For real numbersContinuumA theory resulting from a rigorous description.The real number theory originates fromCalculusThe pursuit of a rigorous theoretical basis of, early human understanding of real numbers is only limited to applications, the understanding of the nature of irrational numbers is not clear, there is no strict definition, after the birth of calculusvariableThe understanding of function and function is gradually clear. For the need of rigorLimit theory、Real number theory。Real number theory is one of the three major parts of the analysis foundation, and the other two parts are limit theory, variables and functions.Limit theory ismathematical analysis And variables and functions are the basic research objects of mathematical analysis.The successful establishment of the real number theory has formed a complete system of analysis basis, which marks thatWeierstrassThe movement of analytical arithmetization advocated was almost completed, thus the second mathematical crisis was also solved in a real sense.[22]
real numberIt is one of the most basic concepts in mathematics.Real number andNumber axisThe points on can be one-to-one correspondence.The function studied by mathematical analysis, whichindependent variableBoth take real values, so understanding and understanding real numbers is an indispensable basis for establishing strict analytical theory（“Analysis Basis”）。Real numbers includeRational numberAndIrrational number, and fromEuclidSince then, people have understood them as commensurability and incommensurability of unit long line segmentsCommensuranceThe length of the segment of.By the 17th century, people had become accustomed to the use of real numbers, and began to understand real numbers abstractly away from their geometric prototypes.However, by the middle of the 19th century, in the process of analysis rigor, some facts could not be proved (for example,CauchyIt is unable to prove the sufficiency of the convergence criteria proposed by itself), and some proofs are wrong (such asPolzanoThe proof of the intermediate value of continuous functions), people found that the understanding of real numbers, especially irrational numbers, was still unclear, which prompted a group of mathematicians to focus on the problem of dealing with irrational numbers.Through their efforts, they finally established a variety of strict real number theories that are different in form but equivalent in essence in nearly half a century.All kinds of constructive real number theories first define irrational numbers from rational numbers, that is,Number axisupperRational pointAll gaps (irrational points) between them can be determined by rational numbers in a certain way.Then it is proved that the real number defined in this way (the original rational number and the newly defined irrational number) has all the properties of the previously known real number, especially continuity.These formal real number theories differ from each other due to different methods of determining gaps. They mainly include:Dedekind Using the method of rational number division,cantor Using the basic column of rational numbers,WeierstrassThe method of infinite (acyclic) decimal and the method of using the endpoint as a rational pointClosed intervalThe method of nested and bounded monotone rational sequence.From the standpoint of modern mathematics, the above methods are all based on the assumption that real numbers have certain characteristics (for example, Dedekin's method assumes the continuity of real numbers, while Cantor's method assumes thatCompleteness, and the method of closed interval sleeve reflectsReal axisUpper boundedClosed setOfCompactness）, and these properties are equivalent in the range of real numbers, so the real numbers defined by these methods are identical.In addition, there is another kind ofTectonic methodCompletely different methods of defining real numbers (i.e“Axiom of real number”）。He listed some basic properties of real numbers as oneAxiomatic systemAnd then define the objects that meet this axiom system as real numbers.The real number theory based on these axioms is also equivalent to the above construction based theory.[1]
Of course, it should also be pointed out that not onlyLimit theoryIt can only be established in the real number system, which is a lot of mathematics in middle schoolElementary function, except polynomial and rationalfractionIn addition, no real number can be defined.takeInfinite acyclic decimalIt is easy for students to accept the definition of irrational number, but in the real number system defined in this wayFour arithmetic operationsHow it is carried out is still completely unclear, and in fact it is not simple.As for index andlogarithm
When
It is more difficult to define when all are real numbers.It can be seen that even for the sake ofElementary functionGiven a strict definition, we also need to answer the question of what is a real number.Of course, this is not a task for middle school mathematics.
historical background
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Pythagorean school
500 BC, ancient GreecePythagoras(Pythagoras）Disciple of SchoolHippersos(Hippasus) discovered an amazing fact that the length of the diagonal of a square and its side is notCommensuranceThe incommensurability of (if the side length of the square is 1, the length of the diagonal is not a rational number) and the "everything is number" (Rational number）Their philosophies are quite different.This discovery made the leaders of the school panicked and angry, believing that it would shake their dominance in the academic world.Hippersos was therefore imprisoned, tortured and finally punished by sinking a boat.
The discovery of Bi's disciples revealed the defects of the rational number system for the first time, proving that it cannot be treated equally with continuous infinite straight lines, and rational numbers are not fullNumber axisThe point on the number axis has "pores" that cannot be expressed by rational numbers.This kind of "pore" has been proved by later generations to be "countless".Therefore, the ancient Greeks regarded rational numbers as the arithmetic of continuous connectionContinuumThe idea of ".The discovery of incommensurability and the famousZeno paradoxIt is also called the second crisis in the history of mathematics (the second mathematical crisis)[22]It has had a profound impact on the development of mathematics for more than 2000 years, prompting people to turn from relying on intuition and experience to relying on proof, promoting the development of axiomatic geometry and logic, and giving birth toCalculusThe sprout of ideas.
