Negative number, the full name of negative real number, is a mathematical term, such as − 3, − 1.5, − 1/2, − 584 and other numbers with "−" in front of positive numbers, which are called negative numbers.0 is neither positive nor negative.[5]
Negative numbers and positive numbers represent quantities with opposite meanings.
For negative numbersminus sign(Minus Sign, which is equivalent to a minus sign) "−" and a positive number mark, such as − 2, represent 2Inverse number。Therefore, any positive number is negative if it is preceded by a negative sign.A negative number is the opposite of its absolute value.stayNumber axisOn the line, negative numbers are on the left side of 0. The earliest record of negative numbers is the ancient Chinese mathematical work Nine Chapters of Arithmetic.stayCalculationAccording to the stipulation of "positive counting is red, negative counting is black", that is, red counting is used to represent positive numbers, and black counting is used to represent negative numbers.Two negative numbers are compared in size, and the one with a larger absolute value is smaller.
People often encounter various quantities with opposite meanings in life.For example, there are surplus and deficit in bookkeeping;When calculating the rice stored in the granary, sometimes it is necessary to record the grain, and sometimes it is necessary to record the grain.For convenience, people consider the number in the opposite meaning.So people introduced the concept of positive and negative numbers, recording the surplus money into grain as positive, and recording the loss of money and grain as negative.It can be seen that positive and negative numbers are produced in production practice.
According to historical records, as early as more than 2000 years ago, China had the concept of positive and negative numbers and mastered the operating principles of positive and negative numbers.When people calculate, they use some small bamboo sticks to put out various figures for calculation.For example, 356 is placed as | | |, 3056 is placed as, etc.These little bamboo sticks are called“Calculation”, Suanqian can also be made from bones and ivory.
Liu Hui, a scholar in the Three Kingdoms Period of China[2]He has made great contributions to the establishment of the concept of negative numbers.Liu Hui first gave the definition of positive and negative numbers. He said: "The gains and losses of today's two calculations are opposite, so we should name them positive and negative." This means that when we encounter quantities with opposite meanings in the calculation process, we should distinguish them with positive and negative numbers.
For the first time, Liu Hui gave a method to distinguish positive and negative numbers.He said: "Positive counts as red, negative counts as black; otherwise, the slant positive counts as different", which means that the number placed by the red stick represents positive numbers, and the number placed by the black stick represents negative numbers;You can also use a slanted stick to represent negative numbers and a straight stick to represent positive numbers.
Famous mathematical monographs in ancient China《Chapter Nine Arithmetic》(It was written in the first century AD), the first rule of addition and subtraction of positive and negative numbers was put forward: "positive and negative numbers say: the same name divides, synonyms benefit each other, positive does not enter negative, negative does not enter positive; its synonyms divide, homonyms benefit each other,[3]The positive does not enter the positive, and the negative does not enter the negative. "Here, "name" is "sign", "division" is "subtraction", "mutual benefit" and "division" are "addition" and "subtraction" of the absolute value of two numbers, and "nothing" is "zero".
The addition and subtraction rule of positive and negative numbers is: subtracting two numbers with the same sign is equal to subtracting their absolute values, and subtracting two numbers with different signs is equal to adding their absolute values.Negative number of zero minus positive number, zeroalleviate excessive burdenThe number is positive.The addition of two numbers with different signs is equal to the subtraction of their absolute values, and the addition of two numbers with the same sign is equal to the addition of their absolute values.Zero plus positive numbers equals positive numbers, zero plus negative numbers equals negative numbers. "
This paragraph about the algorithm of positive and negative numbers is not completely correct. The introduction of negative numbers is one of the outstanding contributions of Chinese mathematicians.
The habit of using numbers in different colors to represent positive and negative numbers, and using red to represent negative numbers. Newspapers published that a country had a deficit in its economy, indicating that expenditure was greater than income, and that it was unable to make ends meet financially.[4]
Negative numbers are positiveInverse number。In real life, positive and negative numbers represent two quantities with opposite meanings.The temperature in Wuhan is as high as 42 ° C in summer. You will think that Wuhan is really like a furnace. In winter, the temperature in Harbin is minus 32 ° C, which makes you feel cold in northern winter.
