up ]
denominations by PARASHAR on 2007-11-20 21:45:59
In a currency there are three denominations 1, 10 and 50. To give a total sum of 107, how many ways 1, 10 and 50 can be arranged to yield 107?
[ reply | up ]
friends and enemies by PARASHAR on 2007-11-20 21:40:00
Consider a set S of pairs(i,j) where 1<=i<j<=n. Now 2 pairs are considered to be friends if they have one element in common and enemies if they have no common element. For eg: consider a set S->((1,2),(1,3),(1,4),(4,5)). In this set, (1,2) and (1,3) are friends and (1,2) and (4,5) are enemies. Now the questions are

1) In a set, how many friends would be there?
2) In a set, consider 2 pairs which are friends, then how many other pairs are there which are common friends of both these pairs. For eg: consider (1,2) and (1,3), then (1,4) is a common friend of (1,2) and (1,3)

Answers would be an eqaution in terms of n
[ reply | up ]
minimum value by PARASHAR on 2007-11-19 21:32:30
Minimum value of x^2 + y^2 + z^2 can only be found out if

i) x+y+z is known
ii) Any 2 from x,y and z are equal
iii)None of the above i.e. cannot be found out with above 2 cases

Which one of above is correct and how?
[ reply | up ]

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