Gauss's achievements cover all fields of mathematics, and he has made pioneering contributions in number theory, non-Euclidean geometry, differential geometry, hypergeometric series, complex variable function theory, elliptic function theory and so on. He attached great importance to the application of mathematics, and emphasized the use of mathematical methods in the research of astronomy, geodesy and magnetism.
When Gauss 18 years old, he discovered the prime number distribution theorem and the least square method. After processing enough measurement data, new probability measurement results can be obtained. On this basis, Gaussian then focuses on the calculation of surfaces and curves, and successfully obtains Gaussian bell curve (normal distribution curve). Its function is named standard normal distribution (or Gaussian distribution), which is widely used in probability calculation.
At the age of 65,438+09, Gauss constructed a regular polygon of 65,438+07 with only scale-free rulers and compasses (neither Archimedes nor Newton drew it), which provided the first important supplement to Euclidean geometry that has been circulating for 2000 years since ancient Greece.
The trajectory Gauss of Ceres calculated by Gauss sums up the application of complex numbers, and strictly proves that every N-order algebraic equation must have N real solutions or complex solutions. In his first masterpiece, Number Theory, he proved the law of quadratic reciprocity, which became an important basis for the continued development of number theory. The first chapter of this book deduces the concept of triangle congruence theorem.
With the help of his survey adjustment theory based on least square method, Gauss calculated the trajectory of celestial bodies. So we found the trajectory of ceres. Ceres was discovered by Italian astronomer Piazi in 180 1 year, but he delayed his observation due to illness and lost the trajectory of this asteroid. Piazi was named after the goddess of harvest (Ceres) in Greek mythology, that is, Planetoiden Ceres, and announced the previously observed position, hoping that astronomers all over the world would look for it together. Gauss calculated the trajectory of Ceres through the previous three observation data. Austrian astronomers successfully discovered this asteroid in the orbit calculated by Gauss, and since then Gauss has become famous all over the world. Gauss wrote this method in his book The Theory of Celestial Motion.
Gauss attaches great importance to the rigor and refinement of scientific expression. He can't tolerate that some previous narratives and proofs can't stand scrutiny, so he is determined to make his works impeccable in this respect. In his letter to a friend, he clearly stated: "You know I write slowly, mainly because I always want to express as many ideas as possible in as few words as possible. It takes more time to write briefly than in detail."
Gauss is full of ideas, so he has to keep a short journal of scientific discoveries. He has no time to write a detailed paper. He said, "What gives me the greatest pleasure is not the achievement, but the process of achieving it. When I have clarified a problem and thoroughly studied it, I will let go to explore unknown areas. " 1898, I found a Gauss notebook with only 19 pages from his grandson's house, saying that his diary recorded 146 mathematical discoveries. It is estimated that if every discovery he made in science is published in a perfect form, it will take several long-lived Gauss all his life. Great achievements have been made in the fields of number theory, function theory, probability statistics, calculus geometry and non-Euclidean geometry.
American mathematician Simmons said, "This is Gauss, a supreme mathematician. His achievements in so many aspects are beyond the reach of an ordinary genius, so that we sometimes have a strange feeling that he is a paradise in the upper world. "