Open lesson in mathematics “Multiplying the number zero and zero. Zero division. The rule for multiplying any number by zero Multiply by 0, we get the rule
Consider an example of multiplying an integer by zero. How much will it be if 2 (two) times 0 (zero)? Any number multiplied by zero equals zero. It doesn't matter if we know this number or not.
According to the generally accepted definition, zero is the number that separates positive numbers from negative ones on the number line. Zero is the most problematic place in mathematics that does not obey logic, and all mathematical operations with zero are based not on logic, but on generally accepted definitions.
Zero is the first digit in all standard number systems. Every month began from day zero in the Mayan calendar. It is interesting that the Maya mathematics used the same zero sign to denote infinity, the second problem of modern mathematics. Zero without a wand. absolute zero. Zero point five. Five times zero equals zero 5 x 0 = 0 See the rule for multiplying by zero above in the text. Chatyri multiply by zero for free - I answer for free that it will be zero. Free help is included - the word "four" is spelled a little differently than you write in your search query.
https://youtu.be/EGpr23Tc8iY
Where zero occurs in mathematics, logic is powerless
If you liked this post and want to know more, please help me with more content. It appeared in the comments and something hooked me. Student's question: And now, dear author, please multiply zero by zero and tell me how much you get as a result?
I have already explained in my article “What is zero” where it can be applied. You just need to take those answers that are written in textbooks: zero multiplied by zero equals zero; Dividing by zero is not allowed. Of all the foreseeable options for multiplication and division by zero, ignorant scientists have chosen the most acceptable and digestible option.
I have no problem with division by zero. I hear about the connection between Heron's formula and 0/0=1 for the first time. However, there is something impure in mathematics. Problems with raising zero to zero and negative powers. But you might as well say that 0^2 doesn't make sense either, because 0^2=0^5/0^3=0/0, and you can't divide by zero.
Zero to the zeroth power is an expression that has no meaning. Zero to the zero power is equal to one - this is how the formulas show. This amount of anything, some real, material things, can be multiplied by a number. In this case, the amount of something is expressed only by zero or a positive number.
Everything in units and in mathematics at this level is in order. This is a convention, degrees cannot be expressed in quantity, so you cannot multiply them by a number. Somewhere on this site there is Durnev with his questions on school curriculum, including mathematics. Maybe it was invented in the same way as zero? To impose certain rules and subordinate them to all other people. What only a person will not do for himself, his beloved.
It is enough that textbooks often write "belongs to the set of natural numbers" even when this is true for all numbers except complex ones. An infinite number of zeros in zero are inventions of shamans for cavemen:) If we close our eyes, then everything we look at will look the same black. Multiplication by zero must be considered from a completely different end. What is multiplication?
It is enough to understand what multiplication is, then the issue with the result of multiplication by zero will be solved by itself. 2 apples, and trying to multiply them by 0 apples, as a result, we lose our 2 apples. Apparently, those who ask this have lost at least one digit at the beginning of each number. 10 and 11 - it is appropriate to talk about percentages here.
And it’s interesting how, when dividing 0 by any number, you can subtract this number at all (even if it’s zero times) ..
It can’t be so easy to become zero from multiplication! So math is not an exact science? Someone once came up with this "rule" is not known for what. Your math is wrong. In practice, this whole mathematical topic with multiplication by 0 cannot be!!! How do you want to multiply something by 10, even by 0 - it will turn out 0 ?? Unless, of course, 0 is a black hole, or 0 as you lose, to nowhere, zero is like emptiness, nothing, but this cannot be ....
If you can’t divide something (the same 5 apples into 0 imaginary baskets), then the result of an integer number is recorded and the remainder in this division ... 0 can be multiplied many times (like I went to the forest 15 times and did not find mushrooms ...
For example, we divide 5 apples by zero people; Calculate how many times 5 degrees Celsius is greater than zero degrees Celsius. Of this, it is most likely impossible to multiply by 0 (since, by the definition of multiplication, this CANNOT be written using the addition operation) and divide 0 itself by something ... since the answer cannot be determined ...
The substitution of concepts occurs when multiplying by zero itself ... Remember, any number or operation with numbers multiplied by zero IS ANNIGATED ... In other words, the operation itself does not occur when multiplied by zero and you can simply “ignore it” ... So, you stole my idea!))) For the first time I meet a more or less clear understanding of multiplication and division by zero. Whether we consider this to be mathematical operations or not, mathematics does not give a damn.
The first example of zero being problematic is the natural numbers. In Russian schools, zero is not natural number, in other schools zero is a natural number. For those who are interested in the question of the origin of zero, I suggest reading the article “The History of Zero” by J. J. O’Connor and E. F. Robertson, translated by I. Yu. Osmolovsky.
