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Carl Friedrich Gauss (1777-1855) was a child prodigy. A month after 19 years, he made an extraordinary discovery. For 2000 years, people have known how to use rulers and compasses to make equilateral triangles and regular pentagons (and other regular polygons whose sides are multiples of 2, 3, and 5), but they do not know how to make regular polygons with prime sides. Gauss proved that a regular heptahedron can also be made with rulers and compasses.

Gauss commemorates his discoveries by keeping a diary in which he wrote down many of his discoveries over the next 18 years. He achieved a lot of success when he was a student. Some of them are rediscoveries of theorems already proved by Euler, Lagrange and other 18th-century mathematicians; many are new discoveries. Among the more important discoveries of his student days, we can pick out the least square method, the proof of the quadratic reciprocity law in number theory, and his study of the basic theorems of algebra. He received his doctorate, and the title of his thesis is "A new proof of the theorem that all rational algebraic entire functions with one variable can be decomposed into primary or quadratic real factors". This is the first of four proofs of the basic theorem of algebra published in his life. in this paper, Gauss emphasizes the importance of proving at least one root in the process of proving this theorem. The following instructions show his way of thinking.

We can use the graphic method to solve the equation.

It is proved that there is a complex value z=a+bi which satisfies this equation. Replace z with a+bi and separate the real part from the imaginary part of the equation, and we get a ^ 2-b ^ 2 = 0 and ab-2=0. Interpret an and b as variables and draw these functions in the same coordinate system, one axis represents the real part a, the other axis represents the imaginary part b, and we have two curves; one is composed of straight lines a+b=0 and a-b=0, and the other is composed of equiaxed hyperbolic ab=+2.

Obviously, the two curves have a point of intersection P in the first quadrant. We should pay special attention to the fact that one branch of the first curve leaves the origin in the direction of θ = 1 π / 4 and θ = 3 π / 4; the branch of the second curve moves asymptotically in the direction of θ = 0 π / 4 and θ = 2 π / 4; the intersection is between the last two directions θ = 0 and θ = π / 2. The coordinates of an and b at this intersection are the real and imaginary parts of a solution of the equation z ^ 2-4i=0. If our initial polynomial equation is cubic instead of quadratic, then one branch of a curve will approach the direction of θ = 1 π / 6 and θ = 3 π / 6, and the other curve will approach the direction of θ = 0 π / 6 and θ = 2 π / 6. In each case, these branches are continuous, so they must intersect somewhere between θ = 0 and θ = π / 3.

For a n-th equation, one branch of a curve has asymptotic directions θ = 1 π / 2n and θ = 3 π / 2n, while the other curve has asymptotic directions θ = 0 π / 2n and θ = 2 π / 2n. These branches must intersect from θ = 0 to θ = π / n, and the coordinates of an and b at this intersection are the real and imaginary parts of the complex number that satisfy the equation. So we see that no matter what the degree of a polynomial equation is, it must have at least one complex root. We will notice that Gauss relies on the diagrams of these curves to prove that they intersect. Admitting this result, it is proved that the polynomial equation can be decomposed into primary or quadratic real factors.

Number Theory Gauss began to write an important number theory work, arithmetic Research, when he was a student at the University of Gottingen, which was one of the great classics in the mathematical literature and was published two years after his doctoral thesis was adopted. The book consists of seven parts. The first four parts are essentially the concentration and reconstruction of number theory in the 18th century. The basic principles discussed are the concepts of congruence and residue classes. Part 5 is devoted to the theory of binary quadratic forms, especially the form such as

The technique developed in this part became the basis of a great deal of work done by later generations of algebraists. Part 6 consists of a variety of different applications. The last part attracted the most attention at first, dealing with the solution of the Secant equation with prime degree.

Gauss called the law of quadratic reciprocity published by Legendre two years ago the golden law. In later works, Gauss tried to derive a similar theorem of the congruence form x ^ n = p (modq) for nails 3 and 4; but in both cases, he found it necessary to expand the meaning of the word "integer" to include so-called Gaussian integers, integers in the form of a+bi, where an and b are integers. Gaussian integers form a ring, just like a real integer ring, but more general. The problem of divisibility becomes more complicated because 5 is no longer a prime and can be decomposed into the product of two "primes" 1x 2i and 1-2i. In fact, any real prime in the form of 4n+1 is not a "Gaussian prime", while a real prime in the shape of 4n-1 is still a prime in a general sense. In Gauss's arithmetic study, it includes the basic theorem of arithmetic, which is one of the basic principles that continue to be valid in the integral ring of Gaussian integers. In fact, any factorization is the only domain today is called a Gaussian domain. One of the contributions of arithmetic Research is the proof of the following theorem, which has been known since the time of Euclid:

Any positive integer can be expressed as the product of primes in one way and only one way.

