decimal system(Decimal System) is a counting method, that is, full decimal one, first digitPosition powerIt is 10 ^ 0, the second digit is 10 ^ 1, and the Nth digit is 10 ^ (N-1). The value of this number is equal to the sum of each (value × bit weight).
The decimal counting method is the most commonly used counting method in daily lifeCounting unitBetweenProgression rateThe counting rule of all ten is the decimal counting method.
Human arithmetic adopts decimal system, which may be related to human having ten fingers.AristotleHe said that the universal use of decimal system is just the result of the anatomical fact that most people are born with 10 fingers.In fact, in the written numeration system independently developed in the ancient world, exceptBabylonian civilizationWedge numbers of are base 60,Mayan numeralsThey are all decimal except 20.However, these decimal numeration systems are not bitwise.
The decimal system, called Decimal System in English, is derived from the Greek Decem, meaning ten.Decimal counting was invented by Hindus 1500 years ago and passed down from Arabs to the 11th century.
The decimal system is based on the two principles of the decimal system and the decimal system, that is, all numbers are represented by 10 basic symbols, and the full decimal system is one. At the same time, the same symbol represents different values at different positions, so the position of the symbol is very important.The basic symbol is ten numbers from 0 to 9.To represent 10 times of these ten numbers, move these numbers to the left by one digit and fill in the blanks with 0, that is, 10, 20, 30,90；To represent 10 times of these ten numbers, continue to move the number to the left, that is, 100200300.To represent 1/10 of a number, move the number to the right, and fill in the blanks with 0 if necessary: 1/10 bit 0.1, 1/100 is 0.01, and 1/1000 is 0.001.
history
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An indispensable part of daily lifeBit value systemIs a great invention of China.At the latestShang dynastyAt that time, China had adopted the decimal value system.From the unearthed pottery inscriptions and oracle bone inscriptions of the Shang Dynasty, it can be seen that 13 numbers, including one, two, three, four, five, six, seven, eight, nine, ten, hundred, thousand and ten thousand, can be used to record any natural number within 100000.Although the shape of these numeration characters has changed in later generations, the numeration method has never been interrupted and has been followed and improved.DecimalNumerationIt is the most advanced and scientific numeration method in the ancient worldWorld ScienceAnd the development of culture.The decimal system is an outstanding creation of the Chinese people and is of great significance in the history of world mathematics.
A group of ceramic measuring tools were unearthed in F901 of the late Yangshao house in Dadiwan, mainly including muddy trough shaped bar plate, long handle twist ear shovel shaped reading with fine sand, muddy single ring ear dustpan shaped reading, and four deep belly pots with mud cover.The volume of the bar plate is about 264.3 cubic centimeters;The natural grain volume of spade copying is about 2650.7 cubic centimeters;The natural grain volume of the dustpan is about 5288.4 cubic centimeters;The volume of four deep belly pots is about 26082.1 cubic centimeters.It can be seen from this that the relationship between the other three pieces is increasing by ten times, except that the dustpan copying is twice as much as the shovel copying.The discovery of these weights and measures also provides valuable material materials for the study of the origin of the ancient Chinese decimal system.
CUBA BILLENAlthough it has the significance of the place value system, it uses the decimal notation, which is very tedious.Ancient EgyptThere are only two number symbols from one to ten, and four number symbols from one hundred to ten million, and these symbols are pictographic, such as using a bird to represent 100000.Because of its advanced geometry, ancient Greece despised calculation and its backward numeration method. It used all Greek letters to represent numbers ranging from one to ten thousand. If there were not enough letters, it would be supplemented by adding symbols such as "'".Ancient Rome used the cumulative method, such as ccc for 300.In ancient India, there were both letters and accumulation methods. The decimal value system was adopted in the seventh century AD, which was probably influenced by China.India in general use——Arabic numeralsAnd numeration, which was introduced to Europe around the tenth century.
In terms of computational mathematics, China hadFour arithmetic operations, toSpring and Autumn Period and Warring States PeriodintegerThe four operations of and fraction are quite complete.Among them, the positive integer appeared in the Spring and Autumn PeriodmultiplicationThe rhyme "Jiujiuge" can be called the crystallization of the combination of advanced decimal counting method and concise Chinese language, which can not be produced by any other counting method and language.Since then, the "Jiujiu Song" has become one of the most basic foundations for the popularization and development of mathematics, and has continued. Its change is only that the ancient "Jiujiu Song" started from "9981" to "22 as four", while the modern "one as one" to "9981".
Some scholars believe that BeijingZhoukou StoreTen thousand years agoMountaintop cavemanThe bone tube unearthed from the site is represented by one dot, two dots juxtaposed to represent 2, three dots juxtaposed to represent 3, five dots arranged in two up and three down represent 5, and the oblong may represent ten.Professor Li Di, a famous Chinese mathematical historian and a communication academician of the International Academy of the History of Science, believes that the human bone tube symbol in the mountaintop cave is "a decimal idea".