At what values of x is the equality true: zero multiplied by x equals zero? - this equality is true for any values of x. This equality is said to have an infinite number of solutions. The math was a little easier. In the most natural way, banal typos are superimposed on my natural illiteracy when typing.
I am opposed to those sermons that mathematicians read to us and to which we all))) refer. This equation was a completely different story. Can this be or can't it be? After a little thought, I "carried out a thought experiment"))) and imagined this situation. Somewhere in the drafts all the calculations about this are lying around. You are cunning What is not accepted in wide circles is not necessarily not true.
How to spell zero or null? The words zero and zero have the same meaning, but differ in usage. Who said zero is a number? Mathematicians? 0 + 5/0… zero and five (zero) in the remainder… and then everything converges and everyone is happy… Yes, in fact, there are not so many difficulties. The problem is how to perceive Zero (as a number or as something empty) and what is meant by multiplication ...
For the first time with such an arithmetic operation as multiplication, students get acquainted on the school bench. The math teacher among the numerous rules raises the topic of "multiplying by zero." Despite the unambiguity of the wording, students have many questions. Let's look at what happens if we multiply by 0.
The rule that you cannot multiply by zero generates a lot of disputes between teachers and their students. It is important to understand that multiplication by zero is a controversial aspect due to its ambiguity.
First of all, attention is focused on the lack of a sufficient level of knowledge among secondary school students. secondary school. Crossing the threshold educational institution, participant educational process in most cases, does not think about the main goal that needs to be pursued.
During the training, the teacher covers various issues. These include the situation, what happens if you multiply by 0. In an effort to anticipate the teacher's narration, some students enter into controversy. They prove, at least they try, that multiplication by 0 is valid. But, unfortunately, this is not the case. Multiplying any number by 0 results in nothing. In some literary sources, there is even a mention that any number multiplied by zero forms a void.
Important! Attentive audience listeners immediately grasp that if the number is multiplied by 0, then the result will be 0. A different development of events can be traced in the case of those students who systematically skip classes. Inattentive or unscrupulous students are more likely than others to think about how much it will be if they multiply by zero.
As a result of the lack of knowledge on the topic, the teacher and the negligent student find themselves opposite sides contradictory situation.
The difference in views on the topic of the dispute lies in the degree of education on the subject of whether it is possible to multiply by 0 or still not. The only acceptable way out of this situation is to try to appeal to logical thinking to find the correct answer.
It is not recommended to use the following example to explain the rule. Vanya has 2 apples in her bag for a snack. At lunch he thought about putting some more apples in his briefcase. But at that moment there was not a single fruit nearby. Vanya did not put anything. In other words, he placed 0 apples to 2 apples.
In terms of arithmetic, in this example, it turns out that if 2 is multiplied by 0, then there is no void. The answer in this case is clear. For this example, the multiplication by zero rule is not relevant. The correct solution is summation. That is why the correct answer is 2 apples.
Otherwise, the teacher has no choice but to compose a series of tasks. The last measure is to re-set the passage of the topic and poll for exceptions in the multiplication.
Essence of action
It is advisable to start studying the algorithm of actions when multiplying by zero by indicating the essence of the arithmetic operation.
The essence of the action to multiply was originally determined exclusively for a natural number. If the mechanism of action is revealed, then a certain number involved in the calculation is added to itself.
It is important to consider the number of additions. Depending on this criterion, different results are obtained. The addition of a number relative to itself determines such a property of it as naturalness.
Let's look at an example. It is necessary to multiply the number 15 by 3. When multiplied by 3, the number 15 increases three times in its value. In other words, the action looks like 15 * 3 = 15 + 15 + 15 = 45. Based on the calculation mechanism, it becomes obvious that if a number is multiplied by another natural number, there is a semblance of addition in a simplified form.
It is advisable to start the algorithm of actions when multiplying by 0 by providing a characteristic by zero.
Note! According to conventional wisdom, zero stands for the whole nothingness. For emptiness of this kind, a designation is provided in arithmetic. Despite this fact, the zero value does not carry anything.
It should be noted that such an opinion in the modern world scientific society differs from the point of view of the ancient Eastern scholars. According to the theory they held, zero was equal to infinity.
In other words, if you multiply by zero, you get a variety of options. In the zero value, scientists considered a kind of depth of the universe.
As confirmation of the possibility of multiplying by 0, mathematicians cited the following fact. If you put 0 next to any natural number, you get a value ten times greater than the original one.
The example given is one of the arguments. In addition to proofs of this kind, there are many other examples. It is they that underlie the ongoing disputes when multiplying by emptiness.
The feasibility of trying
Among students quite often at the beginning of mastering educational material there are attempts to multiply a number by 0. Such an action is a gross mistake.
In essence, nothing will happen from such attempts, but there will be no benefit either. If you multiply by a zero value, you get an unsatisfactory mark in the diary.
