Shulou(Shulou.com)11/24 Report--

Archimedes, Newton and Gauss form their own rank among the great mathematicians, trying to arrange their positions according to their achievements, which is beyond the reach of ordinary people. All three made waves in pure mathematics and applied mathematics: Archimedes rated his pure mathematics as higher than its applied mathematics; Newton applied his mathematical inventions to science; and Gauss declared that it was the same for him to do pure mathematics or applied mathematics. However, Gauss still regarded advanced arithmetic (the most impractical mathematical research of his time) as the queen of all mathematics.

Prince Gauss of mathematics, the son of a poor family, was born on April 30, 1777 in a cottage in Brunswick, Germany. In the whole history of mathematics, there has never been anyone as precocious as Gauss. People don't know when Archimedes showed signs of genius. Newton may not have been noticed when he first showed great mathematical talent. It's hard to believe, but Gauss showed his talent before he was three years old. In his later years, Gauss liked to joke that he knew how to count before he could speak. He has maintained an extraordinary ability to do complex mental calculations all his life.

Gauss entered his first school just after the age of seven. Gauss began to take arithmetic classes at the age of 10. In his early study, Gauss developed a major interest in his life. He soon mastered the binomial theorem

Where n is not necessarily a positive integer, it can be any number. If n is not a positive integer, the series on the right is infinite. In order to show when this series is really equal to (1x) ^ n, we must study what restrictions need to be imposed on x and n in order to make the infinite series converge to a definite finite limit. Because, if we draw the absurd conclusion (1-2) ^-1, that is, (- 1) ^-1, that is,-1, which is equal to 1 ^ 2 + 2 ^ 2 + 2 ^ 3 +. Even infinity; that is to say,-1 equals infinity, which is obviously absurd.

Gauss's early encounter with the binomial theorem inspired him to do some of the greatest work, and he became the first "stricter". When n is not an integer greater than zero, the proof of binomial theorem is beyond the scope of elementary textbooks even today. Gauss was not satisfied with the proof in the book, and Gauss made another proof, which led him to enter into mathematical analysis. The true essence of analysis lies in the correct use of infinite processes.

Gauss is going to change the whole face of mathematics. Newton, Leibniz, Euler, Lagrange, Laplace-- all great mathematicians of their time-- actually had no idea of what is now acceptable proof involving infinite processes. The "proof" that Gauss first saw clearly that could lead to absurd conclusions such as "- 1 equals infinity" is not proof at all. Even in some cases, a formula provides contradictory results, it has no place in mathematics, unless strict conditions are determined, it can continue to produce contradictory results under these conditions.

Under the influence of his own habits and those of his contemporaries, Abel, Cauchy, and his successors, Verstras and Dedkin, Gauss gradually dwarfed other areas of mathematics. Gauss's later mathematics became completely different from that of Newton, Euler and Lagrange.

In a positive sense, Gauss is a revolutionary. By the age of 12 he had looked skeptically at the basis of Euclidean geometry; by the age of 16, he had caught his first glimpse of another kind of geometry different from Euclidean geometry. A year later, he began to tentatively criticize the proofs that his predecessors were satisfied with in number theory, and engaged in filling in the gaps. Arithmetic was the first area of his success and became the battleground for his great works. Gauss has a certain perception of what is the essence of proof, and at the same time has unsurpassed and rich mathematical creative ability. The combination of the two is invincible.

Gauss was strongly attracted by philosophical research, but he soon found a more fascinating attraction in mathematics, which is lucky for science. Gauss had mastered Latin when he entered college, and many of his greatest works were written in Latin.

Gauss studied at Caroline College for three years, during which time he mastered the more important works of Euler and Lagrange, and most importantly, Newton's "principles." The highest praise a great man can receive is from another great man of his rank. As a 17-year-old boy, Gauss never underestimated Newton's achievements. Others-Euler, Laplace, Lagrange, Legendre-appear in Gauss's Latin praise as "brilliant"; Newton is "the highest".

While still at Caroline College, Gauss began his study of advanced arithmetic, which later made him immortal. His extraordinary computing power played a key role. He directly explored the numbers themselves, experimented with them, and used induction to discover some esoteric general theorems, which he even had to prove with great effort. In this way, he rediscovered the "treasure of arithmetic", the "Golden Theorem". Euler also discovered it by induction, which is called the law of quadratic reciprocity, and Gauss was the first to prove it.

The whole study begins with a simple question that many beginners in arithmetic ask themselves: how many numbers are there in each cycle of recurring decimals? In order to find clues to the problem, Gauss calculated the decimal representation of all fractions 1 / n for n from 1 to 1000. He discovered something incomparably great-the law of quadratic reciprocity. Because the statement is simple, we will describe it and introduce a revolutionary improvement invented by Gauss in arithmetical terminology and notation, congruence. All the numbers involved below are integers.

