Prove:

We know that 9=3*3=3+3+3.

Then in the same way, 9=(-3) * (-3) = -(-3)+-(-3) (converted to -3 -3 addition here).

We assume that+is the positive direction and-is the negative direction; Then the opposite direction is the positive direction.

So 9=(-3) * (-3) = 3+3+3.

Such as why (-3) * (3) = -9?

Prove:

We know that 9=3*3=3+3+3.

So it's the same -9=(-3) * (3) = (-3)+(-3)+(-3) (converted to three-three addition here).

We assume that+is the positive direction and-is the negative direction; Because they are in the same direction, they are all in the opposite direction.

So -9=(-3) * (3) = -(3+3+3)

Online example 1:

? Positive and negative numbers and 0*** together form a real number, which is used to distinguish the quantitative relationship between things in the same category and in the opposite direction. Set the amount of similar income as a positive number, zero if there is no money, and the amount of expenditure as a negative number. This income and expenditure are opposite things in the same category. In order to fully understand their income and expenditure, people have the operational relationship of positive number, negative number and 0. When the income and expenditure are equal, the positive and negative numbers are offset to zero, and when the income exceeds the expenditure, the offset is positive, and vice versa. The relationship and result of this addition and subtraction operation are abstracted from the actual cases in life and production, and become the addition and subtraction algorithm of real numbers.

? For multiplication and division, it is just a higher-level movement form of addition and subtraction. For the same positive number, if it is income every time, a * * * earns five times, and this sum is also the same sum of five positive numbers, and the result is naturally positive. This multiplication is a simple operation of addition, and a positive number multiplied by a positive number is also a positive number. If every expenditure is negative, five expenditures are the same, which add up to negative, and the result of multiplication is also negative. Multiplication is also a simple operation of addition and the result is the same. If every expenditure is negative, for example, ten yuan, it will be recorded as negative ten. After spending five times, it is negative 50 yuan. Now suppose this person spends ten dollars every time, and he spends a negative time. How much did he spend? Obviously, there is a difference between spending negative time and spending positive time. As a result, you can only lose ten dollars if you spend ten dollars. This expenditure is negative once, that is, spending once in the opposite direction, that is, earning once and earning ten yuan once, and the result is positive ten yuan. So it can also be said that spending once is negative, and as a result, I earned ten dollars. Spending twice is minus two times minus ten, which means I earned ten dollars twice. This is a true example and truth that negation is affirmation. Summarizing similar mathematical movements into laws means that negative is positive in multiplication.

Online example 2:

?

? Why "negative is positive"? Maybe you haven't considered this question at all, or maybe your explanation is that "the textbook stipulates this." This answer can't satisfy everyone's curiosity and thirst for knowledge. Please understand the development history of "negative gains are positive".

? As we all know, the concept of negative number first appeared in China. In "Nine Chapters of Arithmetic", the addition and subtraction algorithm of positive and negative numbers is given in the equation chapter, and the plus and minus numbers were not given by mathematician Zhu Shijie until the end of 13. Zhu Shijie put forward in "Enlightenment of Arithmetic" (1299): "Ming multiplication and division method, the same name multiplied by positive, different names multiplied by negative." .

For multiplication and division, it is just a higher-level movement form of addition and subtraction. For the same positive number, if it is income every time, a * * * earns five times, and this sum is also the same sum of five positive numbers, and the result is naturally positive. This multiplication is a simple operation of addition, and a positive number multiplied by a positive number is also a positive number. If every expenditure is negative, five expenditures are the same, which add up to negative, and the result of multiplication is also negative. Multiplication is also a simple operation of addition and the result is the same. If every expenditure is negative, for example, ten yuan, it will be recorded as negative ten. After spending five times, it is negative 50 yuan. Now suppose this person spends ten dollars every time, and he spends a negative time. How much did he spend? Obviously, there is a difference between spending negative time and spending positive time. As a result, you can only lose ten dollars if you spend ten dollars. This expenditure is negative once, that is, spending once in the opposite direction, that is, earning once and earning ten yuan once, and the result is positive ten yuan. So it can also be said that spending once is negative, and as a result, I earned ten dollars. Spending twice is minus two times minus ten, which means I earned ten dollars twice. This is a true example and truth that negation is affirmation. Summarizing similar mathematical movements into laws means that negative is positive in multiplication.

? In the 7th century AD, Indian mathematician brahmayup-ta had a clear concept of positive and negative numbers and four arithmetic rules: "Positive and negative numbers are multiplied to get negative numbers, two negative numbers are multiplied to get positive numbers, and two positive numbers get positive numbers."

? Until18th century, some western mathematicians thought that the arithmetic of "negative is positive" was a fallacy. Even in the19th century, some British mathematicians did not accept negative numbers. For example, the British mathematician Fred (1757- 184 1) criticized those who talked about "negative is positive".

