Introduction to negative numbers
Less than zero (
Such as -2, -5.33, -45/77,-π.
See: non-negative, positive, zero, negative/negative sign.
For example 1, we have learned the natural number 1, 2, 3, ... There is no object, so it is represented by 0. Sometimes, measurement and calculation cannot get an integer.
Therefore, it should be expressed in fractions and decimals. Have the students seen other kinds of numbers?
Now there are two thermometers, the thermometer liquid level refers to the sixth scale above 0, and the temperature it indicates is 6℃, so the thermometer liquid level refers to the sixth scale below 0.
Scale, how to express the temperature at this time?
Tip:
If it is expressed by 6℃, it is impossible to tell whether it is 6℃ above zero or 6℃ below zero, so we introduce a new number-negative number.
Reference answer:
Write it down as -6℃.
Description:
In order to distinguish between 6℃ above zero and 6℃ below zero, we introduced the concept of negative number.
Example 2. Let's look at another example. As can be seen from the topographic map of China, there is the highest mountain in the world-Mount Everest, marked 8844.
There is also a Turpan basin marked as-155. Can you tell me their height?
Tip:
On the topographic map of China, we can see that the above two places are marked with their height figures, and the height indicated by the marked figures on the map is relative to the sea level.
It is usually called height. 8844 means the elevation of Mount Everest is 8844 meters, and-155 means the elevation of Turpan Basin 155 meters.
Reference answer:
The height of Mount Everest is 8844 meters above sea level;
The altitude of Turpan basin is-155 meters.
Description:
This example also shows that we introduce negative numbers for practical needs, in order to distinguish the heights above and below the altitude. They also pointed out that
A quantity with opposite meaning.
The altitude of A is 35 meters, that of B is 15 meters, and that of C is -20 meters. Where is the highest? Where is the highest?
Lowest? How much higher is the highest place than the lowest place?
Tip:
What do 35 meters, 15 meters and -20 meters mean respectively?
Reference answer:
A is the highest and C is the lowest. The highest place is 55 meters higher than the lowest place.
Description:
35 meters stands for 35 meters above sea level, 15 meters stands for 15 meters above sea level, and -20 meters stands for 20 meters above sea level, so a is the highest.
C is the lowest, and A is 55 meters higher than C. ..
We already know that quantities with opposite meanings can be represented by positive numbers and negative numbers. For example, 5℃ above zero and 6℃ below zero can be recorded as +5℃ and.
-6℃; Altitude10m and below 8m can be recorded as+10/0m and-8m; Income in 200 yuan and expenditure in 300 yuan can be recorded as
+200 yuan and -300 yuan; 30 meters forward and 40 meters backward can be recorded as +30 meters and -40 meters, and 7 meters upward and 9 meters east can be recorded as
+7 meters and -9 meters?
Tip:
Do the rising amount and the eastward moving amount have opposite meanings?
Reference answer:
Cannot be recorded as+7m and-9m.
Description:
Quantities with opposite meanings must meet two conditions: (1) must be quantities with the same attributes; (2) They have opposite meanings. rise
And decline; Moving eastward and westward are quantities with opposite meanings, because rising and moving eastward are not quantities with opposite meanings, so they cannot be.
Think of it as +7 meters and -9 meters.
-π is a transcendental number, not a rational number
The origin of complex numbers
People often encounter various quantities with opposite meanings in their lives. For example, there are surpluses and deficits in accounting; When calculating the rice stored in the granary, sometimes you should remember the grain and sometimes you should remember the valley. For convenience, people think that numbers have opposite meanings. So people introduced the concepts of positive number and negative number, and recorded the excess money as positive number of grain and the loss of money and grain as negative number. It can be seen that both positive and negative numbers are produced in production practice.
According to historical records, as early as 2000 years ago, China had the concept of positive and negative numbers and mastered the arithmetic of positive and negative numbers. When people calculate, they use some small bamboo sticks to put out various figures to calculate. For example, 356 is placed in |||, 3056 is placed in, and so on. These small bamboo sticks are called "computing chips" and can also be made of bones and ivory.
Liu Hui, a scholar in China during the Three Kingdoms period, made great contributions to the establishment of the concept of negative numbers. Liu Hui first gave the definitions of positive numbers and negative numbers. He said: "Today's gains and losses are the opposite, and positive and negative numbers should be named." In other words, in the process of calculation, positive numbers and negative numbers should be used to distinguish.
Liu Hui gave the method of distinguishing positive and negative numbers for the first time. He said: "The front is red and the negative is black; Otherwise, the number of the red pendulum represents a positive number, and the number of the Hei Bang pendulum represents a negative number; You can also use a stick with an oblique pendulum to represent negative numbers, and a stick with a positive pendulum to represent positive numbers.
In China's famous ancient mathematical monograph "Nine Chapters of Arithmetic" (written in the first century AD), the law of addition and subtraction of positive and negative numbers was put forward for the first time: "Positive and negative numbers say: the same name is divided, different names are beneficial, positive and negative; Its synonyms are divided, the same name is beneficial, and there is no positive or negative. " Here, the name is a number, except subtraction, mutual benefit and division are the absolute values of two numbers, and nothing is zero.
In the present words: "the addition and subtraction of positive and negative numbers is: the subtraction of two numbers with the same sign equals the subtraction of their absolute values, and the subtraction of two numbers with different signs equals the addition of their absolute values." Zero minus a positive number is a negative number, and zero minus a positive number. The addition of two numbers with different signs equals the subtraction of their absolute values, and the addition of two numbers with the same sign equals the addition of their absolute values. Zero plus positive number equals positive number, and zero plus negative number equals negative number. "
This statement about the arithmetic of positive and negative numbers is completely correct and completely in line with the current law! The introduction of negative numbers is one of the outstanding contributions of mathematicians in China.