What is the essence of incommensurability?For a long time, there has been a lot of controversy, and no correct explanation has been obtained. The two incommensurable ratios have also been regarded as unreasonable numbers.The famous Italian painter Da Vinci in the 15th century called it "irrational number", and the German astronomer in the 17th centuryKeplerIt is called "nameless" number.
However, truth cannot be drowned after all, and it is "unreasonable" for Bi School to obliterate truth.People in memoryHippersosThe respectable scholar who devoted himself to the truth named incommensurable quantity“Irrational number”This is the origin of irrational numbers.
Pythagoras
The discovery of irrational numbers is brokenPythagorean school“Everything is numbered”My dream.At the same time, the defects of the rational number system are exposed: although the rational number on a straight line is "dense", it has many "pores", and the "pores" are numerous.Thus, the ancient Greeks regarded rational numbers as the arithmetic of continuous connectionContinuumThe idea of "is completely destroyed.".Its collapse had a profound impact on the development of mathematics in the next two thousand years.What is the essence of incommensurability?For a long time, people have different opinions.The ratio of two incommensurable quantities is also considered as an unreasonable number because it cannot be correctly explained.fifteenth centuryVinci(Leonardo da Vinci, 1452 - 1519) called them "irrational numbers",Kepler(J. Kepler, 1571 - 1630) called them "nameless" numbers.Although these "irrational" and "nameless" numbers are gradually used in later operations, whether they are real numbers has always been a troubling problem.
Ancient Chinese MathematicsWhen dealing with the problem of square root extraction, it is inevitable to encounter irrational roots.For this "endless" number《Chapter Nine Arithmetic》Directly accept it "face to face",Liu HuiIn the annotation, "finding its differential number" is actually to use decimal numbers to approach irrational numbers infinitely.This was the right way to complete the real number system, but Liu Hui's thought was far beyond his time, and failed to attract the attention of future generations.However, traditional Chinese mathematics focuses on the calculation of quantity,logarithmThe essence of the Greek is not very interesting, and the Greeks who are good at probing can not get over this hurdle.Since we cannot overcome it, we have to avoid it.Later Greek mathematicians, such asOdox（Eudoxus）、Euclid(Euclid) In their geometry, they strictly avoid combining numbers withGeometric quantityEqually.Odox's theory of proportion (see《Geometric primitives》Volume 5), which makes geometry logically bypass the impossibilityCommensuranceBut in the long period after that, there was a significant separation between geometry and arithmetic.
French mathematician Cauchy
17. The development of calculus in the 18th century almost attracted the attention of all mathematicians. It was precisely people's attention to the foundation of calculus that madeNumber fieldThe continuity of.Because calculus is variable mathematics based on limit operation, which requires a closed number field.Irrational number is exactlyReal number fieldThe key to continuity.
What is irrational number?French mathematicianCauchy(A. Cauchy, 1789 - 1875) gave the answer: irrational numbers are the limits of rational number sequences.However, according to Cauchy's limit definition, the so-called limit of rational number sequence means that there is a certain number in advance, so that the difference between it and the numbers in the sequence can be arbitrarily small when the sequence tends to infinity.But where does this pre-existing "number" come from?In Cauchy's view, the limit of rational sequence seems to exist a priori.This shows that although Cauchy was a great analyst at that time, he still could not get rid of the influence of the traditional ideas based on geometric intuition for more than 2000 years.
Variable mathematics is independently built and completeNumber fieldIn the second half of the 19th centuryWeierstrass（Weierstrass,1815- 1897）、Dedekind （R.Dedekind1831- 1916）、Cantor(G. Cantor, 1845 - 1918), et al.
1872 is the most memorable year in the history of modern mathematics.This year,Klein(F.Kline, 1849 - 1925) proposed the famous“Erlangen Program ”(Erlanger Program), Weierstrass gave a famous example of continuous but nondifferentiable functions everywhere.It was also in this year that the three major schools of real number theory: DeDekin's "segmentation" theory;Cantor's "basic sequence" theory and Weierstrass's "bounded monotone sequence" theory appeared in Germany at the same time.
German mathematician Klein
The purpose of trying to establish real numbers is to give a formal logical definition, which does not rely on the meaning of geometry and avoids using limits to define irrational numbersLogic error。Based on these definitionslimitThe deduction of the basic theorem of the "," will not have a theoretical cycle.derivativesandintegralThus, it can be directly established on these definitions, eliminating anyPerceptual knowledgeThe nature of the connection.Geometric concepts cannot be fully understood and accurate, which has been proved in the long years of development of calculus.Therefore, the necessary strictness can only pass through the concept of number, and cut off the relationship between the concept of number andGeometric quantityOnly after the connection of ideas can it be fully achieved.here,Dedekind The work of“dedekind cut ”The real number defined is the creation of human intelligence that is completely independent of the intuition of space and time.
The three major schools of real number theory essentially give a strict definition of irrational numbers, thus establishing a completeReal number field。The successful construction of the real number field has completely filled the gap between arithmetic and geometry for more than 2000 years. Irrational numbers are no longer "irrational numbers"ContinuumThe idea of "," has finally been realized in a strict scientific sense.
