Why introduce negative numbers? Please find some quantities in the opposite sense

The origin of negative numbers
People often encounter various quantities with opposite meanings in their lives. For example, when they keep accounts, they have surplus and deficit; when they calculate the amount of rice stored in the granary, sometimes they need to record grain, sometimes they need to record grain. For convenience, people consider numbers with opposite meanings. So people introduce the concept of positive and negative numbers, and record the surplus money into grain as positive, The loss of money, food recorded as negative. It can be seen that positive and negative numbers are produced in production practice
According to historical records, as early as two thousand years ago, China had the concept of positive and negative numbers, and mastered the algorithm of positive and negative numbers. When people calculate, they use some small bamboo sticks to put out various numbers for calculation. For example, 356 sticks are placed in | |, 3056 sticks are placed in |, and so on. These small bamboo sticks are called "calculation chips". Calculation chips can also be made from bones and ivory
Liu Hui, a scholar in the period of the Three Kingdoms in China, made a great contribution 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 that the positive and negative numbers should be named." that is to say, in the process of calculation, we should use positive numbers and negative numbers to distinguish them
For the first time, Liu Hui gave a method to distinguish positive and negative numbers. He said: "positive numbers are red, negative numbers are black; otherwise, evil and positive numbers are different." this means that the numbers put out by the red stick represent positive numbers, and the numbers put out by the black stick represent negative numbers; the numbers put out by the slanting stick represent negative numbers, and the numbers put out by the positive stick represent positive numbers
In the 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 first put forward: "positive and negative numbers say: Division of the same name, mutual benefit of different names, positive without negative, negative without positive; division of different names, mutual benefit of the same name, positive without positive, negative without negative.", "Mutual benefit" and "division" are the absolute values of two numbers
In current words: "the law of addition and subtraction of positive and negative numbers is: subtracting two numbers with the same sign equals to subtracting their absolute values, subtracting two numbers with different signs equals to adding their absolute values. Subtracting positive numbers by zero equals to negative numbers, subtracting negative numbers by zero equals to positive numbers. Adding two numbers with different signs equals to subtracting their absolute values, adding two numbers with the same sign equals to adding their absolute values. Adding positive numbers by zero equals to positive numbers, and adding negative numbers by zero equals to negative numbers."
The introduction of negative numbers is one of the outstanding contributions of Chinese mathematicians
The habit of using numbers of different colors to express positive and negative numbers has been retained until now. Now, red is generally used to express negative numbers. The newspaper reports that a country's economy is in deficit, indicating that its expenditure is greater than its income and that it has lost money financially
Negative numbers are the opposite of positive numbers. In real life, we often use positive numbers and negative numbers to express two quantities with opposite meanings. In summer, the temperature in Wuhan is as high as 42 ° C. you will think that Wuhan is really like a stove. In winter, the temperature in Harbin is - 32 ° c. a negative sign makes you feel cold in winter in the north
In today's primary and secondary school textbooks, the introduction of negative numbers is introduced through the method of arithmetic operation: only a small number minus a large number can get a negative number. This introduction method can give a direct understanding of negative numbers in a special problem situation, Negative numbers are often generated in the process of solving algebraic equations. The study of ancient Babylonian algebra found that the Babylonians did not put forward the concept of negative roots in solving equations, that is, they did not or could not find the concept of negative roots. In the works of Diophantine, a Greek scholar in the third century, only the positive roots of equations were given. However, in traditional Chinese mathematics, the negative roots of equations were not found, Negative numbers and related algorithms have been formed earlier
In addition to the definition of positive and negative operation method in Jiuzhang arithmetic, Liu Qian (206 A.D.) in the late Eastern Han Dynasty and Yang Hui (1261 A.D.) in the Song Dynasty also discussed the addition and subtraction rules of positive and negative numbers, which are completely consistent with Jiuzhang arithmetic, In his algorithmic enlightenment, negative numbers were recognized abroad much later than in China, It was not until 628 that the mathematician brahmogupto realized that negative numbers could be the roots of quadratic equations. However, in Europe, the most successful French mathematician in the 14th century, chukai, regarded negative numbers as absurd numbers. It was not until the 17th century that Gerard (1629), the Dutch, first recognized and used negative numbers to solve geometric problems
Different from ancient Chinese mathematicians, Western mathematicians mostly study the rationality of the existence of negative numbers. In the 16th and 17th centuries, most European mathematicians did not recognize negative numbers as numbers. Pascal thought that subtracting 4 from 0 was pure nonsense. Pascal's friend arunde put forward an interesting argument against negative numbers, saying (- 1): 1 = 1: (- 1), How can the ratio of a smaller number to a larger number be equal to the ratio of a larger number to a smaller number? Until 1712, even Leibniz admitted that it was reasonable. Wally, an English mathematician, admitted that negative numbers were less than zero and greater than infinity (1655). He explained that: because when a > 0, the ratio of a to a larger number is greater than infinity, In 1831, De Morgan, a famous British algebra mathematician, still thought that negative numbers were fictitious. 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 be twice as old as his son?" he formulated the equation 56 + x = 2 (29 + x) and solved x = - 2. He called the solution absurd, There are not many people in Europe who reject 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 is really established
We learned the natural numbers 1,2,3,... In primary school. If there is no object, we use 0 to express it. Sometimes we can't get the integer value by measurement and calculation
As a result, it has to be expressed in fractions and decimals. Have you seen any other kinds of numbers?
Now there are two thermometers. The liquid level of the thermometer refers to the 6th scale above 0, which indicates that the temperature is 6 ℃, so the liquid level of the thermometer refers to the 6th scale below 0
Scale, how to express the temperature at this time?
Tips:
If we use 6 ℃ to express it, we can't distinguish 6 ℃ above zero or 6 ℃ below zero, so we introduce a new number negative number
Reference answer:
It is recorded as - 6 ℃
Note:
We introduce the concept of negative number in order to distinguish the group of quantities with opposite meaning
Now let's take another example. From the topographic map of China, we can see that there is the world's highest peak, Mount Everest, marked 8844;
There is also a Turpan Basin, marked - 155 on the map. Can you tell the height of each of them?
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 the sea level,
8844 is 8844 meters higher than sea level, and - 155 is 155 meters lower than sea level in Turpan Basin
Reference answer:
The height of Mount 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 for practical needs to distinguish the height above sea level from that below sea level. They also represent the height above sea level
A quantity of opposite significance
The altitude of land a is 35 meters, that of land B is 15 meters, and that of land C is - 20 meters. Which place is the highest
The lowest? How much higher is the highest place than the lowest?
Tips:
What are the meanings of 35 meters, 15 meters and - 20 meters?
Reference answer:
Land a is the highest, and land C is the lowest. The highest place is 55 meters higher than the lowest place
Note:
35 meters means 35 meters above sea level, 15 meters means 15 meters above sea level, and - 20 meters means 20 meters below sea level,
C is the lowest, and a is 55 meters higher than C
We have known that quantities with opposite meanings can be expressed by positive and negative numbers. For example, 5 ℃ above zero and 6 ℃ below zero can be recorded as + 5 ℃ and + 5 ℃
- 6 ℃; 10 meters above sea level and 8 meters below sea level can be recorded as + 10 meters and - 8 meters; income of 200 yuan and expenditure of 300 yuan can be recorded as
For example, + 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 eastward can be recorded as + 30 meters and - 40 meters respectively
+ 7m and - 9m?
Tips:
Is rising and eastward movement opposite?
Reference answer:
It can not be recorded as + 7m and - 9m
Note:
Quantities with opposite meanings must satisfy two conditions: (1) they must be quantities with the same attribute; (2) they have opposite meanings
The eastward movement and the westward movement are the opposite quantities, because the upward and eastward movements are not the opposite quantities
Take the mark as + 7m and - 9m
- π is transcendental, not rational

