Chinese name
Value system symbol
Foreign name
positional notation
subject
Mathematics (arithmetic)
Another name
Bit value symbol
related notion
value system
quick
navigate by water/air
Decimal counting method
Development history
Basic introduction
definition
Bitwise notation refers to the method of counting by bit system, that is, the size of a number is represented by a series of continuous numbers, and the size represented by each number depends on its own numerical value; It depends on its location.
Roman numerals are non-value notation, while the usual carry notation is bit notation. The earliest people with the concept of value system were the ancient Babylonians around 2000 BC, but they used sexagesimal. In the world, China first used decimal notation in Shang Dynasty (about 16 and 17 century BC to about 1045 century BC). During the Warring States Period (4th century BC) or earlier, a perfect decimal notation system with spaces (zeros) as the unit was formed (but it was not widely used until10th century). Decimalization was really widely used in India at the end of the 6th century. After the 9th century, the decimal and numerical symbols they used-Arabic numerals-gradually spread to Arab and European countries.
value system
Position system, also known as position system and position system, is developed on the basis of carry system. The so-called position system means that if the same number is in different positions (digits), it will have different position values (also called weights), so the values represented will be different. In fact, it represents a certain multiple of the corresponding level "unit" of this bit, such as 6, which means that the single digit is 6× 1, the decimal digit is 6× 10, and the hundredth digit is 100. More strictly speaking, after selecting the radix b of the carry system, the writing of the number B (numbers) of 0, 1 2, ..., b- 1 is given, so any natural number n can be represented by a polynomial of b, with these numbers as coefficients (can be repeated):
Among them,. So in the "binary" symbol, we remember n as
Obviously, the same number is written in different positions, and the corresponding power of B is different, so the number represented by this number is also different. In addition, in order not to be confused with other radix counting methods, we can omit the brackets of n and the foot code b indicating the radix and write it directly. This is exactly what people do in decimal numerical system and binary numerical system commonly used in electronic computers [2].
Decimal counting method
Decimal notation is the most commonly used notation, that is, decimal-based notation. Any positive integer has only one decimal expression:
Where is a number. N is usually abbreviated as, and decimals are also expressed by decimals.
Development history
In history, the numeration system of the earlier developed value system includes: Babylon's 60-digit value system (not later than BC19th century, see Babylonian Mathematics); The decimal numerical system in ancient China (not later than the 5th century BC, see Ancient Mathematics of China); Mayan decimal system (for centuries BC, see Maya Mathematics); India's decimal numerical system (about the 5th century AD, see "Ancient Indian Mathematics" and "Indo-Arabic Numbers") [2].
Here we define the lower limit of the decimal notation in ancient China in the 5th century BC, that is, at the end of the Spring and Autumn Period, because the decimal notation-based notation system and its operation method were very developed, complete and popular at that time. In fact, decimal notation may have been produced in China much earlier than this lower limit. In the Oracle Bone Inscriptions of Yin Ruins from about14th century BC to about 1 1 th century BC, there have been similar records of "348" and "2656", which are characterized by one, two, three, ... and ninety-nine numbers and characters representing10th power. At each level (bit) corresponding to each power of 10, when the value on this bit is obtained by dividing nine numbers (i.e. 1, 2, ... and nine, using ten, hundred, thousand and ten thousand to represent different powers of 10, which is different from the polynomial representing the binary system (/kloc). The difference between multiplication and cluster number system and numerical system is not only that the former must write the power of B-system of carry system together with its coefficients, but also that when some coefficients are zero, the former doesn't need to mention this item at all, while the latter (numerical system) must leave it blank or add a zero sign in the corresponding position. But on the other hand, the records in Oracle Bone Inscriptions only describe the results of digital operation at that time in words, but do not give the representation of numbers in actual operation, which is obviously different. In fact, although the decimal numeration system was very popular at the latest in the 5th century BC, in the Ten Books of Calculating Classics, which was formed from the Han Dynasty to the Tang Dynasty, the way of describing numbers with words was exactly the same as that of Oracle Bone Inscriptions, even today when we write numbers with Chinese characters.