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 introduces the construction method and basic properties of the Cantor set. By converting the numbers in the Cantor set into ternary fractions containing only 0 and 2, and then establishing a one-to-one correspondence with the binary fractions of real numbers in the interval [0, 1], it explains why there are as many numbers in the Cantor set as there are real numbers.

This article focuses on the question of "why there are far more irrational numbers than rational numbers", and introduces Cantor's proof that the real numbers in the interval (0,1) are as numerous as all real numbers. Then, he uses the method of reductio ad absurdum to prove that the real numbers in the interval (0,1) cannot be arranged into an infinite sequence, concluding that the infinity of real numbers is greater than the infinity of rational numbers, and thus deducing the conclusion that irrational numbers far outnumber rational numbers.

This article focuses on "why there are as many rational numbers as positive integers", introducing the proof process discovered by Cantor in 1873 that rational numbers can be established in a one-to-one correspondence with positive integers, and explaining the reason why the number of the two is the same.

This article vividly explains the concept of infinite sets through the story of the Hilbert Hotel described by Hilbert, illustrating the reason why the fully booked Hilbert Hotel (a hotel with an infinite number of rooms) can still accommodate new guests and even an infinite number of new guests by establishing a one-to-one correspondence between infinite sets.

This article focuses on the question of "Are the parts as numerous as the whole?" and uses one-to-one correspondence as a criterion for judging the equality of the number of two types of things. It introduces the relevant research on infinite sets by Galileo, Bolzano, and Cantor, and points out that the essential characteristic of infinite sets is that their size can be equal to that of their proper subsets.

This article revolves around the question of "Are all real numbers the roots of algebraic equations with integer coefficients?" It introduces the characteristics of algebraic equations with integer coefficients, points out that rational numbers are all the roots of algebraic equations with integer coefficients, and their roots can also be irrational numbers. It describes how French mathematician Liouville first gave a negative answer to this question and constructed Liouville numbers. It explains the definitions of algebraic numbers and transcendental numbers, and shows that it is difficult to prove that specific numbers are transcendental. It has been proven that e and π are transcendental numbers, but whether simple combinations such as e + π are transcendental remains unsolved. It also mentions that German mathematician Cantor proved that transcendental numbers are far more numerous than algebraic numbers, and that almost all real numbers are transcendental. The existence of transcendental numbers also proves that the problem of "squaring the circle" with a ruler and compass is impossible to achieve.

This article focuses on the question "Do all first-order algebraic equations have solutions?" It introduces the definition and classification of first-order algebraic equations and the concept of radical solutions, reviews the exploration and discovery process of radical solutions for first-order to fourth-order algebraic equations, and discusses the historical facts related to Abel's proof that general first-order quintic equations do not have radical solutions and Galois's introduction of group theory and his criterion for the solvability of algebraic equations by radicals. It concludes that generally, first-order quintic and higher-order algebraic equations do not have radical solutions.

This article introduces the basic model of pyramid selling and the tactics used by pyramid sellers to deceive people into joining by exploiting the "multiplication" story of King of India and Sorcerer's Chess Grains. It demonstrates that the "multiplication" myth in pyramid selling propaganda is unreliable from the aspects of membership fees and product issues, the increasing difficulty of developing downlines, low profit margins and unstable downlines, practical population limitations, and the actual development of enterprises that pioneered the pyramid selling model.

This article focuses on the "doubling bet" method in casino gambling, explaining its seemingly foolproof operating logic. By comparing the growth rate of the winning money in an arithmetic progression with the doubling growth rate of the losing money, and analyzing the law that the probability of consecutive losses increases with the increase in the number of bets, combined with specific data and cases, it shows that as long as the gambler's capital is limited, he will eventually lose everything. It explains why the "doubling bet" method is not a foolproof winning strategy.

This article focuses on the question of "how many times can a piece of paper be folded at most". It explains the principle that ordinary A4 paper can't be folded more than 7 times due to its increasing thickness and decreasing width after being folded. It also introduces several origami experiments and related research conclusions, showing the counterintuitive fact that continuous doubling will result in huge numbers after a limited number of folds.