If the difference between two numbers (a ≡ b or b m m) is divisible by the number m, we say that a dint b is congruent with respect to module m, or congruence with module m for short, and we use the symbol a ≡ b (mod m) to denote it. So, 100 ≡ 2 (mod7), 35 ≡ 2 (mod11).

The advantage of this method is that it reminds us of the method of writing algebraic equations. Using a concise notation to denote the divisibility of arithmetic, let's introduce some methods that lead to interesting results in algebra into arithmetic. For example, we can "add" some equations, and we find that if modules are all the same, congruences can also be added up to get other congruences.

Let x denote an unknown number, r and m denote a given number, and r is not divisible by m. Is there an x that makes

If there is, r is called a quadratic residue of m, if not, r is called a quadratic non-residue of m.

If r is a quadratic residue of m, then at least one x must be found, whose square is divided by m r; if r is a quadratic non-residue of m, then there is no x whose square is divided by m. These are the direct conclusions defined above.

For example: is 13 the quadratic surplus of 17? If so, congruence must be found.

Use 1, 2, 2, 3, … To try, we found that xylene 8pm 25pm 42pm 59, … They are all solutions, so 13 is a quadratic residue of 17. But x ^ 2 = 5 (mod17) has no solution, so 5 is a quadratic non-residue of 17.

Now, naturally, what are the quadratic residues and quadratic non-residuals of a given m? That is to say, given m in x ^ 2 ≡ r (modm), when x takes all the numbers 1, 2, 3, … What kind of number r can appear and what kind of number r can't appear?

It doesn't take much effort to show that it is enough to limit r and m to prime numbers to answer this question. So let's restate the problem: if p is a given prime, what kind of prime Q can make congruence x ^ 2 ≡ Q (mod p) solvable? In the current state of arithmetic, this is too much. However, this situation is not hopeless.

There is a wonderful "reciprocity" in the following pair of congruences.

Where p and Q are primes: if Q ≡ 1 mod 4, then the congruence x ^ 2 ≡ p mod q is solvable if and only if x ^ 2 ≡ q mod p is solvable. If Q ≡ 3mod 4 and p ≡ 3mod 4, then the congruence x ^ 2 = p mod q is solvable if and only if x ^ 2 ≡-q mod p is solvable.

It is not easy to prove. In fact, it puzzled Euler and Legendre, and Gauss gave his first proof at the age of 19. Since this law of reciprocity is very important in many advanced parts of higher arithmetic and algebra, Gauss tried to find its root. He pondered it over and over for many years, until he gave a total of six different proofs, one of which depended on the ruler drawing of the regular polygon.

When Gauss left Caroline College and went to Gottingen University in October 1795 at the age of 18, he still could not decide whether to make mathematics or philosophy his life's career. He has discovered the "least square" method (at the age of 18), which is indispensable today in geodesy, in the simplification of observations, and in virtually all work to derive the most likely values from a large number of measurements. Gauss and Legendre shared the honor, and Legendre independently published the method in 1806. This work was the beginning of Gauss's interest in the theory of observation error. The Gaussian law of error normal distribution and the bell curve with it are indispensable in statistics.

The turning point on March 30, 1796 marked a turning point in Gauss's life. On that day, exactly a month before his 20th birthday, Gauss explicitly decided to pursue a career in mathematics. Learning language is still a lifelong hobby, but philosophy lost Gauss forever on this unforgettable day in March.

On the same day, Gauss began to keep his science diaries, which are one of the most valuable documents in the history of mathematics. The first article records his great discovery. It was only in 1898, 43 years after Gauss's death, that the diary was spread in the scientific community, when the Royal Academy of Gottingen borrowed it from one of Gauss's grandsons for forensic research. All of Gauss's discoveries during the prolific period from 1796 to 1814 have not been fully recorded. But many of the hasty notes are enough to establish Gauss's lead in such areas, such as elliptic functions.

Several diaries show that the diary is entirely a private matter of its author. As in the diary of July 10, 1796, it is written

In translation, this is an imitation of Archimedes cheering "Eureka!" It shows that every positive integer is the sum of three triangular numbers, and a triangular number is a series of 0pr 1rec 3pr 6e 10je 15, … One of which (after 0) each has the form 1 + 2n (n = 1), where n is any positive integer. Another theory is that every number in the form of 8n+3 is the sum of three odd squares:

It is not easy to prove it.

What is more difficult to understand is the mysterious item in the diary of October 11, 1796, "Vicimus GE-GAN". What kind of strange dragon did Gauss tie up this time? And, on April 8, 1799, when he circled the REV.GALEN in a neat box, what kind of giant did he conquer? Although the meaning of these things has been lost forever, most of the 144 left are clear enough. In particular, there is one of first importance: the diary of March 19, 1797 shows that Gauss has discovered the biperiodicity of some elliptic functions. He was not yet 20 at that time. In addition, a later diary shows that Gauss has seen the biperiodicity of the general situation. If he publishes the result, it will be enough to make him famous. But he never published it.

