Mathematics

This article focuses on the aspect ratio of A4 paper being √2:1, and explains that the reason why A-series paper adopts this ratio is that the aspect ratio can remain unchanged after folding along the long edge. This ratio can avoid the generation of scrap paper edges during paper cutting, reduce waste, thereby reducing the cost of re-production and improving work efficiency. At the same time, it introduces the specification setting of B-series paper made by folding and cutting in the same ratio.

This article focuses on the question of why real numbers are divided into rational and irrational numbers, and introduces the relevant mathematical history background: the Pythagorean School proposed the belief that "everything is number" and that any line segment can be measured through studies such as music and astronomy. Hippasus discovered a new number (later called irrational number) whose diagonal of a square cannot be measured with its side length, and overturned the belief of the Pythagorean School. The original meaning and translation process of the terms "rational number" and "irrational number" are also introduced in this article.

This article analyzes the question of "whether 0.1 and 0.10 are the same" from two perspectives: exact decimal numbers and approximate decimal numbers. It points out that under the category of exact decimal numbers, the two values are equal, and the zero at the end of 0.10 can be omitted. Under the category of approximate decimal numbers, the range of values and accuracy represented by the two are different. The accuracy of 0.10 is higher, and the zero at the end cannot be arbitrarily removed. It also takes financial accounting rules as an example to illustrate the importance of the zero at the end of approximate decimal numbers.

This article focuses on the question of "why we often use approximate values for numbers". Through multiple examples such as age comparison, height measurement, engineering drawing, radio timekeeping, and the lifetime of hyperons, it demonstrates that different practical problems have different requirements for the accuracy of quantities, so we usually choose to use approximate values according to actual needs.

This article introduces a method for quickly determining whether a natural number can be divided by 3, 9, or 11. To determine whether it can be divided by 3 or 9, we need to check whether the sum of all the digits of the number in decimal form can be divided by 3 or 9. To determine whether it can be divided by 11, we need to check whether the difference between the sum of the digits in the units, hundreds, and tens of thousands places and the sum of the digits in the tens, thousands, and hundreds of thousands places is a multiple of 11.

This article focuses on the question of "why fractions can be converted into finite decimals or infinite recurring decimals", explaining the principle that fractions can be transformed into finite decimals or infinite recurring decimals. It also demonstrates that finite decimals and infinite recurring decimals can be converted into fractions, and introduces the conversion method and provides relevant examples.

This paper focuses on the problem that "decimal points need to be aligned to the decimal point when adding decimal numbers, and decimal numbers need not be aligned". It compares the rule of adding integers to align the number of digits, explains that the logic of decimal addition is "aligned to the left", and mentions the ancient people's naming of decimal places. It explains the reason why multiplication does not need to be aligned, and gives specific examples to prove it.

This paper focuses on the core problem of matrix multiplication of "whether A×B is equal to B×A", introduces the definition and operation rules of matrix multiplication, explains that the definition of matrix multiplication has practical needs and mathematical significance, and mentions non-commutativity in Heisenberg's quantum mechanics. Application and awards in matrix mechanics.

This article explains the reason why a^0=1 (a≠0) is stipulated in mathematics, deduces this conclusion from the algorithm of the power of positive integers, and also mentions that the exponent can be extended to negative integers, fractions, and real numbers. It also explains that the exponent of fractions and irrational numbers has limitations on a, clearly presenting the logic of exponential expansion.

This article explains the rule of "negative results in positive" in multiplication and division operations. First explain the actual meaning of negative numbers (such as lack of money, debt, etc. in "Nine Chapters of Arithmetic") and the meaning of the number axis, and then explain the rationality of this provision from many aspects such as life examples, the nature of the number axis, the law of algebraic distribution, and the negation logic of negation. Answer the question of why it is difficult to understand.

This article focuses on "why fractions must be divided first", explaining that fraction addition cannot be directly added to the numerator and denominator respectively, because fractions represent the overall proportion, and different denominator fractions correspond to different division methods, and the general division needs to be converted to the same denominator before calculation can be carried out. Use examples such as watermelon fractions to explain the relevant principles.

This paper discusses why the mixed four-rule operation stipulates that multiplication and division first followed by addition and subtraction. First, it points out that inconsistent operation order will lead to confusion in the results, and then explains its rationality from the perspective of multiplication being a simple operation of continuous addition. Combined with daily expense calculation examples, it finally extends to the hierarchical rules of number operations.