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Anonymous users2024-02-08"Analysis and Arithmetic" bridges the gap between arithmetic and geometry。In order to make the analysis more rigorous, we must first make the real number system itself more rigorous. Therefore, the most reliable method is to reduce real numbers to integers (rational numbers) based on strict reasoning.
In this way, all analytical concepts can be derived from integers, which makes it possible to fill in the holes and flaws of the past. This is the so-called "analytic and arithmeticization" procedure.
Guided by Will Strass's "analytic arithmeticization" movement, Dedekin, Cantor, and Will Strass all developed their own theories of real numbers. In 1872, the German mathematician Dedekin used the "division" of rational numbers to define irrational numbers based on the requirement of continuity, and established the theory of real numbers on a strict scientific basis. He divides the set of all rational numbers into two non-empty and disjoint subsets a and a', such that each element in set a is smaller than each element in set a'.
Set A is called the divided lower group, Set A'It is called the upper group after the division, and this division is called a|a '。
Dedekin defined this division as the division of rational numbers。Here, Dedekind expands from rational numbers to real numbers, establishing the definition of irrational number theory and continuous pure arithmetic. Cantor also achieved the same goal with the sequence theory of rational numbers.
Both Cantor and Dedekind defined real numbers as some type of "set" of rational numbers. The Dedekin method can be referred to as the ordered completion method, and the Cantor method can be referred to as the measurement completion method. These methods have become typical construction methods in modern mathematics, and have been continuously promoted and developed into powerful tools in mathematical theory by later generations.
Weilstras published the theory of bounded monotonic sequences.
The definition of real numbers and the establishment of their completeness marked the completion of the analytical and arithmetic movement initiated by Will Strass. In this way, the long-term logical loop around the concept of real numbers can be eliminated completely. As a result, the gap between arithmetic and geometry, which existed for more than 2,000 years, was completely filled, and irrational numbers were no longer "irrational numbers".
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Anonymous users2024-02-07I think the arithmeticization of analytics is a great breakthrough for the digital edifice now.
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Anonymous users2024-02-06I think the current development of the digital economy is still very good, that's why I said that.
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Anonymous users2024-02-05It may be the development of the digital economy now. That's where digital analytics comes in.
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Anonymous users2024-02-04This is because with this algorithm, new breakthroughs have been made in mathematics.
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