Why didn't Gauss disclose his great discovery? Gauss said that he engaged in scientific works only out of the deepest inspiration of his nature, and whether they should be published for others was entirely secondary to him. Gauss once said another thing to a friend, explaining his diary and the reasons for his delay in publishing it. Before he was 26, he said, there were so many unstoppable new ideas in his mind that he could hardly control them, and that only a small part of his time could be recorded. The diary contains only a few final brief explanations of the research results that had made him ponder painstakingly for weeks.

Gauss believes that what he leaves behind should be perfect works of art, with more points added and less points reduced. He says a cathedral is not a cathedral until the final scaffolding is removed and removed. With such an ideal job, Gauss would rather refine a masterpiece over and over again than publish a summary of many masterpieces that he could easily write. His motto

Pauca sed matura (less, but mature).

As a result, some of his works have to be explained by gifted interpreters before ordinary mathematicians can understand them and move forward. His contemporaries begged him to relax his stiff and ruthless perfection so that math could move faster. But Gauss never relaxed. It was not until long after his death that people knew how much 19th-century mathematics that Gauss had foreseen and led before 1800. If he publishes the conclusions he knows, it is likely that the current mathematics has advanced half a century or more than it is now. Abel and Jacobi would be able to start where Gauss stopped without having to devote most of their best energy to rediscovering what Gauss knew long before they were born. the creators of non-Euclidean geometry will be able to turn their genius to something else.

Speaking of himself, Gauss said he was "just a mathematician", which was unfair to him, his second motto

Nature, you are my goddess, I am willing to bow to your laws.

It really sums up his life of devoting himself to the mathematics and physical sciences of his time.

The three years at the University of Gottingen were the period in which Gauss wrote the most books in his life (1795Mai 1798). He has been conceiving a great work on number theory since 1795. By 1798, the study of arithmetic had actually been completed. During this period, he also met two mathematicians, Wolfgang Boyer and John Friedrich Pfaff (the most famous mathematician in Germany at that time).

Before describing "arithmetic research", we should take a look at Gauss's doctoral thesis, "A new proof that every rational entire function of a single variable can be decomposed into first-or second-order real factors."

What this paper proves is what we now call the basic theorem of algebra. Gauss proved that all roots of any algebraic equation are numbers in the form of a+bi and I is an imaginary number. This new type of "number" a+bi is called plural.

The word "imaginary number" is the greatest disaster of algebra, but because it has long been recognized, mathematicians cannot cancel it. You shouldn't have used it at all. Many math books use rotation to give a simple explanation to imaginary numbers. Interpret I × c (c is a real number) as the line segment Oc rotates a right angle around point 0, and Oc rotates to OY; then multiply by I, that is, I × I × c, rotate Oc by another right angle, so that the total effect is to rotate Oc by two right angles, resulting in + Oc becoming-Oc. As an operation, the product of I × I multiplication has the same effect as the product of-1, and the product of I multiplication has the same effect as rotating a right angle.

Gauss believes that the theorem that every algebraic equation has a root is very important, so he gives four clear proofs, the last of which was given when he was 70. Today, some people will transfer this theorem from algebra to analysis. Even Gauss assumes that the graph of a polynomial is a continuous curve, and if the polynomial is odd, the graph must intersect the axis at least once. This is obvious to anyone who is a beginner in algebra. But today, it is not obvious without proof, and there are once again the difficulties associated with continuity and infinity in trying to prove it. The root of an equation as simple as x ^ 2-2-0 cannot be accurately calculated in any finite step.

The study of arithmetic is Gauss's first masterpiece, which some people think is his greatest masterpiece. After that, he no longer took mathematics as his only interest. When the work was published in 1801 (Gauss was 24 at the time), he expanded his activities to both mathematical and practical aspects in the fields of astronomy, geodesy and electromagnetism. What he regretted later was that he had never taken the time to write the second volume he planned to write when he was young. This book has seven verses.

The first sentence of the preface describes the general scope of the book.

The research results contained in this work belong to the part of mathematics involving integers and fractions, with the exception of irrational numbers.

The first three sections discuss the theory of congruence, especially binomial congruence.

This wonderful arithmetic theory has many similarities with the algebraic theory of the corresponding binomial equation x ^ n = A, but its unique arithmetic part is incomparably richer and more difficult than algebra which has no resemblance to arithmetic.

In section 4, Gauss developed the theory of quadratic residue. The first published proof of the quadratic reciprocity law can be found here. The proof is amazingly mathematical induction and is an excellent example of the ingenious logic that can be found anywhere.

Section 5 first discusses the binary quadratic form from the arithmetic point of view, then discusses the ternary quadratic form, and finds that it is essential to complete the binary theory. The quadratic reciprocity law plays a very important role in these difficult plans. For the first form, the general problem is to discuss the indefinite equation.

For the second form, the subject of study is the equation.

The integer solution of x ~ y ~ ~ z, where a ~ b ~ m is a given integer. One of the seemingly easy but actually difficult problems in this field is to impose a guaranteed indefinite equation on a _

There are sufficient and necessary limitations for the integer solution x _ (1) y _ (1) Z.

