◆ Contribution to Analytic Geometry
Fermat discovered the basic principles of analytic geometry independently of Rene Descartes.
Before 1629, Fermat began to rewrite the book Plane Trajectory, which was lost by the ancient Greek geometer Apollonius in the third century BC. He supplemented some lost proofs of Apollonius trajectory with algebraic methods, summarized and sorted out ancient Greek geometry, especially Apollonius' conic curve theory, and made a general study of curves. 1630, he wrote an 8-page paper "Introduction to Plane and Solid Trajectory" in Latin.
Fermat began to correspond with the great mathematicians Mei Sen and Robewal at that time in 1636, and talked about his own mathematical work. However, Introduction to Plane and Solid Trajectory was published after Fermat's death 14 years ago, so few people knew Fermat's work before 1679, but now it seems that Fermat's work is groundbreaking.
Fermat's discovery was revealed in Introduction to Plane and Solid Trajectory. He pointed out: "An equation determined by two unknowns corresponds to a trajectory and can describe a straight line or curve." Fermat's discovery was seven years earlier than rene descartes's discovery of the basic principles of analytic geometry. Fermat also discussed the equations of general straight lines and circles, hyperbolas, ellipses and parabolas.
Descartes looks for its equation from the trajectory, while Fermat studies the trajectory from the equation, which are two opposite aspects of the basic principle of analytic geometry.
In a letter from 1643, Fermat also talked about his analytic geometry thought. He talked about cylinder, elliptic paraboloid, hyperboloid and ellipsoid, and pointed out that an equation containing three unknowns represents a surface, and further studied it.
◆ Contribution to Calculus
16 and 17 centuries, calculus is the brightest pearl after analytic geometry. As we all know, Newton and Leibniz were the founders of calculus. Before them, at least dozens of scientists did basic work for the invention of calculus. But among many pioneers, Fermat is worth mentioning, mainly because he provided the inspiration closest to the modern form for the derivation of the concept of calculus, so that in the field of calculus, Fermat, as the founder after Newton and Leibniz, will also be recognized by the mathematical community.
The tangent of a curve and the minimum of a function are one of the origins of calculus. This work is relatively old, dating back to ancient Greece. Archimedes used the exhaustive method to find the area of any figure surrounded by curves. Because method of exhaustion was troublesome and clumsy, he was gradually forgotten, and was not taken seriously until16th century. When exploring the laws of planetary motion, johannes kepler encountered the problem of how to determine the ellipse area and ellipse arc length. The concepts of infinity and infinitesimal are introduced to replace the tedious exhaustive method. Although this method is not perfect, it has opened a very broad thinking space for mathematicians since cavalieri came to Fermat.
Fermat founded tangent method, maximum method, minimum method and definite integral method, which made great contributions to calculus.
◆ Contribution to probability theory
As early as in ancient Greece, the relationship between contingency and inevitability aroused the interest and debate of many philosophers, but it was after15th century to describe and deal with it mathematically. In the early16th century, Italian mathematicians such as cardano studied the game opportunities in dice and explored the division of gambling funds in game points. /kloc-In the 7th century, French Pascal and Fermat studied the abstraction of Italian Pachauri, and established corresponding relations, thus laying the foundation of probability theory.
Fermat considers that there are 2× 2× 2× 2 = 16 possible outcomes for four gambling, except one outcome, that is, the opponent wins all four gambling, and the first gambler wins all other situations. Fermat has not used the word probability yet, but he has come to the conclusion that the probability of the first gambler winning is 15/ 16, that is, the ratio of the number of favorable situations to the number of all possible situations. This condition can generally be met in combination problems, such as card games, throwing silver and modeling balls from cans. This study actually laid a game foundation for the abstraction of probability space, a mathematical model of probability, although this summary was made by Kolmogorov in 1933.
Fermat and Blaise Pascal established the basic principle of probability theory-the concept of mathematical expectation in their correspondence and works. This should start with the mathematical problem of integral: in an interrupted game, how to determine the division of gambling funds between players with the same assumed skills, and how to know the scores of two players when they are interrupted and the scores needed to win the game. Fermat discussed the situation that player A needs 4 points to win and player B needs 3 points to win, which is Fermat's solution to this special situation. Because it can be decided four times at most.
