Number is a basic concept in mathematics and an important part of human civilization. Every expansion of the concept of numbers marks a leap in mathematics. People's understanding and application of numbers and the perfection of number system theory in an era reflect the level of mathematics development at that time. Today, the digital system we use has been constructed so completely and meticulously that it has become a basic language and an indispensable tool in various fields of science and technology and social life. When we enjoy this wealth of human civilization easily, do we think of the twists and turns and hardships that human wisdom has experienced in the historical process of the formation and development of the number system? In the period of ignorance of evolution, human beings have a kind of "ability to recognize numbers", which psychologists call "perception of numbers". Animal behaviorists believe that this "digital perception" is not unique to human beings. The excellence of human intelligence lies in their invention of various counting methods. "Zhouyi under Cohesion" records that "in ancient times, it was ruled by knots, and later saints used books easily." Zheng Xuan in the Eastern Han Dynasty said: "The big event is the big knot; Small things, sum up their tricks. How many knots, the number of things. " In fact, the method of counting by knotting and writing stories is all over the world, such as Greece, Persia, Rome, Palestine, Islam and Central American countries, where there are literature records and physical specimens. It was not until 1826 that the British Treasury decided to stop using the contract as a legal counter. With the progress of human society, digital language is constantly developing and improving. The first milestone in the development of number system appeared: position system notation. The so-called position notation is to use a small number of symbols to represent different numbers through their arrangement of different numbers. What interests historians and historians of mathematics is that under the influence of natural environment and social conditions, different civilizations have created completely different counting methods. Such as Babylonian cuneiform numeral system, Egyptian hieroglyphic numeral system, Greek alphanumeric system, Mayan numeral system, Indo-Arabic numeral system and China's numeration system.
The earliest developed number system should be a simple grouping system. For example, the Egyptian hieroglyphics in 3400 BC are 10 decimal, but they are not positional. From 3000 BC to 2000 BC, the Babylonians developed a 60-base position number system and adopted the position system, but it was not 10. The most important and wonderful notation is 10 decimal position notation.
The famous French mathematician Laplace (Laplace,1749–1827) once wrote:
All numbers are represented by ten symbols, and each symbol has not only an absolute value, but also a position value. This ingenious method comes from India. This is a far-reaching and important idea. It seems so simple today that we ignore its truly great achievements. But it is precisely because of its simplicity and great convenience for all calculations that our arithmetic ranks first among all useful inventions; When we think that it did not attract the attention of the genius thoughts of Archimedes and apollonius, the two greatest figures in ancient times, we feel the greatness of this achievement even more.
Laplace's comments are wonderful, but it's a pity that he is arrogant and blames India for this invention. At present, there are sufficient and conclusive historical data to prove that the decimal notation of 10 originated in China. This is also advocated by some western mathematical historians. Needham once pointed out, "Behind the' Indian Numbers' that the West later became accustomed to, the post system has existed in China for two thousand years." However, the decimal notation of 10 cannot be simply attributed to the wisdom of genius. The progress of notation is related to the improvement of calculation tools. The research shows that the notation of 10 decimal system originated in China, which is closely related to the use of calculation and the evolution of calculation system.
As a vacancy of notation, "0" is indispensable in the civilization of positional notation. The early Babylonian cuneiform and the notation before the Song Dynasty in China all left spaces without symbols. At first, Indians also used spaces to represent zeros, then recorded them as dots, and finally developed them into circular numbers. Indian digital was introduced into Arab countries in the 8th century. /kloc-At the beginning of the 3rd century, Italian businessman Leonardo Fibonacci (1 175- 1250) compiled "Lieber Abachi (1202)", which introduced the complete Indian figures including zero into Europe. Indian numeral and 10 decimal notation have played an important role in the progress of science and civilization in Europe since they were generally accepted by Europeans. The ancient Greeks once asked a question: They thought that the sand in the world was infinite, and no one could write anything more than sand, even if it was not infinite. Archimedes, BC287-2 12+02) Answer: No, in sand counting, Archimedes established a new notation based on myriad, so that any large number can be represented. His practice is: from 1 to 1 billion (originally 1 billion, renamed as 1 billion according to China custom), it is called 1 series; The second series with a unit of 100 million (10 8), from 100 million to 100 million (i.e.10/6) is called the second series; In billions, at most one billion (10 24) is called the third series. Until the last billion in the 65438+ billion series. Archimedes calculated that the amount of sand filling the universe was only10.51. Even if it is extended to the "stellar universe", that is, the celestial sphere with the radius of the distance from the sun to the stars, it can only accommodate 10 63 grains of sand!