Section 6 applies the previous theory to a variety of special cases, such as the integer solution x ^ y of mx ^ 2 + n y ^ 2 = A, where m ^ 2 nline An is an arbitrary integer.

The culmination of this work is Section 7, where Gauss uses previous developments, especially the theory of quadratic congruence, to brilliantly discuss the algebraic equation x ^ n + 1, where n is any given integer. Thus arithmetic, algebra and geometry are woven into a perfect pattern. The equation x ^ n = 1 is an algebraic formula for drawing geometric problems of n-polygons or n-equipartition circles; the arithmetic congruence x ^ m ≡ 1 (mod p) runs through algebra and geometry and gives the pattern a simple clue.

In the past, some people-Fermat, Euler, Lagrange, Legendre and others-have done many of this in other ways, but Gauss discussed it entirely from his personal point of view, adding a lot of his own things. and derived many isolated results from his general formulas and answers to the relevant questions. For example, Fermat used his method of "infinite descent" to prove that every prime in the form of 4n+1 is the square sum of two numbers, and there is only one way to express it; his wonderful conclusion is the natural result of Gauss's general exposition on the binary quadratic form.

Gauss said in his later years that "arithmetic research" is a thing of the past. The publication of the study of arithmetic has put forward a new direction for higher arithmetic, so that the number theory, which was a variety of unrelated special results in the 17th and 18th centuries, has now adopted a unified form. rise to the same status as algebra, analysis and geometry in mathematical science.

Dirichlet, a student of Gauss, has an amazing theorem that every arithmetic series

Contains an infinite number of primes, where a _ () ~ (b) is an integer with no common divisor greater than 1. This is proved by analysis, which is a miracle in itself, because the theorem considers integers and analyzes continuous non-integers.

We may ask why Gauss never solved Fermat's Great Theorem. He answered himself:

I am really not interested in Fermat Theorem as an isolated proposition, because I can easily put forward a lot of such propositions that can neither prove nor prove to be true.

Gauss went on to say that the question reminded him of some of his original ideas for a great extension of advanced arithmetic. This undoubtedly refers to the theory of algebra that Kumer, Dedkin and Cronecker will develop independently.

Ceres

The second great stage of Gauss's life began on the first day of the 19th century, which was also marked in scarlet letters in the history of philosophy and astronomy. Since Sir William Herschel discovered Uranus in 1781, increasing the number of planets known at that time to a philosophically satisfactory seven, astronomers have been searching space tirelessly for other members of the solar family. According to Bode's law, they should exist between the orbits of Mars and Jupiter. The search was fruitless until Giuseppe Piazi observed on the first day of the 19th century that he initially mistook it for a small comet approaching the sun, but it was soon recognized as a new planet-later named Ceres, the first of a large group of very small planets known today.

The discovery of Ceres coincided with a satirical attack by the famous philosopher Friedrich Hegel on astronomers looking for the eighth planet. Hegel asserted that if they paid a little attention to philosophy, they would immediately understand that there could only be seven planets, no more and no less.

On November 1, 1844, Gauss wrote to his friend Schumacher saying:

You see the same thing in the contemporary philosophers Schelling, Hegel, A. Nuis von Eisenbeck and their followers (mathematical incompetence); read the methods used by Plato and others (except Aristotle) in the history of ancient philosophy. But even for Kant himself, it is often not much better. I think his distinction between analytical propositions and comprehensive propositions is either trivial or wrong.

Gauss has fully mastered non-Euclidean geometry at the time of writing this letter, and non-Euclidean geometry itself is enough to refute Kant's theory of "space" and geometry. It must not be assumed that Gauss does not understand philosophy just because of this isolated example of the terminology of pure mathematics. He knows. All philosophical advances have great charm to him, although he often disagrees with the methods used to make them. He once said

Some questions, such as moving ethics, or our relationship with God, or questions about our destiny and our future, I pay more attention to the answers to these questions than to mathematical ones; but these questions are totally beyond our reach, and they are not within the scope of science at all.

Ceres is a disaster for mathematics. To understand why Gauss takes it so seriously, we must remember that in 1801, Newton's colossal image (more than 70 years after his death) still overshadowed mathematics. Contemporary "big" mathematicians are those who tirelessly completed Newton's celestial mechanics mansion, such as Laplace. Mathematics is still regarded as mathematical physics. The illusion of mathematics as an independent discipline seen by Archimedes in the 3rd century BC has disappeared under the brilliance of Newton. Mathematics was not recognized as an independent science until the young Gauss grasped the illusion again. Just as he began to work nervously in the uncultivated wilderness of the mathematical kingdom, the asteroid Ceres attracted his unparalleled wisdom at the age of 24.

A new planet, ceres, has been discovered in a position that is extremely difficult to observe. It would be difficult for Laplace himself to calculate the orbits of the planets from the meagre data available. Newton once declared that this kind of problem was the most difficult problem in mathematical astronomy. It is necessary to establish an orbit accurate enough to ensure that Ceres can be seen with telescopes as it orbits the sun, and today's computers are likely to be baffled by the arithmetic needed to establish this orbit.