The habit of using numbers of different colors to represent positive and negative numbers has been preserved until now. At present, red is generally used to represent negative numbers. The newspaper reports that a country's economy is in deficit, which shows that its expenditure is greater than its income and it has incurred financial losses.
Negative numbers are antonyms of positive numbers. In real life, we often use positive numbers and negative numbers to represent two quantities with opposite meanings. In summer, the temperature in Wuhan is as high as 42℃, and you will feel that Wuhan is really like a stove. The minus sign of the temperature in Harbin in winter is -32℃, which makes you feel the cold in winter in the north.
In the current textbooks for primary and secondary schools, the introduction of negative numbers is through arithmetic operation: a negative number can be obtained by subtracting a larger number from a smaller number. This introduction method can have an intuitive understanding of negative numbers in special problem scenarios. In ancient mathematics, in the process of solving algebraic equations, negative numbers are often produced. The algebraic study of ancient Babylon found that the Babylonians did not put forward the concept of negative root when solving equations, that is, they did not use or find the concept of negative root. In the works of Diophantine, a Greek scholar in the 3rd century, only the positive root of the equation was given. However, in China's traditional mathematics, negative numbers and related arithmetic were formed earlier.
In addition to the positive and negative operation methods defined in Nine Chapters Arithmetic, Liu Hong (AD 206) at the end of the Eastern Han Dynasty and Yang Hui (126 1) in the Song Dynasty also discussed the addition and subtraction principles of positive and negative numbers, all of which were completely consistent with those mentioned in Nine Chapters Arithmetic. In particular, in Yuan Dynasty, Zhu Shijie gave not only the rules of addition and subtraction of positive and negative numbers with the same sign but different signs, but also the rules of multiplication and division of positive and negative numbers. In his algorithm enlightenment, negative numbers were recognized and recognized abroad, much later than in China. In India, it was not until AD 628 that the mathematician Yarlung Zangbo realized that negative numbers can be the root of quadratic equations. In Europe, Qiu Kai, the most successful French mathematician in the14th century, described negative numbers as absurd numbers. It was not until the17th century that the Dutchman Jirar (1629) first realized and used negative numbers to solve geometric problems.
Unlike China's ancient mathematicians, western mathematicians are more concerned about the rationality of the existence of negative numbers. In the 16 and 17 centuries, most mathematicians in Europe did not admit that negative numbers were numbers. Pascal thinks that subtracting 4 from 0 is sheer nonsense. Pascal's friend Ahrend 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 17 12, even Leibniz admitted that this statement was reasonable. Wally, a British mathematician, acknowledged negative numbers and thought that negative numbers were less than zero and greater than infinity (1655). He explained that when a > 0, Augustus de Morgan, a famous British mathematician, still thought that negative numbers were fictitious in 183 1. He used the following example to illustrate this point: "My father is 56 years old and my son is 29 years old. When will the father be twice as big as his son? " Simultaneous equation 56+x=2(29+x) is solved, and x=-2 is obtained. He called the solution absurd. Of course, in Europe in the18th century, not many people refused negative numbers. With the establishment of integer theory in19th century, the logical rationality of negative numbers was really established.
Application of negative numbers
Temperature: MINUS 3 degrees Celsius -3 degrees Celsius
Floor: underground 1- 1.
Altitude: The lowest point in Turpan Basin is below sea level.
155m-the altitude is-155m.
negative number
China introduced the concept of negative numbers and the addition and subtraction of positive and negative numbers in the chapters of Nine Arithmetic and Equation. In some problems, the number sold is positive (because of income) and the number bought is negative (because of payment); The surplus money is positive, and the lack of money is negative. In the calculation of grain, the addition is positive and the subtraction is negative. The terms "positive" and "negative" have been used to this day.
In the chapter "Equation", the law of adding positive and negative numbers introduced is called "addition and subtraction operation". The law of multiplication and division of positive and negative numbers appeared relatively late. In the Enlightenment of Arithmetic written by Zhu Shijie 1299, the laws of addition and subtraction of positive and negative numbers are described in Ming Qian Zheng Shu, which consists of eight articles, which is more clear than Nine Chapters of Arithmetic. There is a saying in Ming Dynasty's Multiplication and Division that "the multiplication of the same name is positive and the multiplication of different names is negative", that is, (A) × (B) =+AB, (A )× (B) =-AB, which is the earliest record in China. At the end of Song Dynasty, Ye Li also used oblique strokes to represent negative numbers. The introduction of the concept of negative number is one of the most outstanding creations of China's ancient mathematics.
The first Indian to put forward negative numbers was Brahmagupta, about 628 (about 598-665). He put forward the arithmetic of negative numbers and marked them with dots or circles to represent negative numbers. The Italian mathematician Fibonacci (1 170- 1250) first put forward the concept of negative numbers in Europe. When solving a profit problem, he said: I will prove that this problem cannot be solved unless I admit that this person can be in debt. /kloc-Shukai in the 0/5th century (1445? - 15 10? ) and Steve in the 6th century (1553) both found negative numbers, but both described them as absurd numbers. Cardan (1545) gave the negative root of the equation, but he described it as a "pseudo number". Vedas knew that negative numbers existed, but he didn't want them at all. Descartes partially accepted negative numbers. He called the negative root of the equation a false root because it was smaller than "nothing".
Harley Otto (1560- 162 1) accidentally wrote a negative number on one side of the equation and indicated it with "-",but he didn't accept negative numbers. Bonberi (1526-1572) gives a clear definition of negative numbers. Steven uses positive and negative coefficients in the equation and accepts negative roots.