The low-order calculation comes from Europe, and it is not just rounding that needs to be rewritten. At that time, Europeans respected written calculation, which we can understand as follows: It is because in Europe, not only the introduction of Arabic numerals and written calculation greatly simplified calculation (compared with calculation with Roman numerals), but also promoted the rapid development of mathematics. But in some European countries, the three-digit reading is 100, then 10, and the two-digit reading is 10. This is what I learned from the parents of German and Belgian students. Most students from the United States, Japan and South Korea study from high to low, just like China. At the end of the Qing Dynasty, it moved into the western academic system and curriculum knowledge structure in an all-round way, and it was all westernized. Mathematics classes in schools are all written, which is not in line with the national conditions of China and most countries in the world. Digital monism thinking, bead code is binary thinking, different brain parts are used, and the thinking effect is different, and the effect is the last word.

Shen Songnian commented on the original address: starting from a high position and starting from a low position. Author: Talking about Liu from a high position and from a low position in mental arithmetic.

I often hear some math teachers say that children are confused when they learn abacus mental arithmetic. Because abacus is calculated from the high position, and the written calculation in school mathematics is calculated from the low position, it is easy for students to get confused because they learn the high position first and then the low position. Therefore, many primary school math teachers use this as an excuse to oppose learning abacus mental arithmetic. Let's talk about the high position and the low position from the perspective of the historical development of mathematics.

First, the formation of written calculation

India has created easy-to-write numbers, which we now call Arabic numerals. This number was actually created by Indians. Just because in history, numbers were introduced into Europe from Arab countries, and Europeans called them Arabic numerals, which spread all over the world). India has a certain form of written calculation, and this number is used in the calculation; After continuous improvement by people all over the world, it has become a well-known written calculation today.

This digital code was introduced into Islamic countries in the 8th century. At that time, there were two Arab cultural centers: one was Baghdad in East Arabia, and the other was Cordoba in West Arabia (southern Spain). At that time, there was no printing, books were copied, and fonts were different everywhere. It may also be because there are great differences between the East and the West in writing digital characters through different channels. After several years of evolution, the difference is getting bigger and bigger. The fonts in East Arabia are gradually fixed, forming unique numbers, which are still used in many Islamic countries. The figures in West Arabia are closer to those commonly used in the world today, and were introduced to Europe by Fibonacci at the beginning of13rd century. At the beginning of Lieber Abac, he put forward an Indian Arabic numeral with the number 0: "Here are nine Indian numerals 98765432 1. With these nine numbers and the Arabic symbol 0, you can write any number. " [1] According to Arab custom, words and numbers are read from left to right. Fibonacci's abacus book popularized Indian Arabic numerals and played an extremely important role in changing the face of European mathematics.

The "written calculation" formed by Arabic numerals is actually just a form of record. Because the "pen" itself can't calculate, what we usually say is actually to record the questions, align the numbers properly, calculate the results bit by bit with the mouth (mental arithmetic), and then record them with Arabic numerals. In short: written calculation is oral calculation plus report card. Because the biggest advantage of Arabic numerals is that most of them can be written in one go (except 4 and 5). So it is easy to record with it.

Second, the origin of low calculation

Because abacus calculation is calculated from high position and pen calculation is calculated from low position, many people regard pen calculation as a standard to measure the quality of an algorithm, and think that an algorithm that is inconsistent with pen calculation is not good. In fact, in the early days of the formation of written calculation, its addition, subtraction, multiplication and division were all calculated from the high position; Even now, the division by writing is still from the high position. From the evolution of written calculation, we can see that the addition, subtraction, multiplication and division of written calculation also starts from the high position, but the previous numbers need to be rewritten when carrying. Therefore, the addition, subtraction, multiplication and division in written calculation gradually changed to the low position. As can be seen from the following formula:

For example: 65 391+3 279+65 438+00 420 = 79 090.

In India at that time, its calculation process was: 65 39 1.

3 279

10 420

78/ 98/0

9 09

This is the addition on Indian sand table in 12 century [2], and this program has been preserved since the high position. This also confirms that the initial stage of written calculation of addition and subtraction also starts from the high position. In the calculation process, in order to avoid the trouble of constantly rewriting Arabic numerals, it is gradually changed to a low position. As for multiplication, there are no more examples.

Third, the comparison between literal calculation and Roman numeral calculation.

It is very troublesome to get together with Roman numerals and symbols. For example, three C's, C's and C's in the above formula cannot be directly synthesized into 300. With Arabic numerals, it is easy to add two codes bit by bit.

Let's analyze and compare the four elements of the algorithm: the simplicity of using Arabic numerals in writing and Roman numerals in operation:

First of all, from the input point of view: pen calculation is only to write Arabic numerals in a bit-by-bit arrangement; Roman numerals, on the other hand, are relatively more complicated, so it is impossible to write them bit by bit. Every number is tired according to the law of "left minus right plus", which is much more complicated than handwriting. Secondly, look at storage: there is not much difference between the two, they are all written on paper, but Roman numerals are written more. Third, look at the operation: written calculation is to add and subtract 162 sentences according to the formula memorized in advance, and calculate the number of answers in your mind; The operation of Roman numerals is much more complicated. For example, the above numerical symbols can clearly see two "X" s and an "L", but they cannot be directly combined. Finally, let's look at the output: the output of pen calculation 663 is much simpler than the output of Roman numeral DCLXIII, no matter from reading or writing. Let's look at multiplication and division:

Due to the introduction of Arabic numerals and characters, the complexity of the four calculations is greatly simplified. There are two main reasons for this: first, Arabic numerals use the decimal system invented by China when representing multiple digits; second, Arabic numerals are easy to write. So as to promote the rapid development and progress of European mathematics during the Renaissance. Promote Arabic numeral and character calculation to become an international common counting method and operation mode.

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