Other scholars have made comments on Qinghai, ChinaLedu CountyMore than 1000 Neolithic unearthed in LiuwanBone sliceThey belong to four Chinese cultural types, namely, Machang, Banshan, Qijia and Xindian.The bone slice is 2-2.4 cm long and 1 mm thick.There are no more than eight nicks on the bone slice. The number of nicks on each bone slice is no more than ten. They believe that there were addition operations and decimal systems in the Neolithic Age.
Other scholars believe that,OracleThe middle horizontal represents 1, the overlapping of two horizontal represents 2, the three horizontal represents 3, and the four horizontal represents 4,X represents five, "human" represents six, "ten" represents seven, ") (" represents eight, "nine" is already nine; | represents ten, | | represents 20, | | | represents thirty, | | | represents forty; in addition, 50, 60, 70, 80, 90100200300400600700800100010002000,...... 900000000...... 40000 have different symbols.Oracle inscriptions of the Shang Dynasty "have formed a complete decimal system".
BeijingMuseum of Chinese HistoryThere is one hiddenYin Xu The ivory ruler unearthed is 15.78cm long and divided into ten inches, which indicates that the decimal system of the Shang Dynasty has been used in length.
The Law of Chronology and Numbers of the Inscriptions in the Zhou Dynasty of China, InheritanceShang dynastyThe decimal system of has also made significant progress. The decimal symbols include ten, one hundred, one thousand, ten thousand, and one hundred million, such asWestern Zhou DynastyIn the inscriptions of Jin Dynasty, "fighting against ghosts... captured thirteen thousand eighty-one people", "King Wu marched all over the country and captured three hundred million people, two hundred and thirty people", there appeared a digit count, such as "three hundred and fifty-five cattle captured", of which three hundred and fifty-five were written as "three complete XX", the front "all" was the "hundred" in the inscriptions of Jin Dynasty, and the latter two XX were fifty-five, omitting the "ten", and the concept of location appeared,But a complete bit value system has not yet been formed.Inscriptions on bronzeShang Yang quantityThe inscription also shows scores.
Around 3400 BC,Ancient EgyptThere is a decimal based notation.But this decimal system has no concept of bit value.
In 800 years before the Vedic era, the Shindharma Sutra in the Indian ritual sutras included a large number of fractions, but there was no evidence that the written numeration system at that time was decimal.[1]
In 300 BC, the Brahmi number in India was decimal.The Brahmi decimal system has no concept of place value.
Unearthed inPakistanOfancient IndiaBucksally manuscript may be the earliest "real" decimal system including 0 in the world, but its specific time is controversial.
Oracle ScriptAccording to records in, people in the Shang Dynasty had learned to use 13 words, namely, one, two, three, four, five, six, seven, eight, nine, ten, hundred, thousand, and ten thousand, to record any number within 100000, but the largest number that can be confirmed at that time was 30000.oracular inscriptions on tortoise shells and bonesAndOdd number、even numbersandmultipleThe concept of.
There is an idiom called“can be counted on one's fingers"It shows that ancient people could not count without fingers, while ordinary people had ten fingers. Therefore, the use of decimal system seems to be extremely natural. But the actual situation is not always the caseBabylonThe 60 carry system is used (this carry system still has traces, such as one minute=60 seconds, etc.), and the 20 carry system is also used.Ancient EgyptThey used the decimal system very early, but they didn't know the bit value system.The so-called bit value system is that what number a number represents depends on its location.The position value system is the crystallization of human wisdom for thousands of years.Zero isPlace value notationThe essence of.But its emergence is not easy.China is the first country to use decimal notation and recognize the carry system.The number expressed in spoken or written Chinese also follows this principle, such as 127.At the same time, I was the first to know 0.
Formal differences
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The Babylonian 60 carry system represents 1 with one big wedge at the top and one small wedge at the bottom, two parallel wedges representing 2, three parallel wedges representing 3, two wedges at the top and two wedges at the bottom representing 4, three wedges at the top and two wedges at the bottom representing 5, three wedges at the top and three wedges at the bottom representing 6, four wedges at the top and three wedges at the bottom representing 7, four wedges at the top and four wedges at the bottom representing 8, and five wedges at the top and four wedges at the bottom representing 9;One left small right large cross wedge represents 10, two cross wedges side by side represent 20, three cross wedges side by side represent 30, and four cross wedges side by side represent 40.
MayaThe 20 carry system uses one point to represent 1, two points to represent 2, three points to represent 3, four points to represent 4, a short horizontal line to represent 5, one point to represent 6, two points to represent 7, three points to represent 8, and four points to represent 9;The upper and lower two horizontal lines represent 10, one point above the upper and lower two horizontal lines represents 11, three overlapping horizontal lines represent 15, and one, two, and three points above the three horizontal lines represent 16, 17, and 18;A dot on a small circle represents 20.