The same problem also appeared in ancient China. Before the Han Dynasty, the number was 10, and10,000 was one billion. Wei explained Article 16 of Guoyu Zhengyu: "Hundreds of millions of things, things predicted by materials, things learned from classics and extreme things". Note, "Count, count also; Material, cut also. Tang Jia said that everything is one billion, and Zheng Housinong said: One hundred trillion is one billion, and one billion is a symbol, since ancient times. " A complete set of naming and three methods of carrying large numbers are recorded in Shu Shu. Numerology says:
Huangdi is the law, with ten levels and three usages. The tenth category is billion, trillion, Beijing, easy, easy, soil, ditch, flow, positive and load; Third-class, that is, upper, middle and lower. The next few. Ten changes, if you say one hundred thousand, one billion, ten million, Beijing. Those in the middle of the number will never change it. If they say 100 million, 100 billion, 10 trillion, they will say Beijing. Poverty changes. You say 100 million, 100 million is trillion, and trillion is Beijing. From 1 billion to loading, it's finally great.
The mathematical significance of "The Method of Large Numbers" in Numerology Notes lies not only in its construction of three counting methods, but more importantly, it reveals the difficult course of people's understanding of logarithm from finite to infinite. The objective needs and the development of mathematics urge people to know and master more and more large numbers. At first, for some large numbers, people can understand them and express them with existing numeration units. However, with the development of people's understanding, these large numbers are also expanding rapidly, and the original counting units are difficult to use. People can't help asking:
Is the number bad?
This is an important proposition that needs to be answered in the development of number system. The dialogue between Xu Yue and his teacher Liu Hong recorded in "The Legacy of Numerology" brilliantly illustrates the profound truth of "if you are poor, change it":
Xu Yue asked: Is this figure poor?
Hui Ji (Liu Hong) replied: I once traveled in Tianmu Mountain. When I saw a hermit, I didn't know his name. I called him Mr. Tianmu, and that's what I asked him. Mr. Wang said: things can't be compared with three and two, so why bother with four dimensions? If you don't know three, just talk about knowing ten. You can know tens of billions without distinguishing the size. The yellow emperor is the law, and there are ten. ..... from billion to load, finally great.
Hui Ji asked: Sir, if there are fewer people on the list, it will change. Since the cloud is ultimately big and its spread is limited, how can it be infinite?
Mr. Wang replied: use numbers, the words change again, the small ones become bigger, and the cycle is added. Circulation and poverty!
Mr. Tianmu's approach is to understand infinity with the help of the "circular theory" of "small and broad", and the important idea guiding this approach is "emphasizing that words will change" Even today, the profound philosophy contained in the simple dialectical thinking of "poverty changes" is still worth pondering. The appearance of position notation indicates that the number language mastered by human beings has developed from a small number of characters to a number system with perfect operation rules. The first number system known to mankind is the "natural number system". However, with the development of human understanding, the defects of natural number system are gradually exposed. First of all, the natural number system is discrete rather than dense [2]. Therefore, as a representation of quantity, it can only be limited to an integer multiple of unit quantity, but not to its part. At the same time, as a means of operation, only addition and multiplication can be carried out in the natural number system, but their inverse operations cannot be carried out freely. These defects are compensated by the appearance of scores and negative numbers.
Interestingly, these scores also have strong regional characteristics. Babylon's score is decimal 60, Egypt uses single score, and Arabia's score is more complicated: single score, unit score and compound score. This complex fractional representation will inevitably lead to complex fractional operation methods, so the European fractional theory has been stagnant for a long time, and it was not until the15th century that modern fractional algorithm was gradually formed. In sharp contrast, China made outstanding contributions to the theory of fractions in ancient times.
