Here we mainly introduce the binary notation.
Binary
In the famous Schlossbiliothke zu Gotha library (Schlossbiliothke zu Gotha) in Turingen, Germany, there is a precious manuscript with the title: "1 and 0 , The miraculous origin of all numbers. This is a wonderful example of the secret of creation, because everything comes from God."
This is the German genius master Leibniz (Gottfried Wilhelm Leibniz, 1646-1716) Handwriting. However, Leibniz has only a few pages of extremely concise descriptions of this wonderful digital system. Using words familiar to modern people, we can explain binary as follows:
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64< /p>
2^7 = 128
And so on.
Add the numbers on the right side of the equal sign to get any natural number. We just need to explain: the number of squares of 2 is used, and the number of squares of 2 is discarded. The binary representation sequence starts from the right, the first digit is 2 to the 0th power, the second digit is 2 to the 1st power, the third digit is 2 to the 2nd power..., and so on. All the square positions of 2 are used, we use "1" to mark, and all the square positions of 2 are discarded, we use "0" to mark. In this way, we get the following sequence:
1 1 1 0 0 1 0 1
2 to the 7th power
2 to the 6th power< /p>
The 5th power of 2
The 4th power of 2
The 3rd power of 2
The second power of 2
2 to the 1st power
2 to the 0 power
128+64+32+0+0+4+0+1=229
< p> In this example, the decimal number "229" can be expressed as the binary "11100101". The leftmost bit of any binary number is "1". In this way, the entire natural number sequence expressed by the ten numbers 1 to 9 and 0 can be replaced by the two numbers 0 and 1. The two numbers 0 and 1 are easily electronic: there is a current is 1; there is no current is 0. This is the fundamental secret of the entire modern computer technology.Gossip
When this manuscript was completed, Leibniz was fifty years old. There is no doubt that he is the inventor of this binary which is the basis of modern computer technology. Moreover, before or at the same time as him, no one seemed to have thought about this problem. This is very rare in the history of mathematics.
Leibniz not only invented binary, but also gave it religious connotations. He wrote to the French Jesuit priest Joachim Bouvet (1662-1732) who was preaching in China at that time:
"The beginning of the first day is 1, which is God. The beginning of the second day was 2.... By the seventh day, everything was there. Therefore, this last day was also the most perfect. Because, at this time, everything in the world has been created. So it was written as '7 ', which is '111' (111 in binary is equal to 7 in decimal), and does not contain 0. Only when we only use 0 and 1 to express this number, we can understand why the seventh day is the most perfect, why 7 is A sacred number. What is particularly noteworthy is its (the seventh day) feature (written as binary 111) and its connection to the Trinity."
Bu Wei is a master of Sinology, and his introduction to China is One of the most important reasons for China's craze in European academic circles in the 17th and 18th centuries. Bouvier is a good friend of Leibniz and has maintained frequent correspondence with him. Leibniz has translated many of Bouvier's articles into German and published them. It was Bouvier who introduced the Zhouyi and the system of gossip to Leibniz, and explained the authoritative status of the Zhouyi in Chinese culture.
Bagua is a divination system composed of eight symbol groups, and these symbols are divided into continuous and discontinuous horizontal lines. These two symbols, later called "yin" and "yang", in Leibniz's eyes, are the Chinese version of his binary system. He felt that the relationship between this symbol system from ancient Chinese culture and his binary system was too obvious, so he asserted that binary system is the most universal and perfect logical language in the world.
Another person who may arouse Leibniz’s interest in gossip is Wilhelm Ernst Tentzel, who was the head of the coin collection room of the Grand Duke of Turingen and Leibniz One of his friends. There is a coin printed with gossip symbols in the coin collection he is in charge of.
Bagua and Binary
Today, there is a general consensus in the Western academic circles: There is no direct relationship between Bagua and Binary. First of all, China's number system is decimal. Secondly, according to the historical data we have today, in the Qin and Han dynasties, China does not yet have the concept of "zero" in Leibniz's binary sense.
If it is said that the yin and yang metamorphosing all things in the cohesive part of Zhouyi is what Leibniz said 0 and 1 are the source of all things, this is difficult to establish. The current edition of Zhouyi can be roughly divided into three parts, the first is the hexagram, the second is the Yao, and the third is the biography, the so-called "Ten Wings". Among them, the part of the hexagram should be the oldest. From pre-Qin documents such as Shang Shu, Zhou Li, Zuo Zhuan, Guo Yu, and later archaeological excavations, we have a preliminary understanding of the tortoise divination in the early years of the Western Zhou Dynasty. However, we hardly have any detailed and reliable information about "Ib". The hexagrams in Zhouyi may be the "yixiang" seen by Han Xuanzi. In any case, we basically cannot see the shadow of yin and yang in the hexagrams and lines. The system of yin and yang is basically developed and described in the Yi Zhuan, although its origin must predate the Yi Zhuan. And "Yi Zhuan" is obviously a decimal system. Through the records of "Han Shu·Lv Li Zhi", we can not only know that in the era of Zhouyi, the calendar calculation used the decimal system, and the key number is not 1, let alone 0, but 2 (yin , Yang) and 3 (Heaven, Earth, Man). (See my book "Confucian Love for Mathematical Geometry")
In addition, the important concept "nothing" in the Taoist philosophy system has no direct relationship with Leibniz's zero. Russell explained "0" in "The Tao of Mathematical Philosophy" as: all classes without molecules. This is exactly the "zero" in Leibniz's mind. Russell’s explanation was inspired by the book Grundlage der Arithmetik ("Basics of Arithmetic") by the famous German language philosopher Gottlob Frege (1848-1925). The "zero" in the number theory system of Flagg and Russell is replaced by Chinese, which is the general term for all "nothing". The "nothing" in Tao philosophy is not but not the sum of many "nothings", but that particular "nothing" is the essence of that "Tao".
