Author | WU Wenjun (Researcher, Institute of Systems Science, Chinese Academy of Sciences, Academician of Chinese Academy of Sciences)
Source: | Science (bimonthly), March 2003 (Vol. 55, No. 2)
Previous: Chinese Ancient Arithmetic and the Real Number System (I)
Keywords: real number system position-value system "Nine chapters of arithmetic" Liu Hui "Nine children's arithmetic notes"
Unlike the ancient Greek Euclid system, which disconnected from shapes, shapes and numbers were inseparable in ancient Chinese mathematics. Line segments are always assigned length, and both planar and solid are assigned area and volume. It is precisely because of the calculations required for length and various other measures that the system of rational numbers, decimals, and real numbers is produced.
From square to irrational number
Chapter 4 of "Nine Chapters of Arithmetic" (hereinafter referred to as "Nine Chapters"), "Shao Guang", deals with problems such as the accumulation of powers and square circles, with a total of 24 questions. Topics 12 to 16 are the area of a known square, finding the length of the sides, which is equivalent to the modern square of the square. Questions 19 to 22 are known to be the volume of the cube , finding the side length, which is equivalent to the modern opening square. Among them, the method of opening the square is "opening the square technique", and the text is as follows: "The opening technique is: put the product into reality, borrow one calculation, step it, super one, etc." The income of the negotiation shall be calculated by multiplying one by one and dividing by one. Divide it by the law of times. Its repetition and division, fold and go down. Reposition and borrowing, step by step, to reconsider one multiplication, the obtained vice, to add the method, to divide. Obtain the subordinate law. Compound division fold down as before. If it is inexhaustible, it is impossible to open, and it should be ordered by face. ”
The "fa", "real", "borrow", "step", "etc.", "division", etc. in the art text are all technical terms used at that time, and the "division" in the art text is equivalent to the modern "subtraction" in ancient times, not the modern "division".
In Liu Hui's "Nine Chapters of Arithmetic Notes" (hereinafter referred to as "Liu Note"), the algorithm of the open (flat) side has a clear geometric background. The "Liu Note" not only has a law, but also a map, and each block in the figure is also painted with different colors such as zhu, yellow, and qing, and marked with A and B to show the difference, such as yellow Jia, yellow yi, qing mi, etc.
These maps have been lost, but according to the "Liu Note", the general situation of the color maps can be deduced.
The search in question 12 of "Chapter 9 Shaoguang" is to find the side length of a square with an area of 55225. The method is to cut the largest small square from the lower left corner according to the decimal value system. First of all, the side length should be in hundreds, "hundred" is equivalent to the "etc" in the text, after trying ("negotiation"), you can cut a square with a side length of 200, but the side length cannot be taken as 300, so cut the square yellow armor with a side length of 200 in the lower left; Again with 10 ("etc" is 10), try again, so the yellow armor expands to a square made of yellow armor + 2 zhu power + yellow yi, and its side length is 230; Then take "etc" as 1 and try it, knowing that if you take 235 as the edge as a square, the original square will be exhausted. From this, it is known that the side lengths of yellow armor, zhu mi, and green power are 200, 30, 5, and the square root of the sought side length is 200+30+5=235.
In the above example, the prescription is exhausted, which is generally not the case, but the principle of calculation is the same. Take an example, cut the gradually increasing square from the lower left corner of the square with an area of 20000, and its side length is 100, 100+40=140 and 100+40+1=141, but there is still one remaining curved side, with an area of 20000-=119.
Regarding the situation where the integer square is inexhaustible, the "Liu Note" said: "The technique may be ordered by borrowing the addition method, although the thickness is similar, it cannot be used." Where the open product is square, the square is multiplied by its own multiplication and its integral points, and now the fate points without borrowing are often slightly less, and the fate points of the addition of borrowing are slightly more. The number cannot be obtained, so it is only ordered by the face so that the ear is not lost. ”
"Liu Note" pointed out that when the old method (that is, "art") was not exhaustive, a fraction ("life") was often added as the square root, and in the modern form, it was to take a+r/(2a) or a+r/(2a+1) as the value. Although this value is similar to the true value ("coarsely similar"), it is not correct ("unusable also"), because the square root should be squared and reverted to the original value, and the above value after adding fractions is either more or less squared, so it can only be used as an approximation. Because of this, only a new type of number can be introduced, and the true square root is defined as a new type of number, called "face" ("to face life"). "Liu Note" also takes fractions as an example, saying that "for example, three is divided by ten, and the rest is one-third, and the plural can be cited", which shows that the introduction of a new type of number due to the needs of the opening side is called "surface", and the introduction of a new type of number "fraction" due to the completion of division is the same reason.
"Chapter 9" and "Liu Note" based on the geometric problem of finding the side length of the known square area, through the geometric background to derive the algorithm of finding the side length, is called "open (flat) square technique". This introduced a new type of number " face" different from the rational number , which is equivalent to a modern ( radical ) irrational number . This kind of introduction is both simple and natural, because the combination of numbers and shapes, fundamentally excludes the possibility of mathematical crisis, this different way of thinking and consequences of dealing with problems in the East and the West, is quite intriguing and worth pondering.
The "Liu Note" said that "where the open product is square, the square is multiplied by itself and its integral", which is essentially equivalent to the definition of the new irrational number of "surface", and is also equivalent to the following formula
where A is a non-negative integer , but can also be a non-rational number. In fact, the text "Open (Ping) Fangshu" says: "If there is a division of reality, the division of the inner son is the definite reality." And open, open, open its mother to report. He also said, "If the mother cannot be opened, then the mother multiplies the truth, and the mother is opened, so that the mother is one." "It's the equivalent of a formula
In other parts of Liu's Notes, there are some similar formulas.