In today's primary and secondary school textbooks, the introduction of negative numbers is introduced through arithmetic operations: you can get a negative number by subtracting a larger number from a smaller number.This introduction method can give intuitive understanding of negative numbers in some special problem situations.And inAncient mathematicsIn, negative numbers are oftenalgebraic equationGenerated in the process of solving.To ancient BabylonianAlgebraThe study found that the Babylonians did not put forward the concept of negative roots in solving the equation, that is, they did not use or failed to find the concept of negative roots.In the works of Diophantus, a Greek scholar in the third century, only the positive root of the equation was given.However, in Chinese traditional mathematics, negative numbers and related algorithms have been formed earlier.
In addition to the positive and negative operation methods defined in Nine Chapters of ArithmeticLiu Yao(206 AD) and Yang Hui (1261 AD) of the Song Dynasty also talked about the addition and subtraction rules of positive and negative numbers, which are completely consistent with the nine chapter arithmetic.It is worth mentioning that Zhu Shijie of the Yuan Dynasty, in addition to explicitly giving the addition and subtraction rules of positive and negative numbers with the same sign but different signs, also gave the multiplication and division rules of positive and negative numbers.Negative numbers were recognized and recognized abroad much later than in China.In India, mathematiciansBrahmaguptaOnly in 628 did we know that negative numbers can beQuadratic equationThe root of.The most successful French mathematician in Europe in the 14th century, Chukai, described negative numbers as absurd.It was not until the 17th century that the Dutch Rilar (1629) first recognized and used negative numbers to solve geometric problems.
Different from ancient Chinese mathematicians, western mathematicians mostly studied the rationality of the existence of negative numbers.16. In the 17th century, most European mathematicians did not recognize negative numbers as numbers.Pascal thinks that subtracting 4 from 0 is pure nonsense.Pascal's friend Arunde put forward an interesting argument against negative numbers. He said (- 1): 1=1: (- 1), then how can the ratio of smaller numbers to larger numbers be equal to the ratio of larger numbers to smaller numbers?Until 1712, even Leibniz admitted that this statement was reasonable.British mathematician Wallace admitted that negative numbers are less than zero and greater thanInfinity(1655).He explained that when a>0, the famous British algebra writer De Morgan still believed that negative numbers were fictitious in 1831.He illustrated this point with the following example: "The father is 56 years old, and his son is 29 years old. When will the father's age be twice that of his son?" He formulated the equation 56+x=2 (29+x), and solved x=- 2.He called this solution absurd.Of course, few people in Europe rejected negative numbers in the 18th century.With the establishment of the theoretical basis of integers in the 19th century, the logical rationality of negative numbers was really established.
The Chinese people began to use negative numbers very early. The famous ancient Chinese mathematical work "Nine Chapters of Arithmetic", in the chapter of "equation", formally introduced negative numbers and their addition and subtraction algorithms for the first time in the history of world mathematics, and gave the algorithm known as "positive and negative technique". Liu Hui, a mathematician in the Wei and Jin dynasties, used different colored arithmetic chips (counting tools in the shape of small sticks) in his book "Nine Chapters of Arithmetic Notes"Represent positive and negative numbers respectively (red is positive, black is negative, horizontal is ten, vertical is one)
"Positive negative technique" is the addition and subtraction rule of positive negative technique.One of them is "the division of the same name, the benefit of the different name, the positive does not enter the negative, the negative does not enter the positive." In fact, he is the addition and subtraction rule. Taking the modern formula as an example, this passage can be explained as follows:
"Divide by the same name" means that when two numbers with the same sign are subtracted, the sign of the minuend is in front of the bracket, and the absolute value of the minuend minus the absolute value of the subtraction is in the bracket.For example:
（+5）－（－3）=+（5+3）
（－5）－（－3）=－（5－3）
"The synonyms are mutually beneficial", that is, when two numbers with different signs are subtracted, the sign in front of the parentheses is the minuend, and the absolute value of the minuend plus the absolute value of the minuend in the parentheses.For example:
（+5）－（－3）=+（5+3）
（－5）－（+3）=－（5+3）
"Positive does not enter negative, negative does not enter positive", that is, 0 minus positive is negative, and 0 minus negative is positive.For example:
0－（+3）=－3
0－（－3）=+3
Historical data prove that Chinese people began to use negative numbers and applied them to production and life more than 200 years ago.For example, in ancient commercial activities, income was positive and expenditure was negative;Surplus is positive, deficit is negative. In ancient agricultural activities, increase in production was positive and decrease in production was negative.The Chinese use negative numbers for the first time in the world.
