Difference between revisions of "Real number"
(least upper bound property!) 
(Much improvement) 

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{{Mathm}}  {{Mathm}}  
−  The '''real numbers''' are a set of numbers with extremely important theoretical and practical properties. They can be considered the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. They are the 4<sup>th</sup> item in this hierarchy of types of [[number]]s:  +  The '''real numbers''' are a set of numbers with extremely important theoretical and practical properties. They can be considered to be the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. They are the 4<sup>th</sup> item in this hierarchy of types of [[number]]s: 
*The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)  *The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)  
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==Formal definition==  ==Formal definition==  
−  Formally, real numbers are defined as the unique [[Field (mathematics)field]] which is [[ordered]], [[Complete (mathematics)complete]], and [[Archimedean]]. The reals can be constructed from the  +  Formally, real numbers are defined as the unique [[Field (mathematics)field]] which is [[ordered]], [[Complete (mathematics)metrically complete]], and [[Archimedean]]. The reals can be constructed from the rationals by means of [[Dedekind cut]]s or [[Cauchy sequence]]s, as outlined below. 
==Real line==  ==Real line==  
−  The real numbers can be thought of as a [[line]], called the '''real line'''. Each real number represents a point on the real line. However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a [[dense set]], so points accumulate around each other.<ref>http://abstractmath.org/MM/MMRealNumbers.htm</ref>  +  The real numbers can be thought of as a [[line]], called the '''real line'''. Each real number represents a point on the real line. <! [This is unnecessary and misleading. Mentioning it at all can lead people into wrong thinking that they wouldn't otherwise do. Just don't say it. And I have no idea what "points accumulate around each other" means. In any case, the reference covers this, if anyone cares.] However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a [[dense set]], so points accumulate around each other.><ref>http://abstractmath.org/MM/MMRealNumbers.htm</ref> 
The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[xaxis]] and [[yaxis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]].  The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[xaxis]] and [[yaxis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]].  
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Any realworld measurement that anyone could possibly make, one can make as accurately as one wants with rational numbers. For example, one can calculate the ratio of the circumference of a circle to its diameter to within one part is a trillion using the number 3.1415926535898 (<math>\pi\,</math> itself is irrational.) Put another way, you never have to worry about the difference between the rationals and the reals in a lumber yard or a laboratory. The technical term that topologists use for this state of affairs is that the rationals are [[dense subsetdense]].  Any realworld measurement that anyone could possibly make, one can make as accurately as one wants with rational numbers. For example, one can calculate the ratio of the circumference of a circle to its diameter to within one part is a trillion using the number 3.1415926535898 (<math>\pi\,</math> itself is irrational.) Put another way, you never have to worry about the difference between the rationals and the reals in a lumber yard or a laboratory. The technical term that topologists use for this state of affairs is that the rationals are [[dense subsetdense]].  
−  The shortcoming of the rationals, that is overcome by defining the reals, is a somewhat subtle theoretical point. The most direct example is that, if one lived in a world with only rational numbers, 2 has no square root, even though it obviously should.  +  The shortcoming of the rationals, that is overcome by defining the reals, is a somewhat subtle theoretical point. The most direct example is that, if one lived in a world with only rational numbers, 2 has no square root, even though it obviously should have one. 
::One can easily prove that there is no rational number m/n such that (m/n)<sup>2</sup> = 2. The factors of m<sup>2</sup> all come in pairs, as do the factors of n<sup>2</sup>. But the factors of m<sup>2</sup> must be the same as the factors of n<sup>2</sup> except for a single extra factor of 2.  ::One can easily prove that there is no rational number m/n such that (m/n)<sup>2</sup> = 2. The factors of m<sup>2</sup> all come in pairs, as do the factors of n<sup>2</sup>. But the factors of m<sup>2</sup> must be the same as the factors of n<sup>2</sup> except for a single extra factor of 2.  
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:Definition: A number L is a ''least upper bound'' (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval <math>(3, 6)\,</math>. 3 is its glb. 6 and 3 are also the lub and glb of the closed interval <math>[3, 6]\,</math>—the inclusion of the endpoints makes no difference.  :Definition: A number L is a ''least upper bound'' (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval <math>(3, 6)\,</math>. 3 is its glb. 6 and 3 are also the lub and glb of the closed interval <math>[3, 6]\,</math>—the inclusion of the endpoints makes no difference.  
