1. Gauss elimination method: Gauss invented an algorithm for solving linear equations, namely Gauss elimination method. This method is still widely used in modern computer science.
2. Least square method: Gauss proposed the least square method, which is a mathematical method to find the best fitting straight line by minimizing the sum of squares of errors. This method is widely used in statistics, physics and engineering.
3. Gaussian distribution: Gaussian distribution, also known as normal distribution, was proposed by Gaussian. This is a continuous probability distribution, which is used to describe the probability distribution of many random variables in natural science and social science.
4. Number theory: Gauss also made important contributions to number theory, including the proof of quadratic reciprocity law and the popularization of prime number theorem.
5. Complex analysis: Gauss also made an important contribution to complex analysis. He proposed Gaussian integral and Gaussian function, which are still widely used in modern mathematics.
6. Study on the Earth's magnetic field: Gauss has also made important contributions to the study of the Earth's magnetic field. He discovered the law of motion of the earth's magnetic poles and put forward the theory of the earth's magnetic field.
7. Astronomy: Gauss also made important contributions to astronomy. He put forward Gauss's law, which is the basic law describing the propagation of light through turbulent media.
Generally speaking, Gauss's mathematical achievements are rich and varied, and he has made important contributions in many fields, which has a far-reaching impact on later generations.