Real number isReal number theoryThe core research object of.The set of all real numbers can be calledReal number system(real number system) or real number continuum.Any complete Archimedean ordered field can be called a real number system.It is unique in the sense of order preserving isomorphism and is often represented by R.Since R is a defined arithmeticoperationIt has the name of real number system.
Real numbers can be usedmeasureContinuous quantity.theoryAbove, any real number can be usedinfinite decimal Means that the right side of the decimal point is an infiniteseries(can beloopCan also be acyclic).In practical application, real numbers are often approximated to a finite decimal (n is reserved after the decimal pointpositive integer）。staycomputerSince computers can only store a limited number of decimal places, real numbers are often usedFloating point numberTo represent.[1]
Around 500 BC, Greek mathematicians led by Pythagoras realized that rational numbers could not meet the needs in geometry, but Pythagoras himself did not recognize the existence of irrational numbers.It was not until the 17th century that real numbers were widely accepted in Europe.In the 18th century,CalculusIt develops on the basis of real numbers.1871, German mathematiciancantor The strict definition of real number is proposed for the first time.
Based on daily experience,Rational number setOn the number line, it seems that“dense”Therefore, the ancients always believed that rational numbers could meet the actual needs of measurement.Take a square with a side length of 1cm as an example, how long is its diagonal?Under the specified precision (such aserrorLess than 0.001 cm), a rational number can always be used to represent a sufficiently accurate measurement result (such as 1.414 cm).But,ancient GreekPythagorean schoolMathematicians found that the length of this diagonal line cannot be completely and accurately expressed only by rational numbers, which completely hit their mathematical concept. They thought that the ratio of any two line segments (length) could be expressed by the ratio of natural numbers.As a result of this,PythagorasI even have the belief that "all things are counted". Here, number refers to natural number (1, 2, 3,...), and all positive rational numbers can be obtained from the ratio of natural numbers. The fact that there is a "gap" in the set of rational numbers was a great blow to many mathematicians at that time (seeThe first mathematical crisis)。
fromancient GreekIt was not until the 17th century that mathematicians gradually accepted the existence of irrational numbers and regarded them as numbers on an equal footing with rational numbers;Laterimaginary numberThe introduction of the concept is called "real number" for differentiation, which means "real number".At that time, although imaginary numbers had appeared and been widely used, the strict definition of real numbers was still a difficult problemfunction、limitandastringencyOnly after the concept ofDedekind 、CantorThe real number was strictly treated by others.
Basic operation
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The basic operations that can be realized by real numbers areplus、reduce、ride、except、PowerWait, yesNonnegative number(i.e. positive number and 0) can also bePrescriptionOperation.The result of adding, subtracting, multiplying, dividing (the divisor is not zero), and squaring a real number is still a real number.Any real number can be opened to the odd power, and the result is still a real number. Only non negative real numbers can be opened to the even power, and the result is still a real number.
nature
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Closure
The four operations of addition, subtraction, multiplication and division (divisor is not zero) of real number set haveClosureThat is, the sum, difference, product and quotient (divisor is not zero) of any two real numbers are still real numbers.
Orderliness
The set of real numbers is ordered, that is, any two real numbers
、
Must meet and only meet one of the following three relationships:
，
，
。
Transitivity
Real number size is transitive, that is, if
, and
, there is
。
Archimedean property
Real numbers have Archimedean property, that is
，
, if
, then positive integer
，
。
Density
The set of real numbers is dense, that is, there must be another real number between two unequal real numbers, both rational and irrational.
The set of rational numbers is not a complete space.For example, (1, 1.4, 1.41, 1.414, 1.4142, 1.41421,...) is rationalCauchy sequence, but no rational numberlimit。In fact, it has a real limit
。
Real number is the completion of rational number, which is also a method to construct the set of real numbers.
The existence of limits isCalculusThe foundation of.The completeness of real numbers is equivalent toEuclidean geometryThere is no "gap" in the straight line of.
2、 "Complete ordered field"
The set of real numbers is usually described as a "complete ordered field", which can be explained in several ways.
First, an ordered field can be a complete lattice.However, it is easy to find that no ordered field can be a complete lattice.This is because the ordered field does not have the largest element (for any element
，
Will be larger).Therefore, "complete" here does not mean complete lattice.
In addition, the ordered domain satisfies DeDekin completeness, which has been defined in the above axiom.The uniqueness of the above also shows that "complete" here meansDeDekin completenessMeans.The meaning of completeness is very close to adoptingdedekind cut The method of constructing real numbers is to start from the (rational number) order field and establish DeDekin's completeness through the standard method.