Gauss's discovery of primes is not all included in the study of arithmetic. When he was a 14-year-old boy, Gauss wrote this obscure line in German on the back of a logarithm table:

This line refers to a famous prime theorem: the number of primes less than a given integer a tends to be a / lna when an increases infinitely.

As we have seen, Legendre was close to discovering the theorem in advance; but strangely, as we speculated, Gauss wrote it, but he kept this ingenious conclusion secret. We don't know if he proved the theorem, or even when he wrote a statement of the theorem. The distribution of prime numbers has a strong attraction to mathematicians.

In 1845, when Gauss was an old man, Joseph L.F., a professor in Paris. Bertrand proposed such a conjecture: if n > 3, then at least one prime number is included between n and 2n (or, more accurately, 2n-2). This conjecture, known as the Bertrand axiom, was proved by Pavnudi Chebyshev of St. Petersburg University in 1850. Chebyshev, as the leading Russian mathematician of his time and a competitor to Robachevsky, later became a foreign academician of the French Academy of Sciences and the Royal Society. Chebyshev obviously did not know Gauss's work on primes, and he was able to prove that if π (n) (lnn) / n approaches a limit when n increases infinitely, then the limit must be 1; but he cannot prove the existence of a limit. It was not until two years after Chebyshev's death that a proof was widely known.

The problem of the number and distribution of primes has fascinated many mathematicians from the Euclidean era to the present. There is a theorem that Gauss himself gives a startling example in arithmetic Research to illustrate the fact that the properties of primes invade the field of geometry even in the most unexpected way.

At the end of arithmetic Research, Gauss included his earliest important discovery in the field of mathematics: the method of regular heptahedron. He brought the subject to its logical result by proving which of the infinitely many possible regular polygons can and cannot be done. A general theorem, such as what Gauss is proving right now, is far more valuable than a special case, no matter how spectacular it is. We should remember that Fermat once believed that

The number of Fermat numbers is prime, and Euler later proved this hypothesis wrong. Gauss has proved that regular 17 polygons can be made, and the question arises naturally: whether regular 257 polygons and regular 65537 polygons can be made with Euclid's tools. In arithmetic Research, Gauss answered this question in the affirmative, and he proved that as long as N is

(where m is any positive integer and pn is a different Fermat prime), then a regular N-polygon can be made. There is one aspect of this question that Gauss has not answered, and so far no one has answered, and that is:

Is the number of Fermat primes finite or infinite?

We already know that Fermat numbers are not primes for nasty 5, 6, 7, 8 and 9, but it seems quite possible that there are and only five kinds of regular polygons with prime sides that can be made with rulers and compasses, two of which were known in ancient times, and the other three were discovered by Gauss. One person that Gauss admires very much is Ferdinand Gothold Eisenstein, a math teacher in Berlin. He added a new conjecture about prime numbers. At that time, he boldly came up with an idea that had not yet been proved.

The number of et cetera is prime. Gauss is said to have commented: "there are only three epoch-making mathematicians: Archimedes, Newton and Eisenstein." It's a pity that Eisenstein died before the age of 30.

Gauss's arithmetic study was in a deep sleep until the 1820s, when C.G.J. Jacobi and Dirichlet reveal for the first time that some of the more profound results come from this work.

Gauss's contribution to astronomy on January 1, 1801, Josep Piazi, director of the Palermo Observatory, discovered the new asteroid Ceres; but a few weeks later, the asteroid was out of sight. Believing that he had extraordinary computing power and the additional advantages of the least square method, Gauss accepted the challenge to calculate the planet's orbit from a small amount of recorded observations. In order to complete the task of calculating the orbit from limited observation data, he designed a method, called the Gaussian method, which is still used to track satellites. The result was a remarkable success, and the planet was rediscovered at the end of the year, very close to his calculated position. Gauss's orbital calculation attracted the attention of astronomers around the world and soon earned him a prominent reputation among German mathematical scientists, most of whom were engaged in astronomy and geodesy at the time. In 1807, he was appointed director of the Gottingen Observatory, a position he retained for nearly half a century. Two years later, his classic work on theoretical astronomy, the Theory of Celestial Motion, was published. This book provides a clear guide to orbital calculation, and by the time of his death, it had been translated into English, French and German.