+  
+  ::The least upper bound is also sometimes called the "supremum", abbreviated "sup". The greatest lower bound is also sometimes called the "infimum", abbreviated "inf". For simple cases like the real numbers, the terms "maximum" and "minimum" may also be used.  
A set has the ''least upper bound property'' if every set that has an upper bound has a least upper bound. There is also a ''greatest lower bound property'', and any reasonable set having one property has the other.  A set has the ''least upper bound property'' if every set that has an upper bound has a least upper bound. There is also a ''greatest lower bound property'', and any reasonable set having one property has the other.  
The least upper bound property is extremely important in caclulus and analysis. It is essential for many theorems, notably the ''[[mean value theorem]]'' and the ''intermediate value theorem''.  The least upper bound property is extremely important in caclulus and analysis. It is essential for many theorems, notably the ''[[mean value theorem]]'' and the ''intermediate value theorem''.  
+  
+  ::''The rational numbers do not satisfy the least upper bound property.''  
+  
+  For example, if we can only use rational numbers, the set of numbers that have squares less then 2 has no rational least uper bound. 1.4142136 is an upper bound, but 1.41421357 is a smaller one. The exact square root of 2 is the least uper bound that we need, but it isn't rational.  
+  
+  ==Two ways to define the reals formally==  
+  There are two ways of formally constructing the reals from the rationals. The simpler way is as [[Dedekind cut]]s, which see. A Dedekind cut could be thought of as a formal least upper bound. That is, the real number <math>\sqrt{2}</math> is, in effect, '''defined''' as "the least upper bound of the set of numbers whose squares are less than 2".  
+  :(This is a common motif in theoretical mathematics—you define something as the abstract set of things that have the properties that you want, and then show that they obey all the familiar properties of the original set.)  
+  The set thus created is "Dedekind complete", which is the same as having the least upper bound and greatest lower bound properties.  
+  
+  The second way is as [[Cauchy sequence]]s, which see. The rationals are not "metrically complete" or "Cauchy complete", in that Cauchy sequences do not necessarily converge. The reals can be, in effect, '''defined''' as "the things that Cauchy sequences would converge to".  
+  
+  The reals are both Dedekind complete and metrically complete. The rationals are neither. (In general, the two properties are not the same—the complex numbers are metrically complete but not Dedekind complete.)  
==Infinity==  ==Infinity==  
−  The real numbers ''do not'' include  +  The real numbers ''do not'' include [[infinity]]. Every real number is finite, though the set of reals is an infinite set. 
+  
+  ::'''INFINITY IS NOT A NUMBER!'''  
−  +  However, there are nonstandard models of real numbers which include <math>\infty</math> or include both <math>\infty</math> and <math>\infty</math>.  
−  +  There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·10<sup>23</sup>) to it, and, similarly, no smallest real number.  
−  It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 7541985; this system is built in to modern processor chips. It provides for values which print out as INF and INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."  +  <! [This paragraph is all wrong, and doesn't belong here in any case. The topic should be covered in the compter arithmetic page, perhaps pointing here.] It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 7541985; this system is built in to modern processor chips. It provides for values which print out as INF and INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."> 
==History==  ==History== 
Revision as of 19:36, 15 October 2009

This article/section deals with mathematical concepts appropriate for a student in mid to late high school. 
The real numbers are a set of numbers with extremely important theoretical and practical properties. They can be considered to be the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. They are the 4^{th} item in this hierarchy of types of numbers:
 The "natural numbers", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
 The "integers"—positive, negative, and zero
 The "rational numbers", or fractions, like 355/113
 The "real numbers", including irrational numbers
 The "complex numbers, which give solutions to polynomial equations
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are irrational. Examples are and . These decimal representations neither repeat nor terminate.
Contents
Formal definition
Formally, real numbers are defined as the unique field which is ordered, metrically complete, and Archimedean. The reals can be constructed from the rationals by means of Dedekind cuts or Cauchy sequences, as outlined below.