The original concept of fraction comes from the division of quantity. For example, Shuo Wen Ba Bu explains "fen": "fen, not also. From the perspective of eight knives, the knives are also separate. " The fraction in "Nine Chapters Arithmetic" is introduced from division operation. His "Unity and Separation" has a cloud: "It is as real as the first law. Those who are dissatisfied with the law will be sanctioned by law. " Divide the dividend by the divisor. If inseparable, define a score. The genius of China's ancient fractional theory lies in that it grasped the essence of fractional algorithm with the help of "homology": general fraction. Liu Hui said in the Nine Chapters Arithmetic Note:
Many points are mixed and not detailed. Take it away, so pass it on. If it passes, it can be merged. Where the mother multiplies, it is called qi, and the multiplication of group mothers is called the same. Same person, same phase, same mother. Qi, the mother and son are qi, and the potential cannot be lost.
In the same way, you can divide the scores into the same kind, disguised and opposite. Liu Hui knew the secret and said, "However, you must have the same skills. Do you still admire the complexity and dynamic harmony? Untie the knot and ignore it. Multiplying by dispersion is about and connected with poly. This is a discipline. "
It is easy to prove that the fractional system is a dense system, and it is closed for addition, multiplication and division. In order to make subtraction work smoothly in the digital system, the appearance of negative numbers is inevitable. Surplus and deficiency, income and expenditure, increase and decrease are all examples of the concept of negative numbers in life, and textbooks often follow this path when teaching students negative numbers. This leads to a misunderstanding: it seems that human beings have introduced negative numbers from this understanding of quantities with opposite meanings. Historical facts show that negative numbers were first introduced by mathematicians in China, which was determined by the characteristics of highly developed algorithm and mechanized calculation in ancient traditional mathematics in China. The concept and algorithm of negative numbers first appeared in the Equation of Nine Chapters Arithmetic, because when adding, subtracting, multiplying and dividing between two lines of the Equation, it is necessary to introduce negative numbers and establish an algorithm of positive and negative numbers. Liu Hui's notes profoundly clarify this point: the gains and losses of the two calculations are opposite, and the pros and cons should be corrected. Positive is red, negative is black, otherwise, oblique is different. The equation has its own skills of taking red and black phases and deducing left and right numbers. However, the decreasing trend cannot be widely communicated, so the red and black phases are eliminated. ..... So the mixture of red and black is enough to set the course up and down. Although the loss of profit is extremely enough to exceed the left and right numbers, the difference is enough to meet the slip rate. However, there is nothing wrong, there is nothing wrong, and there is nothing wrong with its rate.
Although negative numbers spread to Europe through Arabic works, most mathematicians in 16 and 17 centuries did not recognize them as numbers, or even if they did, they did not consider them as the roots of equations. For example, Nicolas Chuquet and Stifel both described negative numbers as absurd numbers and "absurdly below zero". Cardan (150 1- 1576) regards negative numbers as the roots of equations, but thinks that they are impossible solutions and just symbols. He called the negative root an imaginary root. Veda (Vieta, 1540- 1630) doesn't want negative numbers at all, while Pascal (1623- 1662) thinks that subtracting 4 from 0 is pure nonsense.
Negative numbers are the first time that human beings have crossed the range of positive numbers, and all previous experiences are completely useless in front of negative numbers. In the historical process of the development of number system, practical experience is sometimes not only useless, but also an obstacle. As we will see, negative numbers are not the only example. The discovery of irrational numbers shattered the Pythagorean dream that "everything is valuable". At the same time, it also exposes the defects of the rational number system: although the rational numbers on the straight line are "dense", there are many "pores" and many "uncountable numbers", so the ancient Greeks' assumption that rational numbers are a continuous arithmetic continuum is completely shattered. Its collapse will have a far-reaching impact on the development of mathematics in the next two thousand years. What is the essence of incommensurability? There have been many different opinions for a long time. The ratio of two incommensurable measures is also considered an unreasonable number because it cannot be correctly explained. Leonardo da Vinci (1452- 15 19) called them "irrational numbers", although Kepler (J. Kepler, 157 1-65438) gradually used these "irrational numbers" in later operations.