These two concepts of completeness ignore the structure of the domain.However, ordered groups (fields are special groups) can define uniform spaces, and uniform spaces have the concept of complete spaces.The above completeness is only a special case.(The completeness concept in the uniform space is adopted here, rather than the completeness of the relevant well-known metric space, because the definition of the metric space depends on the properties of real numbers.) Of course, it is not the only uniformly complete ordered field, but it is the only uniformly complete Archimedes field.In fact, "complete Archimedean domain" is more common than "complete ordered domain".It can be proved that any uniformly complete Archimedean domain must be complete by Dedekin (and vice versa, of course).The meaning of this completeness is very close to the method of constructing real numbers by Cauchy sequence, that is, starting from the (rational number) Archimedes field, to establish uniform completeness by standard methods.
The "Complete Archimedean Domain" was originally created byHilbertHe also wanted to express something different from the above.He believes that real numbers constitute the largest Archimedean domain, that is, all other Archimedean domains are
The child domain of.such
"Complete" means that adding any element in it will make it no longer an Archimedean domain.The meaning of completeness is very similar to that of usingHyperreal numberTo construct real numbers, that is, from acontainStarting from the pure class of all (hyper real) ordered fields, find the largest Archimedean field from its sub fields.
Corresponding to the number axis
If it is determined on a straight line (usually a horizontal line)
As the origin, specify a direction as the positive direction (usually the direction pointing to the right is specified as the positive direction), and specify a unit length, then this line is calledNumber axis。Any real number corresponds to the only point on the number axis;On the contrary, every point on the number axis also uniquely represents a real number.Therefore, there is a one-to-one correspondence between the set of real numbers and the points on the number axis.
Advanced nature
The set of real numbers is uncountable, that is, the number of real numbers is strictly more than the number of natural numbers (although both areInfinity）。This can be achieved bycantor diagonalMethod demonstration.In fact, the potential of the set of real numbers is
(SeeContinuumPotential), that isSet of natural numbersOfPower setPotential.Because there are only countable elements in the real number setAlgebraic number, most real numbers areTranscendental number。In the subset of the real number set, there is no set whose potential is strictly greater than that of the natural number set and strictly less than that of the real number set. This isContinuum hypothesis。In fact, this hypothesis is independent of ZFC set theory, which can neither prove it nor deduce its negation.
All nonNegative real numberOfsquare rootIt belongs to R, but this pair of negative numbers does not hold.This shows thatRThe order on is determined by itsalgebraic structure affirmatory.And all odd timespolynomialAt least one root belongs to R.These two properties make it the most important instance of real closed domain.To prove this is rightFundamental theorem of algebraThe first half of the proof.
The set of real numbers has a canonical measure, namelyLebesgue measure。
The supremum axiom of the real number set uses thesubset, this is a kind ofSecond-order logicStatement of.It is impossible to use only first-order logic to describe the set of real numbers:
1. L ö wenheim – Skolem theorem theorem shows that there is a countable dense subset of the real number set, which in the first-order logic exactly meets the requirements of theproposition；
2. The set of hyperreal numbers is far larger than R, but it also satisfies the same first-order logical proposition as R.Meet and RThe ordered field of the same first-order logical proposition is called RNon-standard model。This is it.Non-standard analysisThe research content of, proving first-order logical propositions in non-standard models (may be simpler than proving in), so as to determine that these propositions are also true in R.
Topological property
The set of real numbers constitutes a metricspace：
and
The distance between is set as absolute value
。As a totally ordered set, it also has an ordertopology。Here, the topology obtained from the metric and order relations is the same.The set of real numbers is also a one-dimensional contractible space (so it is alsoConnected space）, local compact space, separable space, Bailey space.But the set of real numbers is notCompact space。These can be determined by specific properties. For example, an infinitely continuous separable ordered topology must be homeomorphic to the set of real numbers.The following is an overview of the topological properties of real numbers:
1. Order
Is a real number.
The neighborhood of is a real number set, including a segment containing
The defined sequence is constructed as the complement of rational number.Real numbers can be constructed from rational numbers in different ways.HereAxiomatic method.
The last one is the key to distinguish between real numbers and rational numbers.For example, for allsquareA set of rational numbers less than 2, which has an upper bound in the set of rational numbers, such as 1.5;But there is no supremum in the set of rational numbers (because
Is not a rational number).
Real numbers are uniquely determined by the above properties.More precisely, given any two ordered domains
and
, exists from
reach
The only domain isomorphism of, that is, structurally they can be regarded as the same.