Gauss, the unprecedented god of mathematics, described the faint and elusive things in his diary. Ceres was rediscovered, exactly where the young Gauss was expected to find it after extremely ingenious and detailed calculations. It took Euler three days to complete the calculation (which is said to have blinded him), and Gauss only a few hours. For nearly 20 years, he spent most of his own time on astronomical computing.

But even such dull and tedious work does not erase Gauss's creative genius. In 1809 he published his second masterpiece, the Theory of Celestial bodies moving around the Sun along a Cone Section, which discussed in detail the determination of the orbits of planets and comets based on observed data, including difficult perturbation analysis. the laws that govern computational astronomy and practical astronomy for many years to come.

In 1807, Gauss was appointed director of the Gottingen Observatory. The Gottingen Observatory was able to pay Gauss a small salary at the time, but it was enough to meet the simple needs of Gauss and his family. As his friend Von Walter Shawson wrote:

As when he was young, he remained a simple Gauss throughout his old age until the day he died. A small study, a small workbench covered with a green tablecloth, a necessary desk painted white, a single sofa, and after the age of 70, there was an armchair, a lamp with a lampshade, an unlit bedroom, a simple diet, a dressing gown and a velvet cap, these were enough to meet all his needs.

Analytic function if Gauss disclosed a discovery he confided to Bessel, 1811 might be a mathematical milestone comparable to 1801 (the year arithmetic Research was published). Gauss has fully understood complex numbers and their geometric representations of points on the plane of analytic geometry, and he has posed to himself the problem of studying such numbers, which is now called analytic functions.

Plural z=x+iy. The point z moves on the plane when xPerry y takes its own real values in any specified continuous way. When you assign a value to z, take any single-valued expression that contains z, such as z or 1 / z, and so on, called a single-valued function of z. We use f (z) to denote such a function. Therefore, if f (z) is a specific function z, such that

So obviously when any value is assigned to z, for example, x, 2, 2, and 3, this f (z) therefore determines exactly a value, z, z, 5, 2, 12i.

Not all single-valued functions f (z) have to be studied in the theory of functions of single and complex variables; only single functions are selected for detailed discussion.

Move z to another location z'. The function f (z) takes another value f (x') and is replaced by x'. Now use the difference between the new value and the old value of the variable to divide the difference between the new value and the old value of the function f (z')-f (z), so that there is

Just as we do when calculating the slope of a graph to find the derivative of the function represented by the graph, here we make z 'infinitely close to z so that f (z') is close to f (z) at the same time. But there is a noteworthy new phenomenon here.

There is no unified way for x'to move to coincide with z, because z 'can move in the plural plane through an infinite number of different paths before it coincides with z. We can't expect that (f (z')-f (z)) / (zonal colors z) have the same limit values for all these paths when z 'coincides with z, generally speaking, they are different.

But if f (z) makes the limit just described the same for all the paths that z 'takes when moving to coincident with z, then f (z) is said to be solo at z (or at the point that represents z).

Consistency and singularity are the special characteristics of analytic functions of single complex variables.

The broad field of fluid motion theory is dealt with by univariate analytic functions, from which some important significance of analytic functions can be inferred. Suppose such a function f (z) is divided into a "real" part and a "imaginary" part, such as f (z) = U+iV. For the special analytic function z ^ 2, we have

Imagine a fluid layer flowing on a plane. If there is no eddy current in the motion of the fluid, the streamline of the motion can be obtained by drawing the curve Usinga, where an is an arbitrary real number, which is obtained by some analytic function f (z), and the equipotential line can also be obtained from Vedoub. If we let aforme b change, we will get a complete motion graph, which can be as big as we want. For a given case, such as the case of a fluid flowing around an obstacle, the difficult part of the problem is what kind of analytical function to choose. So the whole thing is the other way around: study some simple functions and find the physical problems they are suitable for. Oddly enough, many of these man-made problems have proved to be the most useful in other practical applications of aerodynamics and fluid motion theory.

The theory of analytic function of single complex variable is one of the greatest fields of success in mathematics in the 19th century. In his letter to Bessel, Gauss explained how important the basic theorem in this theory was, but he did not make it public and left it to Cauchy and later Weierstras to rediscover. Since this is a milestone in the history of mathematical analysis, we will briefly describe it.

Imagine the simple and complex variable z moving on a closed curve of finite length without kink. Mark n points on the curve, P 1 and P 2, … , Pn . Make P1P2, P2P3, P3P4, … Each segment of the PnP1 does not exceed a pre-specified finite length l On each such segment, select a point at both ends of an off-line segment, form a value of f (z) for the value of z corresponding to that point, and multiply this value by the length of the segment in which the point is located. Do this for all segments and add up the results. Finally, when the number of segments increases infinitely, take the limit of this sum. This gives the "line integral" of f (z) for the curve.