The situation of opening square is similar to that of open square, and the corresponding opening square technique in Chapter 9 is similar to open (flat) square technique, which is also based on the problem of finding one side of a cube of known volume, and forms an algorithm through geometric considerations, but this process from geometry to algorithm is not simple.
The problem is what to do with the fourth power, the fifth power, etc., at this time the geometric background completely disappears, if the concept of multidimensional space in ancient times is inconceivable, it can never be true, but in China to the Song Dynasty, the original diagram of the opening method made by Jiaxian of the Northern Song Dynasty (about the 11th century) is essentially the same as the Pascal triangle of later generations on binomial coefficients. Jia Xian gave the algorithm of opening the high power accordingly, obviously he did not derive this algorithm from geometric considerations, but from the algorithm of opening square and opening square techniques, the process is unknown, but the depth of his thinking and the subtlety of his skills cannot but be heartbreaking.
The introduction of infinitesimals
For Kaiping Cube, Liu Hui did not stop at introducing "surface", but further developed a significant innovation, considering another method: "Do not order it with face, add the method as before, and find its differential." The unknown number thinks that the molecule retreats to ten as the mother, and then retreats to the hundred as the mother. If the retreat is submerged, the division is fine, and although Zhu Mi has abandoned the number, it is not enough to say. ”
Still taking 20000 kelvins as an example, =141 + remaining fractions. The past has come to an end, but Liu Hui further pointed out that the Chinese decimal value system notation can also continue the above operations, taking "equal" as 1/10, 1/100, etc., to find that the cut square side length will be 141+4/10, 141+4/10+1/100, this process can be carried out indefinitely, and the discarded "fractions" will become smaller and smaller to the point of being negligible ("not enough to say"). In modern language, this new type of number "face" is the limit of a certain sequence.
= limit (141, 141.4, 141.41, ··).
From this, Liu Hui not only introduced decimals, but also introduced infinite decimals through the limit process, and used them for the specific calculation of pi π. The chapter "Chapter 9 Fang Tian" deals with the calculation of the area of fields and acres, and the "Liu Note" details the principles and methods of the "round field technique" applied therein, and also includes the calculation of π.
Under the condition that the radius of the circle is known, Liu Hui starts from the inner regular hexagon ("hexagon") of the circle, and gradually obtains the decimal approximation of the circumference of the inscribed regular dodecagon, the regular twenty-four to the regular 192 gon, and the area approximation of the regular 96 and the regular 192 gon. There are detailed calculations in the annotated texts in "Cutting Six Yao Techniques for Twelve Yao Techniques" and even "Cutting Forty-Eight Yao for Ninety-Six Yao Techniques".
The Pythagorean theorem is applied to the above calculation, and this calculation process can be carried out indefinitely with the number of sides doubling successively. Liu Hui pointed out in particular: "If you cut it fine, you will lose little, and if you cut it and cut it again, so that it cannot be cut, it will be combined with the circle, and nothing will be lost." In other words, if A is the area of the circle and An is the area of the circle inscribed by the regular n-gon, then
A-A6,A-A12,A-A24,...
will be reduced indefinitely so that it can be omitted ("and nothing is lost"). In modern language , A-A3·2 has a limit of 0 as n increases, or A3·2 has a limit of A as n increases ( " with a circle " ) .
This limit concept and method is not only found in the calculation of square and pi, but also in the calculation of solid volume. In the "Yang Equestrian Technique" of "Chapter 9 Shang Gong", there is: "Half of the mi, the rest of the mi, the subtle, the micro is invisible, so in words, the rest is taken." "It's also an extreme process. From this, Liu Hui established the very prominent volume theory in Chinese arithmetic, which is the most glorious page of ancient Chinese arithmetic.
Liu Hui's introduction of decimals is due to the ancient Chinese decimal notation of the place-value system. In modern notation , he defines a real number as an infinitesimal number
where 9 or 0 is the bit value. It is the limit value of a monotonically increasing or decreasing sequence ,,...,.
Weierstrass defined real numbers in terms of monotonic sequences of increasing or subtracting rational numbers , similar to Liu Hui's method. But comparing the two, the definition of the former is general and abstract, while Liu Hui's definition method is special and concrete. Therefore, the various laws of real numbers: the equivalence or equivalence of real numbers, the law of real number operations, the completeness of real numbers, etc., are not without tedious proofs in the Weierstrass sense, but they are much straightforward to prove in the sense of Liu Hui (including rational numbers equivalent to circular decimals). By the way, Liu Hui's specific representation method of real numbers was once used by Cantor to prove that the set of real numbers cannot be equivalent to the countable set, and at this end alone, it can be seen that this method contains a power that is difficult to match by other methods.
Based on the above, it can be concluded that as early as 263 AD, Liu Hui had completed the system of real numbers in the modern sense through decimals and limit processes.
Finally, it must be pointed out that it was Li Jimin who first understood that the so-called "face order" in "Chapter 9" means to define a certain irrational number and call it "face", and he also pointed out that there are many calculation methods and general rules for such irrational number "face" in "Chapter 9" and "Liu Note".
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(This article is compiled by the author based on a public report at the 2002 Beijing International Congress of Mathematicians)