The concept of generalized probability space is a thorough axiomatization of people's intuitive ideas about concepts. From the point of view of pure mathematics, the finite probability space seems unremarkable. But once random variables and mathematical expectations are introduced, it becomes a magical world. This is Fermat's contribution.
◆ Contribution to number theory
/kloc-At the beginning of the 7th century, the book Arithmetic written by Diophantu, an ancient Greek mathematician in the 3rd century A.D., spread in Europe. Ma Fei bought this book in Paris. He studied the indefinite equations in the book in his spare time. Fermat limited the study of indefinite equations to the range of integers, thus creating a mathematical branch of number theory.
Fermat's achievements in the field of number theory are enormous, including:
Fermat's last theorem: n>2 is an integer, then the equation X N+Y N = Z N does not have an integer solution satisfying xyz≠0. This is an indefinite equation, which has been proved by the British mathematician wiles (1995), and the process of proof is quite difficult!
Fermat's last theorem: a p-a ≡ 0 (mod p), where p is a prime number and a is a positive integer, its proof is relatively simple. Actually, it is a special case of euler theorem. euler theorem said: a φ(n)- 1 ≡ 0 (mod n), where both a and n are positive integers, and φ(n) is an Euler function, indicating the number of positive integers less than n and n coprime (its expression Euler has been obtained, so we can see "Euler formula").
And:
(1) All prime numbers can be divided into 4n+ 1 and 4n+3.
(2) The prime number in the form of 4n+1can and can only be expressed as the sum of two squares in one direction.
(3) No prime number in the form of 4n+3 can be expressed as the sum of two squares.
(4) The prime number in the form of 4n+1can and can only be used as the hypotenuse of a right triangle with integer right angles; The square of 4n+ 1 is and can only be the hypotenuse of two such right triangles; Similarly, the m power of 4n+ 1 is and can only be the hypotenuse of m such right triangles.
(5) The area of a right triangle with a rational number side length cannot be a square number.
(6) The prime number of 4n+1and its square can only be expressed as the sum of two squares in one direction; Its cubic and fourth power can only be expressed as the sum of two squares in two ways; The 5th power and 6th power can only be expressed as the sum of two squares in three ways, and so on until infinity.
(7) Find the second pair of affinity numbers: 17296 and 184 16.
In the sixteenth century, people thought that there was only one pair of affinity numbers in natural numbers: 220 and 284. Some boring people even add superstition or mystery to the number of relatives and make up many fairy tales. It is also publicized that this affinity number plays an important role in magic, magic, astrology, divination and so on.
More than 2500 years after the birth of the first affinity number, the wheel of history turned to the seventeenth century. 1636, Fermat, the king of French amateur mathematicians, discovered the second pair of affinity numbers 17296 and 184 16, rekindled the torch to find affinity numbers and found light in the dark. Two years later, rene descartes, the "father of analytic geometry", announced that he had discovered the third pair of affinity numbers 9437506 and 9363584 on March 3 1 65438. Fermat and Descartes broke the silence of more than two thousand years in two years, which once again aroused the waves of searching for affinity numbers in mathematics.
◆ Contribution to optics
Fermat's outstanding contribution to optics is that he put forward principle of least action, also called the principle of shortest time action. This principle has a long history. As early as ancient Greece, Euclid put forward the laws of linear propagation and phase reflection of light. Later, Helen revealed the theoretical essence of these two laws-light takes the shortest path. After several years, this law was gradually extended to natural law, and then became a philosophical concept. In the end, a more general conclusion was drawn that "nature works in the shortest possible way" and influenced Fermat. Fermat's genius lies in turning this philosophical concept into a scientific theory.
Fermat also discussed the situation that the path of light takes the minimum curve when it propagates in a point-by-point changing medium. Some problems are explained by principle of least action. This has greatly inspired many mathematicians. Leonhard euler, in particular, used the variational technique and this principle to find the extreme value of the function. This leads directly to Lagrange's achievement and gives the concrete form of principle of least action: for a particle, the integral of the product of its mass, velocity and the distance between two fixed points is a maximum and a minimum; That is to say, for the actual path taken by particles, it must be the maximum or minimum.