When will this line integral be zero? In order to make the line integral zero, the sufficient condition is that f (z) is analytic (consistent and monotonic) at every point z on and within the curve.

Beyond the geometric series, this is the great theorem that Gauss told Bessel in 1811. It and another theorem of the same type, in the hands of Cauchy who independently rediscovered it, will produce many important results in analysis in the form of inferences.

In 1812, Napoleon's army struggled desperately to fight guards across the frozen plains, and it was in that year that Gauss published another great work about transcending geometric series.

Where the imaginary point indicates that the series continues indefinitely according to the law shown, the next item is

This research report is another milestone. As has already been pointed out, Gauss is the first strict in modern times. In this work, in order to make this series converge, it is necessary to impose some restrictions on a ~ (?) As a special case, it includes many important series in analysis, such as the series used in the calculation and tabulation of logarithms, trigonometric functions and other functions that appear repeatedly in Newtonian astronomy and mathematical physics; the generalized binomial theorem is also a special case. By studying the general form of this series, Gauss solved many problems at one stroke. From this work, many applications of differential equations in 19th century physics have been developed.

Although it is impossible to discuss many examples of Gauss' contribution to pure mathematics due to the limitation of space, even in the simplest overview, one example can not be ignored, that is, the work on the law of biquadratic reciprocity. Its importance is that it provides a completely unexpected new direction for advanced arithmetic.

Now that the problem of quadratic reciprocity has been solved, it is natural for Gauss to consider the general problem of binomial congruence of any number of times. Let m be a given integer that is not divisible by prime p, and let n be a known positive integer. If an integer x can be found, such that

Then m is called an n-th residue of p; when n is 4, m is a biquadratic residue of p.

In the case of quadratic binomial congruence (nasty 2), there is almost no hint when n exceeds 2. Gauss wants to discuss these higher order congruences and study the corresponding reciprocity laws, that is, the relationship between x ^ n ≡ p (mod Q) and x ^ n ≡ Q (mod p) (about solvable or unsolvable). In particular, the case of nasty 3 and 4 needs to be studied.

The paper of 1825 broke new ground. After many unbearable mistakes, Gauss found that rational integers, 1, 2, 3, … It is not suitable for the discussion of biquadratic reciprocity law; a new class of integers must be invented. These are called Gaussian complex numbers, and they are all those complex numbers in the form of a+bi, where aforme b is rational. In order to explain the biquadratic reciprocity law, the arithmetic divisibility of these complex integers must be discussed in detail. Gauss made such a discussion, so he began the theory of algebra. In the same way, he found the right path for triple reciprocity (nasty 3).

Gauss's favorite disciple Eisenstein solved the problem of reciprocity three times. He also found an amazing connection between the biquadratic reciprocity law and some parts of elliptic function theory, in which Gauss has done in-depth research.

Gauss has also made equally important progress in the application of geometry and mathematics to geodesy, Newtonian theory of gravity and electromagnetism. How can one accomplish such a large amount of work of the highest level? Gauss said, "if other people think about mathematical truth as deeply and persistently as I do, then they can also make these discoveries that I have made."

Gauss couldn't help focusing on mathematical ideas. When he was talking to his friends, he would suddenly fall silent, immerse himself in thoughts beyond his control, and stand motionless, gazing blankly at everything around him. Then he controlled his mind and consciously devoted all his strength to solving a difficult problem until he succeeded. Once he catches a problem, he won't let go until he conquers it, although he may focus on several problems at the same time.

In one such example, he describes how he has spent almost a week for four years without spending some time trying to figure out whether a definite symbol is positive or negative, and the answer suddenly appears. Often after days or weeks of fruitless research, Gauss continued to work after a sleepless night when he found that the obstacles disappeared and all the answers flashed clearly in his mind. His ability to concentrate nervously and persistently is one of his strengths.

Gauss's ability to forget himself in his own thinking world is similar to Archimedes and Newton. He is also on a par with them in the other two aspects: he has a gift for precise observation and scientific originality. These talents enabled him to design the instruments necessary for his scientific research. The reflector in geodesy is attributed to Gauss, which is an ingenious device in which signals can be transmitted instantly on the ground by reflected light. The reflector was a great progress at that time. In Gauss's hands, the astronomical instruments he used have also been significantly improved. In order to use it in his important research on electromagnetism, Gauss invented the double-wire magnetometer. Finally, he invented the Telegraph in 1833 and used it with William Weber, who worked with him, to send messages. The combination of mathematical genius and first-rate experimental talent is an extremely rare situation in all science.