essential information
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Negative numbers are smaller than zero, then negative numbers are smaller than positive numbers.Zero is neither positive nor negative.Then - a
Number axis
There is no minimum or maximum number in negative numbers.
The minus sign before removing a negative number is equal to the absolute value of the negative number.
For example, - 2, - 5.33, - 45, etc.: the absolute value of - 2 is 2, the absolute value of - 5.33 is 5.33, and the absolute value of - 45 is 45, etc.
Scores can also be negative, such as - 2/5
Negativesquare rootuseImaginary unit"I" means.(Negative numbers within the real number range have no square root)
The maximum negative integer is: - 1
There is no minimum negative number.
Example
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Example 1
We learned natural numbers in primary school;If there is no object, it is represented by 0. Sometimes, measurement and calculation cannot getintegerThe result ofdecimalexpress.Have you seen other kinds of numbers?
There are two thermometers. The liquid level of the thermometer refers to the sixth scale above 0, which indicates the temperature of 6 ℃. So the liquid level of the thermometer refers to the sixth scale below 0. How to express the temperature at this time?[1]
Note: If it is also expressed in 6 ℃, it is impossible to distinguish between 6 ℃ above zero and 6 ℃ below zero, so we introduce a new number - negative number.
Reference answer: recorded as - 6 ℃.
Note: We introduced the concept of negative number in order to distinguish the group of quantities with opposite meanings, namely, 6 ℃ above zero and 6 ℃ below zero.
Example 2
As can be seen from the topographic map of China, there is a world's highest peak -Mount Qomolangma, 8844M is marked on the drawing;
One moreTurpanBasin, marked - 155M.Can you tell how high they are?
Tips:
As can be seen from the topographic map of China, the above two places are marked with their height numbers. The height indicated by the numbers on the map is relative to sea level,
It is usually called altitude.8844 means that Zhumulang is 8844m higher than sea level, and - 155 meansTurpan Basin155 meters below sea level.
Reference answer: Everest is 8844 meters above sea level;The height of Turpan Basin is - 155 meters above sea level.
Note: This example also shows that we introduce negative numbers in order to distinguish between heights above sea level and those below sea level. They also represent quantities with opposite meanings.
application
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Negative numbers can be widely used in temperature, floor, altitude, water level, profitability, production increase/decrease, expenditure/income, score/deduction, etc.It is now in Grade 6 of primary school.(I also learned it on the first day of junior high.).
Calculation rules
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+
Negative number 1+negative number 2=- (negative number 1+negative number 2)=negative
Negative number+positive number=sign taking the sign of the addend with the larger absolute value, and the value taking the value obtained by "subtracting the smaller absolute value from the larger absolute value"
－
Negative number 1 - negative number 2=negative number 1+(positive number 2)=negative number 1 plus the opposite number of negative number 2, and then calculate by adding negative number plus positive number
Negative number - positive number=- (positive number+negative number)=negativeSubtracting two numbers with different signs is equal to adding their absolute values
×
Negative number 1 × negative number 2=(negative number 1 × negative number 2)=Positive number
Negative number × positive number=- (positive number × negative number)=negative
÷
Negative 1 ÷ Negative 2=(Negative 1 ÷ Negative 2)=Positive number
Negative number ÷ positive number=- (negative number ÷ positive number)=negative
To sum up, the division of the same sign is equal to a positive number,Different signDividing is equal to negative numbers.
matters needing attention
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For the concepts of positive numbers and negative numbers, it cannot be simply understood that the number with "+" is positive, and the number with "−" is negative. For example, for − a, when a is positive, − a must be negative;When a is 0, − a is 0;When a is negative, − a is a positive number.[5]