Because the content of the real number theory is too large and the processing methods are different, its related theories are also scattered in various literatures. The following is the definition of the real number systemJournals reviewed。
Journals reviewed
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axiomatic method
The so-called axiomatic method originated from ancient Greek mathematiciansEuclidOf《Geometric primitives》。In this book, a few axioms are proposed for geometry, and then all other theorems are obtained by logical reasoning, so as to build the whole geometry into a very clear and strict logical system.As long as the axiom is good, the truth of all the obtained theorems will be no problem.The so-calledaxiomIt sounds abstract, but in fact, it is just a few facts that everyone can accept and have no doubt about their correctness.
The same is true for the axiomatic method of the so-called real number system. We list as axioms the minimum number of independent properties that real numbers should have, so that other properties can be deduced from axioms. This is an axiomatic system（“Axiom of real number”）。
Hilbert's axiomatic method describes what the real number system we need is like. It solves many remaining problems about real numbers in middle school mathematics, such as what is the addition and multiplication of real numbers, and why the addition of real numbers meetsCommutative law、Associative law, multiplication also satisfies commutative law, associative law, etc., which can be understood as axiomatic. In fact, if more basic assumptions are provided (such as on the basis of rational numbers), theseOperational lawIt can be proved.It also guaranteesBasic Theorem of Real Number SystemIs established in mathematical analysisLimit theoryThe development of provides the necessary stage.The existence of the real number system satisfying these axioms depends on the followingConstruction methodThe solution is to give the concrete method of generating real number system, and prove that all axioms listed in the axiomatic method are satisfied in it.For the method of axiomatization, seeZorich Of《Mathematical Analysis (Volume I)》。[2]
Existence
The existence of real number system is throughTectonic methodThe following are three methods for constructing the real number system (mainly defining irrational numbers from rational numbers).
Dadkin's method of segmentation is often introduced in the works on mathematical analysis.The most classic narrative is a small book specially written by Landau for this purpose《Analysis Basis》[3]The subtitle of the book is "Operation of Integer, Rational Number, Real Number and Complex Number". The book starts from natural number and continues to definecomplex, showing the complete number system definition.
In western textbooks,SpivakCalculus[9]At the beginning, two chapters introduce the axioms of the number system in detail, and at the end of the book, three chapters explain how to construct real numbers;Luddin's《Principles of Mathematical Analysis》[10]The first chapter and appendix of have a brief description of the real number theory.These two textbooks have changed the method of DeDekin's segmentation[11]I know that this is basically the real number definition method proposed by Russell.
In various methods of introducing real number system,dedekind cut The method has been highly evaluated and is called a creation of human intelligence that does not rely on the intuition of space and time.
threeWeierstrassMethod of starting from decimal representation
German mathematician Weierstrass
This method is different from the previous two methods. It does not need to introduce new mathematical objects as irrational numbers. Instead, it starts from the existing definitions in middle school, that is, it recognizes thatdecimal systemFinite decimals and infinityRecurring decimalIs a rational number, while the decimal infinite acyclic decimal is an irrational number.This is easier for middle school students to accept.Therefore, it is also calledReal Number Theory of Middle School Students。
But why decimal infinite acyclic decimal?This inevitably involves the problem of limits.With the Cauchy Criterion, we cansequence limit orInfinite seriesTo understand decimal infinite acyclic decimal.But before the establishment of the real number system, it cannot be understood in this way, otherwise it will be the same mistake as Cauchy in history.
Therefore, in order to avoid the logical cycle definition, in thedecimal systemInfinite nonRecurring decimalWhen it is defined as irrational number, it is impossible to regard it as the sum of an infinite series at first, but only as a pure sign, a mathematical object whose meaning is not clear.Then add and multiply in the set composed of all decimal decimals, specify the order between any two decimals, and verify that it meets the field axiom, order axiomArchimedean axiomAnd continuity axioms.Of course, there are many steps to infer.In fact, it is also a mathematical abstraction to think that such a sign represents real numbers, and this is another kind of continuity axiomEquivalent form, HistoricallyWallisIn 1696, rational numbers were equated with recurring decimals.andStoltzIn 1886, it was proposed that the decimal infinite nonRecurring decimalDefinition as irrational number[16]However, a satisfactory real number theory has not yet been established.
Another approach that can be included in this approach is to introduce theClosed intervalThe nested principle is a substitute for the continuity axiom.It is not only intuitive, but also a good way to avoid the decimal infinite acyclic decimal which is difficult to explain at the beginning.[3][18][21]
uniqueness
First of all, understand the exact meaning of uniqueness here, which means thatisomorphismThe uniqueness in the sense, specifically, is to prove that allAxiom of real numberThe real number system models of are isomorphic.
according toDedekind Methods After establishing the real number system, we can refer to the discussion on its uniqueness in the isomorphic senseSpivakCalculus[9]The last chapter is "the uniqueness of real numbers".For the discussion on the uniqueness of establishing the real number system according to Cantor's Cauchy series method, refer to Xu Shaopu, Song Guozhu, etc《mathematical analysis 》[14]The last part of the fifth chapter.
