People often encounter various quantities with opposite meanings in their lives. For example, there are surpluses and deficits in bookkeeping; When calculating the rice stored in the granary, sometimes you have to remember the grain, and sometimes you have to remember the grain. For convenience, people consider using numbers with opposite meanings. So people introduced the concepts of positive number and negative number, and recorded the excess money in grain as positive number 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 rules of positive and negative numbers. When people calculate, they use some small bamboo sticks to calculate various numbers. For example, 356 is put in |||, 3056 is put 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, the gains and losses of the two calculations are opposite, so the positive and negative numbers should be named." This means that positive numbers and negative numbers should be used to distinguish between quantities with opposite meanings in the calculation process.

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 oblique sticks to represent negative numbers and positive sticks 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 mutually divided, homophones complement each other, and negation means negation. Here, the name is a number, except subtraction, mutual benefit and division are the addition and subtraction of the absolute values of two numbers, and nothing is zero.

In the present words: "The addition and subtraction of positive and negative numbers means that two numbers with the same sign are subtracted, which means that their absolute values are subtracted, and two numbers with different signs are subtracted, which means that their absolute values are added. Zero minus positive number means negative number, and zero minus number means positive number. Adding two numbers with different signs means that their absolute values are subtracted, and adding two numbers with the same sign means that their absolute values are added. Zero plus positive number means positive number, and zero plus negative number means 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. Now, red is generally used to represent negative numbers. Newspapers publish that a country has an economic deficit, which means that expenditure exceeds income and financial losses occur.

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 temperature in Harbin in winter is -32℃, and a minus sign makes you feel the cold in winter in the north.

In today's textbooks for primary and secondary schools, negative numbers are introduced through arithmetic operations: a smaller number can be subtracted from a larger number to get a negative number. This introduction method can have an intuitive understanding of negative numbers in special problem situations. In ancient mathematics, in the process of solving algebraic equations, negative numbers are often produced. The study of ancient Babylonian algebra found that Babylonians did not put forward the concept of negative root when solving equations. In other words, the concept of negative roots has not been used or discovered. 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 algorithms 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 talked about the addition and subtraction rules of positive and negative numbers, which are completely consistent with those mentioned in Nine Chapters Arithmetic. It is particularly worth mentioning that Zhu Shijie in Yuan Dynasty gave the addition and subtraction rules of positive and negative numbers with the same sign but different signs. Negative numbers are recognized and recognized abroad, much later than at home. In India, the mathematician Brahmaputra didn't realize that negative numbers can be the root of quadratic equations until AD 628. However, the most accomplished French mathematician in Europe in the14th century described negative numbers as absurd numbers. It was not until17th century that Jural (1629), a Dutchman, 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, saying (-1 = 65438. Until 17 12, even Leibniz admitted that this statement was reasonable. British mathematician Wally admitted that negative numbers are less than zero and greater than infinity (1655). He explained that because a > 0, Augustus de Morgan, a famous British mathematician, still thought negative numbers were fictitious in 183 1. He listed the equation 56+x=2(29+x) and got x=-2. 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.

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 the sum of +5℃.

-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) Their meanings are opposite. Stand up.

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