Gauss himself paid little attention to the possible practical use of his invention. Like Archimedes, he preferred mathematics to all the kingdoms of the earth. But Weber clearly saw what Gottingen's little telegram meant to civilization. We remember that railways first appeared in the early 1930s, and Weber predicted in 1835, "when the world is covered with a network of railways and telegrams, the service provided by this network can be comparable to that of the human nervous system." partly as a means of transport, partly as a way to spread ideas and events at lightning speed. "

Gauss and Legendre

One experience made Legendre the lifelong enemy of Gauss. Gauss mentioned the least square method which he discovered a long time ago in his Theory of Celestial Motion. Legendre published this method in 1806 before Gauss. He wrote to Gauss with great anger, actually accusing him of dishonesty and complaining that Gauss had so rich discoveries that he could have considered decency without stealing the least square, which Legendre regarded as his own most cherished thing. Laplace joined the quarrel. He did not say whether he believed that Gauss was 10 years or earlier ahead of Legendre, but he maintained his usual gentle attitude.

Gauss obviously disdained to argue any further on the matter. But in a letter to a friend, he pointed out the evidence that would have ended the argument if Gauss had not been so "arrogant and disdainful to quarrel." "I told Obers the whole problem in 1802," he said. " And if Legendre had doubts, he could have asked Obus, who had a manuscript.

This argument was very detrimental to the later development of mathematics because Legendre told Jacobi his unfounded suspicions, which prevented Jacobi from establishing a close relationship with Gauss. What is particularly regrettable in this misunderstanding is that Legendre is a man of noble character and he himself is extremely fair. He was destined to be surpassed by mathematicians who were more imaginative than he was in some fields in which most of his long and hard-working life was spent, and his hard work was proved superfluous by young people-Gauss, Abel and Jacobi. Gauss was ahead of Legendre at every step. However, when Legendre accused Gauss of being unfair, Gauss felt that he himself was in trouble. He wrote to Schumacher complaining that

It seems that I was destined to crash with Legendre in almost all my theoretical work. In advanced arithmetic, this is true in the study of transcendental functions related to the ellipse length method (the process of finding the arc length of a curve), in the basis of geometry, but now, in the least square method, …... It's also used in Legendre's work, and it's really beautiful.

Gauss is criticized for his lack of enthusiasm for the great work of others, especially the work of younger people. When Cauchy began to publish his brilliant discoveries in the theory of functions of single and complex variables, Gauss turned a deaf ear to them. Gauss did not say a word of praise or encouragement to Cauchy, because Gauss himself had reached the core of the problem many years before Cauchy began the work. Also, when Hamilton's work on quaternions came to his attention in 1852, he said nothing, because the crux of the problem had already been written down in his notes more than 30 years ago. He kept silent and did not raise his priority. Just like his leading position in the theory of functions of single complex variables, elliptic functions and non-Euclidean geometry, Gauss was content to do this work.

Other great contributions

To illustrate all the outstanding contributions of Gauss to mathematics, pure mathematics and applied mathematics, it is necessary to write a very thick book. Here we can only consider some important work that has not yet been mentioned, and we will choose those that have added new methods to mathematics or successfully solved outstanding problems. From a rough but convenient schedule, we summarize the main areas of interest that Gauss was interested in after 1800:

1800 Murray 1820, astronomy

1820 Murray 1830, geodesy, surface theory, conformal mapping

Mathematical physics, especially electromagnetism, geomagnetism, and the theory of gravity based on Newton's laws, 1830-1840,

1841 Murray 1855, topology, geometry associated with functions of a single complex variable.

In 1821 Murray, in 1848, Gauss was a scientific adviser to the governments of Hanover and Denmark for large-scale geodetic surveys. Gauss took an active part in the work. His least square method and his skills in designing formats for dealing with large amounts of numerical data have been brought into full play, but more importantly, the problems in the accurate measurement of some geodetic surfaces, no doubt raises deeper and more general questions related to all surfaces. These studies will lead to the mathematics of relativity. Several of Gauss's predecessors, especially Euler, Lagrange and Monge, have studied some types of surface geometry, but it is still up to Gauss to solve all general problems. the first great period of differential geometry came into being from his research.

Differential geometry can be roughly described as the study of the properties of curves, surfaces, and so on near a point (so close that the power of the distance higher than the quadratic power can be omitted). Inspired by this work, Riemann wrote a classic paper on the assumptions that form the basis of geometry in 1854, and then began the second great period of differential geometry, which is now used in mathematical physics, especially general relativity.

In his works on surfaces, Gauss considered three problems and put forward theories of great significance to mathematics and science. these three problems are curvature measurement, conformal representation (that is, mapping) and surface attachability.

The "curved" spacetime is an extension of the pure mathematics of the curvature commonly seen in a "space" described by four coordinates instead of two. This not-so-mysterious generalization is a natural development of Gauss's work on surfaces. One of his definitions illustrates the reasonableness of all this. The problem is to imagine some precise ways to describe how the "curvature" of a surface changes from one point on the surface to another; this description must be consistent with our intuitive sense of "bending much" and "not bending much".