? In fact, until the middle of19th century, the negative operation was not correctly explained in algebra textbooks. The French writer Stendhal (1783- 1843) was troubled by this rule for a long time when he was a student. His two math teachers, Mr. Dupuy and Chabel, failed to give him a convincing explanation, so Stendhal gave him a convincing explanation from math and math teachers. Obviously, in order to reduce the difficulty of students' understanding of negative multiplication, it is not advisable to directly introduce the law of negative to positive by blunt "regulation" method. The following are the introduction methods to help students understand.

? Every child grows up listening to stories. Therefore, they should have more interest and enthusiasm for stories. For students, they will be deeply impressed by relatively strong concepts, such as good and bad, good and evil, etc. The following model should give students a more intuitive feeling.

Story model

Good people (positive numbers) or bad people (negative numbers) are all good (positive numbers) when they go into the city (positive numbers) or go out of the city (negative numbers). ) and bad (negative). If good people (+) enter the city (+), it is also good for the town (+). So (+) × () =+:If a good person (+) goes out of town (-).

"Liabilities" model

M (short for meter) Klein thinks that "if you remember the physical meaning, the mixed operation of negative operation and positive and negative numbers is easy to understand". He solved the problem that "the product of two debts is magic income" that has puzzled people for many years.

A person owes $5 a day and $65,438+05 ($0) after a given date. If the debt of $5 is recorded as -5, then the debt of $5 per day for three days can be expressed mathematically: 3× (-5) =-65,438+05. The same person owes $5 a day, so three days before the given date ($0), his

Motion model

When a person walks on the expressway, the rule is this: if the right direction is chosen as the positive direction, then the left direction is the negative direction. That is, the right direction is positive and the left direction is negative. In chronological order, the future time is positive, the past time is negative, and the initial position of people is zero.

+4 × -3 = - 12

Measurement model

A weather station measured that the temperature dropped by 0.6 degrees every time the altitude increased by 1 km, and the temperature of the observation site was zero. What is the temperature 3 kilometers below the observation point? We stipulate that the temperature rise is positive and the temperature drop is negative. It is negative below the observation point and positive above the observation point. It is easy to get the formula of the above problem as (-0.6) ×(-3)= 1.8.

Hands-on model

In this mode, we need a camera as a prop, and we also hope that students can understand the truth of "practicing knowledge" from the process of doing it themselves. Suppose a clean plastic water tank has a transparent drainage pipe, and the drainage speed of the drainage pipe is 3 gallons per minute. Take a picture of the drainage process of the drain pipe for a few minutes with a camera (the "drainage" here is regarded as a negative number. If we play it for 2 minutes, we can see that the water in the water tank is reduced by 6 gallons, but after 3 minutes, the water is reduced.

How to explain "negative is positive"

The realistic model is not enough to convince a clever boy like Stendhal. At this time, we can also explain why "negative is positive" in the following ways.

The first method is to use the algorithm directly:

(- 1)×(- 1)=(- 1)×(- 1)+0×(- 1)

=(- 1)×(- 1)+[(- 1)+ 1] × 1

=(- 1)×(- 1)+(- 1) × 1+ 1× 1

=(- 1) ×(- 1+ 1)+ 1

= 1

The second is the reduction to absurdity: assuming that the negation is affirmative, it is assumed that:

(- 1)×(- 1)=[2+(- 1)]

=(- 1) ×2+(- 1) ( 1)

On the other hand:?

(- 1)×(+ 1)=[ 1+(-2)] ×(+ 1)= 1+(-2) × 1 (2)

If both positive and negative numbers are negative, it is impossible to get-1 =-3 from (1); If both positive and negative are positive, it is impossible to get 1=3 from (2). That is to say, no matter whether the product of a positive number and a negative number is positive or negative, the above conclusion is not valid. This 6544.

Strictly speaking, the above "proof" is only based on two explanations, because our basis is that the arithmetic of positive number and zero satisfaction includes: 0+a=a, 0×a = 0；; a+b = b+ a； a×b = b×a； /kloc-hankel, a German mathematician in the 20th century, told us that "negation is affirmation" can't be proved in formal arithmetic. Klein, a great mathematician, also gave advice: Don't try to prove the logical inevitability of symbolic laws, "Don't make the impossible proof look true". In fact, the above "proof" shows that when we apply the algorithm satisfied by non-negative integers to negative numbers, the result of multiplication of two negative numbers can only be a positive number. One of the principles followed by the expansion of number sets is the non-contradiction of operation rules. It is true that you can stipulate that "negative is positive", but in doing so, you must at least give up an algorithm satisfied by the set of positive integers. This may be the last card we can show Tom Da. However, the research results of mathematics education show that children's knowledge is not constructed by deductive reasoning. But through the means of experience collection, result comparison, induction and summary. Just telling students the operation rate can't get the desired effect, because students are unwilling to use these operation rates. This is consistent with the enlightenment of history. There is no doubt that realistic model is an indispensable teaching method for us.