When dealing with the root problem, ancient mathematics in China will inevitably encounter unreasonable roots. This "inexhaustible" number is directly accepted in Nine Chapters of Arithmetic, and the "finding the difference" in Liu Hui's note actually uses the decimal of 10 to infinitely approach the irrational number. This is a correct way to complete the real number system, but Liu Hui's thought far exceeded his time and failed to attract the attention of future generations. However, China's traditional mathematics pays attention to the calculation of quantity and has little interest in the nature of logarithm. Greeks who are good at asking questions can't get through this hurdle. Since we can't overcome it, we must avoid it. Since then, Greek mathematicians such as eudoxus and Euclid have strictly avoided equating numbers with geometric quantities in their geometry. Eudoxus's proportional theory (see the fifth volume of Elements of Geometry) made geometry logically bypass the obstacle of incommensurability, but for a long time after that, it formed a significant separation between geometry and arithmetic.
The development of calculus in 17 and 18 centuries attracted the attention of almost all mathematicians, and it was precisely people's attention to the foundation of calculus that made the continuity of real number field stand out again. Because calculus is variable mathematics based on limit operation, and limit operation needs a closed number field. Irrational number is the key to the continuity of real number field.
What is an irrational number? French mathematician Cauchy (A.Cauchy, 1789- 1875) gave the answer: irrational number is the limit of rational number sequence. But according to Cauchy's definition of limit, 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 number in the sequence can be arbitrarily small when the sequence tends to infinity. But where did this pre-existing number come from? In Cauchy's view, the limit of rational number 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 concept based on geometric intuition for more than two thousand years.
The historical task of constructing a complete number field independently from variable mathematics was finally completed by Wilstras (18 15- 1897), Dai Dejin (R. Dedekind1831-
1872 is the most memorable year in the history of modern mathematics. This year, F. Kline (1849-1925) put forward the famous Erlanger program, and Wilstras gave a famous example of a function that is continuous everywhere but differentiable everywhere. It was also in this year that three schools of real number theory appeared: Dai Dejin's "division" theory; Cantor's "basic sequence" theory and Wilstrass's "bounded monotone sequence" theory both appeared in German.
The purpose of trying to establish real numbers is to give a formal logical definition, which does not depend on the meaning of geometry and avoids the logical error of defining irrational numbers with limits. With these definitions as the basis, there will be no theoretical cycle in the derivation of the basic theorem of limit in calculus. Derivative and integral can therefore be directly based on these definitions, without any properties associated with perceptual knowledge. The concept of geometry cannot be fully understood and accurate, which has been proved in the long years of the development of calculus. Therefore, the necessary strictness can only be achieved through the concept of number and after cutting off the relationship between the concepts of number and geometric quantity. Here, Dai Dejin's work is highly appraised, because the real number defined by "Dai Dejin's Division" is an intuitive creation of human wisdom completely independent of space and time.
In essence, the theory of three schools of real numbers gives a strict definition of irrational numbers, thus establishing a complete real number field. The successful construction of real number field has completely bridged the gap between arithmetic and geometry for more than two thousand years, and irrational numbers are no longer "irrational numbers". The ancient Greeks' idea of arithmetic continuum was finally realized in a strict scientific sense. The evolution of the concept of complex number is the most peculiar chapter in the history of mathematics, that is, the historical development of number system is completely inconsistent with the logical continuity described in textbooks. People didn't wait until the logical foundation of real numbers was established, so they began to try a new journey. In the historical process of the expansion of number system, many intermediate fields have not been fully understood, and the intuition of genius has reached the distant outpost with the pace of the brave.
1545, Europeans did not fully understand negative numbers and irrational numbers, but their intelligence was challenged by a new "monster". For example, Kadan put forward a problem in the book Important Art (1545): divide 10 into two parts, so that its product is 40. This requires solving the equation x (10-x) = 40, and the roots he got are 5-√- 15 and 5+√- 15. Then he said that "no matter how many consciences are, they will be blamed" and put 5+√- 15 and 5-. So he said, "Arithmetic is going on so wonderfully, and its goal, as the saying goes, is exquisite and useless." Descartes (1596- 1650) also abandoned plural roots and created the name "imaginary number". Leibniz (1646- 17 16) can best represent the understanding of complex numbers: "The Holy Spirit has found an extraordinary display in the spectacle of analysis, which is the end of the ideal world and an amphibian between existence and non-existence, and we call it virtual -65438+.