The total curvature of any part of a surface surrounded by a closed curve C without knot is defined as follows. The normal of a surface at a given point is a straight line passing through that point, which is perpendicular to the plane tangent to the surface at the given point. There is a normal of the surface at every point of C. Imagine all these drawn normals. Now, imagine a ball with a radius of unit length, drawing all the rays parallel to the normal of C from the center of the ball. These rays will hand over a curve on a sphere with a unit radius, such as C'. The area of the part of a sphere surrounded by C'is defined as the total curvature of that part of a given surface surrounded by C. If you imagine it a little bit, you can see that this definition is consistent with the general concept required.

Another basic concept developed by Gauss in the study of surfaces is parametric representation.

Represents a special point on the plane and requires two coordinates. The same is true on a sphere or a sphere like the earth: in this case coordinates can be thought of as longitude and latitude. This shows what a two-dimensional manifold means. Generally speaking, if you want to specifically represent each special member of a class of things (points, sounds, colors, lines), exactly n is sufficient and necessary, then the class is said to be an n-dimensional manifold. In such a representation, it is agreed that only certain characteristics of members of this class should be assigned numbers. For example, if we only consider the pitch of the sound, we have an one-dimensional manifold, because a number, that is, the vibration frequency of the sound, is enough to determine the pitch; if we add the volume, the sound is now a two-dimensional manifold, and so on. If we now think of a surface as made up of points, we can see that it is a two-dimensional manifold. We find that it is convenient to use geometric language to describe any two-dimensional manifold as a "surface" and apply geometric reasoning to the manifold-hoping to find something interesting.

The above considerations lead to the parametric representation of the surface. In Cartesian geometry, an equation between three coordinates represents a surface. Let the coordinates of (Descartes) be x ~ y ~ ~ z. We now use three equations to represent the surface instead of a single equation of x _ mai y _ r Z:

Where f (ugraine v), g (umeny v), h (ureco v) are the functions of the new variable uPersonv. When these variables are eliminated, the surface equation of x ~ (~ ()) ~ (()) is obtained. The parameter equation of the surface is called the parameter equation of the surface, and the three equations x _ r _ f (u _ r _ v), y _ r _ g (u _ r _ v) and z _ r _ h (u _ r _ v) are called the parameter equation of the surface. This method of representing surfaces is much better than the Cartesian method when it is used to study the curvature and other properties of surfaces that change rapidly between points.

Note that the parameter representation is intrinsic; its coordinates refer to the surface itself, rather than to a set of external axes independent of the surface, as in the Cartesian method. It should also be noted that the two parameters UBI v directly indicate the two-dimensional properties of the surface. Longitudes and latitudes on Earth are examples of these inherent, "natural" coordinates.

Another advantage of this method is that it can be easily extended to any dimensional space. As long as you increase the number of parameters, it is sufficient to do so as before. These simple ideas led to the generalization of the metric geometry of Pythagoras and Euclid. The foundation of this generalization was laid by Gauss, but their significance for mathematical and physical sciences was not fully valued until the 20th century.

The study of geodesy also suggests to Gauss another powerful method in geometry, that is, the development of conformal mapping. A mapping that maintains an angle is called a conformal mapping. In this kind of mapping, the analytic function theory of single complex variable is the most useful tool. The whole subject of conformal mapping is often used in mathematical physics and its applications, such as electrostatics, hydrodynamics and its branch aerodynamics. In the last discipline, it plays an important role in wing theory.

Another area of geometry in which Gauss has worked carefully and succeeded is the suitability of surfaces, which requires determining what kind of surface can be attached to another given surface without stretching, tearing, or bending. Here, the method invented by Gauss is universal and has a wide range of applications.

Gauss has also done important research in other areas of science, such as electromagnetism (including geomagnetism), capillarity, the attraction between ellipsoids in the law of gravity (planets are special types of ellipsoids), and refraction, especially the mathematical theory of refraction of lens groups. Finally, this department provided him with an opportunity to apply his pure abstract method (continued fraction), which he developed in his youth to satisfy the curiosity of logarithm theory.

Gauss not only made all these things extremely mathematical, but he was also good at applying mathematics to other disciplines with his hands and eyes. Many of the special theorems he discovered, especially those he discovered in the study of electromagnetism and the theory of gravity, have become indispensable tools for all people in physical science. Gauss, with the help of his friend Weber, searched for a satisfactory theory for all electromagnetic phenomena for many years. He gave up the attempt because he could not find a theory that he was satisfied with. If he discovers the Clark Maxwell equation in the field of electromagnetism, he may be satisfied.

Finally, we must mention topology, which he did not publish except by the way in his paper in 1799, but he predicted that it would become a major topic of concern in mathematics.

Gauss's last few years were full of honor, but he did not get the happiness he was entitled to enjoy. A few months before his death, when the deadly disease showed its first symptoms, Gauss was as agile and creative as he used to be. However, he worked as long as he could, and in spite of his hand spasms, his graceful and clear writing was finally illegible. The last letter he wrote was for Sir David Brewster, talking about the invention of the Telegraph.

He was almost awake until the end, and after a struggle to survive, he died peacefully in the early morning of February 23, 1855 at the age of 78. He lives everywhere in math.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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