However, orbital calculation is not the only field of astronomy in which Gauss has earned himself a reputation and paved the way for future generations. In the first decade of the 19th century, he spent much of his time studying perturbed problems. The problem of perturbation became the focus of astronomers after Goss's good friend, Heyheim Aubers, rediscovered the asteroid Jupiter in 1802. The eccentricity of Jupiter is relatively large, especially affected by the gravity of other planets such as Jupiter and Saturn. Determining the effects of these gravitational forces is a special case of the n-body problem (Euler and Lagrange have studied the case of nasty 2 or 3).

Gauss has consciously followed in the footsteps of these two geniuses since his early years, and it is particularly fascinating for him to find a recent solution. Although he believes that only part of his work has reached the quality of publication, his study of this problem has led not only to some astronomical papers, but also to two classic papers, one is infinite series, and the other is a new method of numerical analysis. The previous paper was submitted to the Gottingen Society in 1812 and devoted itself to the study of hypergeometric series. Because the convergence criterion proposed in this paper is often regarded as opening up a new era of rigor in mathematical analysis. It should be noted, however, that a deeper understanding of convergence did not prevent Gauss and other great mathematicians of the time from using divergent series when solving physical problems, as long as they thought they could do so "with confidence".

The beginning of differential geometry the branch of geometry that Gauss began in 1827 is called differential geometry, which probably belongs more to analysis than to the traditional field of geometry. Since the time of Newton and Leibniz, calculus has been applied to the study of curves in two-dimensional space. In a sense, this work constitutes the embryonic form of differential geometry. Euler and Monge extended this application to the analytical study of surfaces; therefore, they are sometimes regarded as the fathers of differential geometry. However, it was not until Gauss's classic work the General study of surfaces was published that there was a comprehensive work completely focused on this subject. Roughly speaking, orthodox geometry is interested in the whole of a given geometry, while differential geometry is concerned with the properties of the adjacent region of a curve or surface at a point on it. In this direction, Gauss extends Huygens and Clare's work on the curvature of a planar or asymmetric curve by defining the curvature of a surface at a point-"Gaussian curvature" or "total curvature".

If the normal N of S is made by a point P on a well-formed surface S, then the plane beam passing through N will intersect with the surface S in a cluster of planar curves, where each curve has a radius of curvature. The directions of curves with maximum and minimum radii of curvature (R and r) are called the principal directions of S at point P, and they are always perpendicular to each other. The values of R and r are called the principal curvature radius of S at point P, and the Gaussian curvature of S at point P is defined as Kenz1 / rR. The value of quantity is

It is called the mean curvature of S at point P. Gauss gives a formula for calculating the Gaussian curvature K according to the condition of the partial derivative of a surface for different coordinate systems (curvilinear coordinate system and Cartesian coordinate system). He also found some theorems about the properties of curve clusters (such as geodesic lines) drawn on surfaces, which even he considered as "eye-catching theorems".

Gauss began to deal with the surface by using a parametric equation of the surface proposed by Euler. Gauss proved that the properties of a surface depend only on E, F and G.

This leads to a lot of results. In particular, it makes it easy to say that the properties of a surface are constant. It was on the basis of Gauss's work that Riemann and later geometries changed the theme of differential geometry.

The late work of Gauss contributed two important essays: one is the proof of Harriot's Theorem in Algebra, and the other contains Gauss's minimum constraint principle. Historians often quote the first paper (published in 1832) because it contains a geometric representation of the plural of Gauss. The importance of this paper as a whole lies in the fact that it points out a way to expand number theory from real numbers to the plural field, and even further. As already pointed out above, this is very important in the work of later researchers in the field of number theory.

In the last 20 years of his life, Gauss published only two important papers of mathematical significance. One is his fourth proof of the basic theorem of algebra, which he issued on the anniversary of his doctorate in 1849, 50 years after he published his first proof. Another influential paper on potential theory was published in 1840. Geomagnetism took up much of his time in the 1930s and early 1940s, and in the late 1930s he devoted a lot of time to issues related to weight and measurement. Most of the publications in the last decade of his life are related to the work of the observatory; topics include newly discovered asteroids and observations of Neptune.

On February 23, 1855, Gauss died of a heart attack.

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

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