It was not until18th century that mathematicians established some confidence in complex numbers. Because, in the middle step of mathematical reasoning, no matter where complex numbers are used, the results are proved to be correct. Especially in 1799, the proof of "Basic Theorem of Algebra" by Gauss (1777- 1855) must depend on the understanding of complex numbers, thus further consolidating the position of complex numbers. Of course, this does not mean that people's worries about "plural" have been completely eliminated. Even in 183 1 year, de Morgan (1806- 187 1) still thinks in his book "Research and Difficulties in Mathematics":
……
Facts have proved that scores are meaningless, even contradictory and absurd. However, through these symbols, an extremely useful part of algebra is established, which depends on the fact that the general laws of algebra can be applied to these formulas (complex numbers).
……
We know that18th century is a "heroic century" in the history of mathematics. People's enthusiasm is how to exert the power of calculus and expand the territory of mathematics. No one will worry about the logical basis of real number system and complex number system. Since complex numbers are at least intuitive and reliable in arithmetic, why bother?
1797, Norwegian wessel (C. Wessel, 1745- 18 18) wrote an essay "Analytic Representation of Direction", trying to express complex numbers with vectors. Unfortunately, the great value of this article was not translated into French until 1897. The Swiss Arganda (J. Argand, 1768- 1822) gave a slightly different geometric interpretation of complex numbers. He noticed that a negative number is an extension of a positive number, which is a combination of direction and size. His idea is: can we expand the real number system by adding some new concepts? Gauss's work is more effective in getting people to accept complex numbers. He not only expressed a+ bi as a point (A, B) on the complex plane, but also expounded the geometric addition and multiplication of complex numbers. He also said that if the sum of 1,-1 is not called positive, negative and imaginary units, but called straight, negative and horizontal units, then people may not have all kinds of dark and mysterious impressions on these numbers. He said that geometric representations can really give people a new view of imaginary numbers. He introduced the term "complex number" to oppose imaginary number and replaced it with I.
Hamilton (1805–1865), an Irish mathematician, is very important in clarifying the concept of complex numbers. Hamilton is concerned with arithmetic logic and is not satisfied with geometric intuition. He pointed out that the complex number a+ bi is not the real sum in the sense of 2+3, the use of the plus sign is a historical accident, and bi cannot be added to A. The complex number a+ bi is just an ordered number pair (A, B) of real numbers, and four operations of the ordered number pair are given. At the same time, these operations satisfy the law of association, exchange rate and distribution rate. In this view, not only complex numbers are logically based on real numbers, but also the mysterious square root of-1 has been completely eliminated. Looking back at the historical development of the number system, it seems to give people the impression that every expansion of the number system is to add new elements to the old number system. If integers add fractions, positive numbers add negative numbers, rational numbers add irrational numbers, and real numbers add complex numbers. However, from the point of view of modern mathematics, the expansion of number system is not to add new elements to the old number system, but to build a new algebraic system outside the old number system. Its elements can be completely different from the old ones in form, but it contains a subset that is isomorphic to the old algebraic system, and this subset must maintain the same algebraic structure between the old and new algebraic systems. When people define the concept of complex number, the new question is: Can we extend it while maintaining its basic properties? The answer is no, when Hamilton tried to find the simulation of complex numbers in three-dimensional space, he found himself forced to make two concessions: first, his new number should contain four components; Second, he must sacrifice the multiplicative commutative law. These two characteristics are a revolution to the traditional number system. He called this new number quaternion. The appearance of "quaternion" marks the end of the expansion of number system under the traditional concept. In 1878, Fubini (F. Frobenius,1849–1917) proved that the algebra of real coefficients with finite primitive units and multiplication units is first of all an associative law. If it obeys the associative law, there are only algebras of real numbers, complex numbers and real quaternions.
Once mathematical thinking breaks through the traditional mode, it will produce immeasurable creativity. Hamilton's invention of quaternion made mathematicians realize that since we can abandon the interchangeability of real numbers and complex numbers to construct a meaningful and effective new "number system", then we can freely consider or even deviate from the algebraic construction of real numbers and complex numbers. Although the expansion of the number system ended here, the door of abstract algebra was opened.
202 1 year personal summary of dentists 1
-The Department of Stomatology of the Second Affiliated Hospital of College is a g