The numbers other than 0 are divided into positive numbers and negative numbers, which represent two opposite quantities: (1) if a represents a positive number, what number does - a represent
(2) If a is negative, what does - a mean?
(3) Is - a negative?

·1. A is a positive number and - A is a negative number
2. A is a negative number, and - A is a positive number because it is negative
3, - A does not necessarily mean a negative number

What is a general term formula of sequence 0,3,8,15,24?

n^2-1

In triangle ABC, C is a right angle. Given AC = 2, CD = 2, CB = 3, am = BM, what is the area of triangle amn (shadow part)?

We can see from the meaning that BD = bc-cd = 3-2 = 1, because am = MB, & nbsp; so gmbd = 12, GM = 12, so gmcd = 0.52 = 14, because △ NGM ∽ ndcmncn = gmcd = 14, s △ ABC = 2 × 3 △ 2 = 3, so s △ ACM = 12S △ ABC = 32

Solution equation: (x-500) × 80% + 500 × 90% = 494

(x-500)*80%+500*90%=494
(x-500)*80%﹢450=494
(x-500)*80%=494‐450
x-500=44÷80%
x=55+500
x=555

The bottom of a triangle glass is 45 decimeters and the height is 36 decimeters. If the price of glass per square decimeter is 65 yuan, how much does it cost to buy this glass?

Area = 45 × 36 × 1 / 2 = 810 square decimeters
Price = 810 × 65 = 52650 yuan

1:1 in 2002 plus 2:2 in 2002 has been added to 2001:2 in 2002

1 + 1 / 2002 + 2 + 2 / 2002 +. + 2001 + 2001 / 2002 = (1 + 2 +... + 2001) + (1 + 2 +... + 2001) / 20021 + 2 +

If the circumference of an isosceles triangle is 24, then the functional relationship between the waist length y and the bottom x is, where the value range of the dependent variable y is

2y+x=24
So y = - X / 2 + 12
Because the sum of the two sides is greater than the third side
So y + Y > x
x=-2y+24
So 2Y > - 2Y + 24
y>6
x=-2y+24>0
y

It is known that a, B, C belong to R +, a + B + C = 1, the proof is: 1 / A + 1 / B + 1 / C > = 9

If we know Cauchy inequality, direct 1 / A + 1 / B + 1 / C = (a + B + C) (1 / A + 1 / B + 1 / C) ≥ (1 + 1 + 1) &# = 9
If we only know the mean inequality, we expand 1 / A + 1 / B + 1 / C = (a + B + C) (1 / A + 1 / B + 1 / C)
= 3+(a/b+b/a)+(b/c+c/b)+(c/a+a/c) ≥ 3+2+2+2 = 9.

Why introduce negative numbers? Please find some quantities in the opposite sense