To put it simply, in the three hundred years since Leibniz, Western scientists and philosophers have done countless studies, and they have not found any substantial connection between binary and gossip. In China, below the Qin and Han dynasties, apart from the efforts to establish a philosophical system using special explanations of gossip, we basically cannot see a convincing explanation for it.
Binary number
Introduction
Binary is a number system widely used in computing technology. Binary numbers are numbers represented by two digits, 0 and 1. Its base is 2, the carry rule is "every two enters one", and the borrow rule is "borrow one to become two". Binary numbers also use position counting, and their position weight is a power of 2 as the base. For example, for the binary number 110.11, the order of the bit weight is 4, 2, 1, 2-1, 2-2.
Representation
For a binary number with n-digit integers and m-digit decimals, it is expressed by a weighted coefficient expansion, which can be written as:
(N)2= an-1×2n-1+an-2×2n-2+…… +a1×21+a0×20+a-1×2-1+a-2×2-2+……+a-m×2 -m
In the formula, aj represents the coefficient of the jth position, which is a number between 0 and 1.
Binary numbers can generally be written as: (an-1an-2...a1a0a-1a-2...a-m)2.
Example solution
[Example 1] Write the binary number 111.01 in the form of weighting coefficient.
Solution: (111.01)2=1×22+l×21+1×20+1×2-2
addition and multiplication
The basic laws of arithmetic operation of binary numbers and the operation of decimal numbers are very Similar. The most commonly used are addition and multiplication operations.
Binary addition
There are four cases: 0+0=0
0+1=1
1+0=1
1+1=0 Carry to 1
[Example 2] Find (1101)2+(1011)2 Sum
Solution: 1 1 0 1
+ 1 0 1 1
1 1 0 0 0
Binary multiplication h3>
There are four cases: 0×0=0
1×0=0
0×1=0
1×1= 1
[Example 3] Find the product of (1110)2 multiplied by (101)2
Solution: 1 1 1 0
× 1 0 1
1 1 1 0
0 0 0 0
+ 1 1 1 0
1 0 0 0 1 1 0
Place value thinking
The current Indian-Arabic numerals used in counting use the decimal value system principle. Among them, the decimal system is influenced by natural phenomena, and it is recognized that it is related to the ten fingers in life; while the place value system is a subjective product. Looking back at the history of notation, we can find that the importance of place value system in counting is far greater than that of decimal system. It was once compared to the importance of letters in words by historians of mathematics. There are many ways to express place value, and its formation process is also long.
The idea of place value in the notation is that the exponential code symbol not only has the size of the number it is intended to represent, but also depends on its location to determine the exact value of the number in the entire number. For example, India-the Arabic number 121, the number 1 on the right represents the number 1, the 2 in the middle represents 20 because it is on the 10th place, and the same number 1 on the left represents 100 because it is on the hundred's place. Each digit is combined by addition, and the entire number represents one hundred and twenty-one. Another example is the Roman numeral Ⅳ. The Ⅴ on the right represents 5 and the I on the left represents -1. The numbers are also combined by addition, and the entire number represents 4.
India now prevailing --- Arabic numerals use decimal notation, and any natural number can be expressed as an·10n+ an-1·10 n-1 + ……+ a1·10 + a0 Form. 10 is called the carry base, a0 , a1, …, an is 1, 2, …,9 ,0 these 10 digits One of them. The so-called place value system is to omit the power of 10 and the plus sign when writing. As above, 121 is shorthand for 1·102 + 2·10 + 1. Its characteristic is that any natural number can be expressed by only these 10 digits. Counting from the right, the position of the number is called the ones place, tens place, hundreds place and so on. What value a number represents depends on its position. This is the meaning of "place value" (place value or positional value).
In the ancient notation, the place-value system is mainly Babylonian cuneiform notation, Mayan notation, Chinese arithmetic notation and India-Arabic numeral notation. Among them, Babylon uses 60-carry counting, Maya has 20-carry and 18-carry mixed counting, and Chinese arithmetic chips and India---Arabic numbers both use 10-carry. The Maya counts from bottom to top, and the bottom is the ones digit, the higher the digit is, the higher the digit; the rest of the place-value system notation is to increase the digits from right to left. Although the carry base and the number arrangement are not the same, they are all consistent in the meaning of place value, which reflects the commonality of the development of human mathematics.