In the ancient Egyptian decimal system, one vertical channel represents 1, two parallel vertical channels represent 2, three vertical channels represent 3, one horizontal channel represents 4, left two left and right vertical channels represent 5, upper three left and lower three left represent 6, upper two lower three left represent 8, four "(" side by side represents 9, one "herringbone" represents 10, and "human" representsAdding a horizontal line to the top represents 20, adding a dot to the left of 20 represents 30, adding a dot to the top of the horizontal line represents 40, adding three vertical lines to the horizontal line (such as 8 in Chinese planning) represents 60, adding four vertical lines to the horizontal line represents 80 (similar to 9 in Chinese planning) represents 80, and adding three vertical lines to two horizontal lines represents 90.
Greek decimal, 1 to 9, 10 to 90, 100 to 900 each have different single letter representations.
In the Kharoshi decimal system of ancient India, one vertical channel represents 1, two parallel vertical channels represent 2, three vertical channels represent 3, one X represents 4, IX represents 5, | | X represents 6, XX represents 8, 10, and 20 have single characters.
The ancient Indian and Brahmi decimal systems are similar to the Greek decimal system, with 1 to 9, 10 to 90, 100 to 900 each having different single letter representations.There are many symbols.
According to the textual research of some scholars, there are two types of decimal system in ancient China: written system and arithmetic system.
Mo is the right way to write, but Mo is not the right way to write.
Smaller than desert, it is fromTianzhuThe number on the Buddhist sutra of.These "sutra numbers" have become "ancient usage".
And weights and measures
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China's decade system has a long history of measurement.In the 6th century BC, a ruler of Zhou Dynasty was carved with one tenth of an inch and one hundredth of a cent.
usurper who founded the Xin dynastyThe capacity standard of one hundred pairs of bronze is set by the official. One Hu=ten buckets, one bucket=ten liters, and one liter=10 combinations.
traditionWeights and measuresThe decimal system is not used completely, for example, 1 jin is equal to 16 liang, 1 foot is equal to 12 inches, etc.MetricFully use decimal system to make conversion more direct.Government of the Republic of ChinaIn the 1920s, the city system was introduced to connect with the metric system.In the 1980s, the Hong Kong government vigorously promoted the benefits of decimal system. At that time, there were slogans such as "using decimal system, fair and easy to calculate" or "decimal system, easy to calculate", but people still used old system, British system and other non decimal conversion.[2]
Supplementary notes
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Peer games《Dongfang Red Devil Village》One side BOSSRumia Nicknamed "decimal", it comes from the dialogue of Morisha Line: "Why do you always straighten your arms?" "Does it look like Jesus was nailed to the cross?" "It looks like humans have adopted decimal"
And binary
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Internal use of computerBinaryThe conversion between binary and decimal numbers is complicated.For example, if you want the computer to calculate 50+50, you should firstDecimal number50 converted toBinary number110010, this process needs to be done many timesdivision, and the computer is the slowest for division.The conversion of decimal 50 to binary 110010 is not complete. After the result of 1100100 is calculated, it needs to be converted to decimal 100. This is a process of multiplication. Although it is simpler for computers than division, the calculation speed is not fast.The reason for wasting many steps of what was originally completed in one step is that the decimal system used by people is not suitable for modern information equipment and is not the best information counting method.If people use binary to represent numbers, it will not only be easier to communicate with computers, but also only need to remember how to write 0 and 1 to count, which is 80% easier than learning ten numbers in decimal system.This is not all. For example, the decimal 0.8 should be written as 0.11001100 in binary... There are countless 1100, or in other words, decimal 1100terminating decimal Conversion to binary cannot guarantee accurate conversion, and conversion from binary decimal to decimal also encounters the same problem.This also brings inconvenience to information processing.Even in order to convert decimal numbers and binary numbers quickly, special circuits and statements were added to complete this process when designing the processor, which caused a waste of processor design.Therefore, it can be said that the decimal system is not suitable for modern information equipment.
Binary to decimal
The basic way to convert binary numbers into decimal numbers is to first write binary numbers asweightingCoefficient expansion, and then sum according to decimal addition rules.This practice is called the "addition by rights" method.Example 1105 converts binary number 110.11 into decimal number.
Decimal to binary
When a decimal number is converted to a binary number, because the conversion method of integer and decimal is different, the decimal number'sInteger partAnd decimal parts are converted respectively and then merged.
1. The decimal integer is converted to binary integer. The decimal integer is converted to binary integer using the method of "divide by 2 and take remainder, reverse order".The specific method is: use 2 to remove decimalinteger, you can get a quotient sumremainder；Then remove the quotient by 2, and a quotient and remainder will be obtained. This is done until the quotient is zero. Then take the remainder obtained first as the low significant bit of the binary number, and the remainder obtained later as the high significant bit of the binary number, and arrange them in sequence.
The decimal decimal is converted to binary decimal by the method of "multiplying by 2 to round and arranging in order".The specific method is: multiply 2 by decimal decimal to get the productInteger partTake out, then multiply the remaining decimal part by 2 to get another product, and then take out the integral part of the product, and do so until the decimal part of the product is zero or reaches the required precision.
Then arrange the integer parts in order. The integer taken first is the high significant digit of the binary decimal, and the integer taken later is the low significant digit.[3]
