Mathematics

Claude's breakthrough in the direction of Riemann's conjecture is not that AI has solved the problem, but that AI is moving from a scientific research tool to a participant in scientific discovery. In the future, if AI can combine automated laboratories, robots and supercomputing to achieve a closed loop of "proposing hypothesis-experimenting-verifying", scientific exploration will shift from relying on a few talents to being replicable, parallelized and large-scale, which may significantly accelerate human progress.

This article focuses on the question of why some washing machines are so "intelligent", and introduces the working principle of intelligent washing machines that adopt fuzzy control technology. By sensing the degree of dirtiness of clothes, following the reasoning rules that simulate human judgment and decision-making, and issuing commands after mathematical processing, intelligent washing machines achieve intelligent washing.

This article takes the question "Is 1 a prime number?" as the core issue. By introducing the definition of prime numbers, the fundamental theorem of arithmetic, and the uniqueness of prime factorization, it explains why mathematicians stipulate that 1 is not a prime number. At the same time, it points out that prime numbers are the basic elements of integers and that the list of prime numbers is infinite.

The article "Why Imaginary Numbers Aren't Imaginary" introduces the history of imaginary numbers, from when mathematicians initially regarded them as fictional numbers, to when Wessel, Argand, Gauss, and others gave them geometric meaning and practical models by constructing the complex plane, and then to the subsequent development of disciplines such as complex analysis and its widespread application in multiple fields. It explains that imaginary numbers are not abstract and intangible, but rather have significant practical value.

This article focuses on the question of "why we need complex numbers", introduces the definition of complex numbers, points out that the introduction of complex numbers in middle school mathematics textbooks due to the discriminant of quadratic equations being less than 0 may seem unnecessary, elaborates on the historical background of Italian mathematician Cardano's introduction of complex numbers in the 16th century to solve cubic equations (even if seeking real roots, it is impossible to avoid the operation of taking the square root of negative numbers), mentions Euler's improvement of the representation of complex numbers, the application of complex numbers in trigonometric functions and other fields, and French mathematician Hadamard's classic evaluation that "the shortest path between two truths in the real number field is through the complex number field".

This article explains the reasons for introducing logarithmic scales, including handling large-range numerical changes, facilitating written calculations, and aligning with humans' logarithmic perception (Fechners' law), by citing examples of logarithmic scales in daily life, such as the Richter scale, star magnitude classification, and decibel measurement of sound. It demonstrates that logarithmic scales can be found everywhere in our daily lives.

This article focuses on the question of "why the emergence of logarithms represents a revolution in mathematical methods". It introduces that logarithms originated from the heavy computational demands in the European scientific community before the invention of calculators. They were proposed by Napier based on the corresponding relationships between arithmetic progressions and geometric progressions, and served as the inverse operation of exponential operations. Logarithms can transform multiplication and division, as well as power operations, into addition and subtraction, and multiplication operations. By referring to logarithmic tables and inverse logarithmic tables, complex calculations can be greatly simplified. Even in the era of widespread use of electronic computers, logarithms and logarithmic functions still hold great value in mathematical theory and natural sciences. Laplace and Galileo both highly praised the significance of logarithms.

This article focuses on the question of "why the logarithm with the base e is called the natural logarithm", and reviews the development history of logarithms. It introduces the background and role of Napier's invention of logarithms, the definition of the logarithmic function, and the logarithmic function defined by the area of a hyperbola with c=1 corresponding to the simplest logarithm, which is called the natural logarithm. Euler proposed the concept of the base of the logarithmic function after associating the logarithmic function with the exponential function, and found that the base of the natural logarithm is exactly the number e.

This article focuses on the question of "why the number e has such a wide range of applications". It introduces the origin of the number e through the ideal compound interest model, introduces the relevant research of mathematicians such as Jacob Bernoulli and Euler, and explains the unique properties of the exponential function with e as the base and its applications in many fields, including natural phenomena, population research, calculus calculations, and engineering calculations.

This article focuses on the question of "what kind of rectangle looks the most beautiful", and introduces the most beautiful rectangles with edge length ratios close to the golden ratio selected by psychologists through experiments. It also explains the definition and characteristics of the golden rectangle, and introduces geometric figures closely related to the golden ratio, such as the golden triangle, the golden ellipse, and the golden hyperbola.

This article focuses on the theme of "Why is the Golden Ratio ubiquitous in life?" It introduces the definition of the Golden Ratio and the history of its cognitive development, lists its applications in various fields such as architecture, stage performances, photographic composition, five-pointed stars, human body proportions, and artistic creation, and points out that there's no need to mechanically demand complete compliance with the Golden Ratio principles. It's enough to roughly conform to them.

This article focuses on the theme of "How amazing the Fibonacci sequence is", introducing the Fibonacci sequence originating from the rabbit breeding problem in Fibonacci's "Liber Abaci", explaining its recursive rules, sequence definition, and various mathematical properties. It also lists numerous manifestations of the sequence in the plant world, demonstrating the amazing characteristics of the Fibonacci sequence.