Real line
The real numbers can be thought of as a line, called the real line. Each real number represents a point on the real line. ^{[1]}
The real line is useful as a coordinate system for graphing functions. Thus, the xaxis and yaxis are both instances of the real line. The real line is the basis for geometric measurements, and more generally for ideas in metric topology.
What is the problem? Aren't rational numbers good enough?
Any realworld measurement that anyone could possibly make, one can make as accurately as one wants with rational numbers. For example, one can calculate the ratio of the circumference of a circle to its diameter to within one part is a trillion using the number 3.1415926535898 ( itself is irrational.) Put another way, you never have to worry about the difference between the rationals and the reals in a lumber yard or a laboratory. The technical term that topologists use for this state of affairs is that the rationals are dense.
The shortcoming of the rationals, that is overcome by defining the reals, is a somewhat subtle theoretical point. The most direct example is that, if one lived in a world with only rational numbers, 2 has no square root, even though it obviously should have one.
 One can easily prove that there is no rational number m/n such that (m/n)^{2} = 2. The factors of m^{2} all come in pairs, as do the factors of n^{2}. But the factors of m^{2} must be the same as the factors of n^{2} except for a single extra factor of 2.
The theoretical property that the rational numbers lack is called the least upper bound property.
 Definition: A number B is an upper bound for a set of numbers if no element of the set is greater than B. (There is also the notion of a lower bound.)
For example, 10 is an upper bound for the open interval . 7 is also an upper bound, as is 6. 5 is not. 2 is a lower bound.
Some sets do not have upper bounds. For example, all rational or real numbers, or all odd integers.
 Definition: A number L is a least upper bound (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval . 3 is its glb. 6 and 3 are also the lub and glb of the closed interval —the inclusion of the endpoints makes no difference.
 The least upper bound is also sometimes called the "supremum", abbreviated "sup". The greatest lower bound is also sometimes called the "infimum", abbreviated "inf". For simple cases like the real numbers, the terms "maximum" and "minimum" may also be used.
A set has the least upper bound property if every set that has an upper bound has a least upper bound. There is also a greatest lower bound property, and any reasonable set having one property has the other.
The least upper bound property is extremely important in caclulus and analysis. It is essential for many theorems, notably the mean value theorem and the intermediate value theorem.
 The rational numbers do not satisfy the least upper bound property.
For example, if we can only use rational numbers, the set of numbers that have squares less then 2 has no rational least uper bound. 1.4142136 is an upper bound, but 1.41421357 is a smaller one. The exact square root of 2 is the least uper bound that we need, but it isn't rational.
Two ways to define the reals formally
There are two ways of formally constructing the reals from the rationals. The simpler way is as Dedekind cuts, which see. A Dedekind cut could be thought of as a formal least upper bound. That is, the real number is, in effect, defined as "the least upper bound of the set of numbers whose squares are less than 2".
 (This is a common motif in theoretical mathematics—you define something as the abstract set of things that have the properties that you want, and then show that they obey all the familiar properties of the original set.)
The set thus created is "Dedekind complete", which is the same as having the least upper bound and greatest lower bound properties.
The second way is as Cauchy sequences, which see. The rationals are not "metrically complete" or "Cauchy complete", in that Cauchy sequences do not necessarily converge. The reals can be, in effect, defined as "the things that Cauchy sequences would converge to".
The reals are both Dedekind complete and metrically complete. The rationals are neither. (In general, the two properties are not the same—the complex numbers are metrically complete but not Dedekind complete.)
Infinity
The real numbers do not include infinity. Every real number is finite, though the set of reals is an infinite set.
 INFINITY IS NOT A NUMBER!
However, there are nonstandard models of real numbers which include or include both and .
There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·10^{23}) to it, and, similarly, no smallest real number.
History
The ancient Greek mathematicians (Archimedes, Euclid, Pappus, Pythagoras and Zeno) are perhaps the first people to have created the abstract notion of a "number" (a real number; not an integer) to represent a geometrical measurement. They developed the correspondence between numbers and measurements such as distances, areas, and angles. To honor Archimedes' contribution, real analysts have named a property of the real numbers the Archimedean property. Real analysis remained in geometry's shadow until the development of the subfield of calculus. This subject subsumed all geometry known at the time, creating the field of analytic geometry.