The only thought that should arise when multiplying by emptiness is the impossibility of action. memorization in this case plays an important role. Having learned the rule once and for all, the student prevents the appearance of controversial situations.
As an example to be used when multiplying by zero, the following situation is allowed to be used. Sasha decided to buy apples. While she was in the supermarket, she chose 5 large ripe apples. Going to the department of dairy products, she felt that this would not be enough for her. The girl put 5 more pieces in her basket.
After thinking a little more, she took 5 more. As a result, at the checkout, Sasha got: 5 * 3 = 5 + 5 + 5 = 15 apples. If she put 5 apples only 2 times, then it would be 5 * 2 = 5 + 5 = 10. In the event that Sasha did not put 5 apples in the basket, it would be 5 * 0 = 0 + 0 + 0 + 0 + 0 = 0. In other words, buying apples 0 times means not buying any.
The number 0 can be represented as a kind of border separating the world of real numbers from imaginary or negative ones. Due to the ambiguous position, many operations with this numerical value not subject to mathematical logic. The impossibility of dividing by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.
History of Zero
Zero is the reference point in all standard number systems. Europeans began to use this number relatively recently, but the sages of ancient India used zero for a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Maya numerical system. This American people used the duodecimal system, and they began the first day of each month with a zero. Interestingly, among the Maya, the sign for "zero" completely coincided with the sign for "infinity". Thus, the ancient Maya concluded that these quantities were identical and unknowable.
Math operations with zero
Standard mathematical operations with zero can be reduced to a few rules.
Addition: if you add zero to an arbitrary number, then it will not change its value (0+x=x).
Subtraction: when subtracting zero from any number, the value of the subtracted remains unchanged (x-0=x).
Multiplication: any number multiplied by 0 gives 0 in the product (a*0=0).
Division: Zero can be divided by any non-zero number. In this case, the value of such a fraction will be 0. And division by zero is prohibited.
Exponentiation. This action can be performed with any number. An arbitrary number raised to the power of zero will give 1 (x 0 =1).
Zero to any power is equal to 0 (0 a \u003d 0).
In this case, a contradiction immediately arises: the expression 0 0 does not make sense.
Paradoxes of mathematics
The fact that division by zero is impossible, many people know from school. But for some reason it is not possible to explain the reason for such a ban. Indeed, why does the division-by-zero formula not exist, but other actions with this number are quite reasonable and possible? The answer to this question is given by mathematicians.
The thing is that the usual arithmetic operations that schoolchildren study in primary school are actually not as equal as we think. All simple operations with numbers can be reduced to two: addition and multiplication. These operations are the essence of the very concept of a number, and the rest of the operations are based on the use of these two.
Addition and multiplication
Let's take a standard subtraction example: 10-2=8. At school, it is considered simply: if two are taken away from ten objects, eight remain. But mathematicians look at this operation quite differently. After all, there is no such operation as subtraction for them. This example can be written in another way: x+2=10. For mathematicians, the unknown difference is simply the number that must be added to two to make eight. And no subtraction is required here, you just need to find a suitable numerical value.
Multiplication and division are treated in the same way. In the example of 12:4=3, it can be understood that we are talking about the division of eight objects into two equal piles. But in reality, this is just an inverted formula for writing 3x4 \u003d 12. Such examples for division can be given endlessly.
Examples for dividing by 0
This is where it becomes a little clear why it is impossible to divide by zero. Multiplication and division by zero have their own rules. All examples per division of this quantity can be formulated as 6:0=x. But this is an inverted expression of the expression 6 * x = 0. But, as you know, any number multiplied by 0 gives only 0 in the product. This property is inherent in the very concept of a zero value.
It turns out that such a number, which, when multiplied by 0, gives any tangible value, does not exist, that is, this problem has no solution. One should not be afraid of such an answer, it is a natural answer for problems of this type. Just writing 6:0 doesn't make any sense, and it can't explain anything. In short, this expression can be explained by the immortal "no division by zero".
Is there a 0:0 operation? Indeed, if the operation of multiplying by 0 is legal, can zero be divided by zero? After all, an equation of the form 0x5=0 is quite legal. Instead of the number 5, you can put 0, the product will not change from this.
Indeed, 0x0=0. But you still can't divide by 0. As said, division is just the inverse of multiplication. Thus, if in the example 0x5=0, you need to determine the second factor, we get 0x0=5. Or 10. Or infinity. Dividing infinity by zero - how do you like it?
But if any number fits into the expression, then it does not make sense, we cannot choose one from an infinite set of numbers. And if so, it means that the expression 0:0 does not make sense. It turns out that even zero itself cannot be divided by zero.
higher mathematics
Division by zero is headache for school mathematics. Mathematical analysis studied in technical universities slightly expands the concept of problems that have no solution. For example, to the already known expression 0:0, new ones are added that have no solution in school courses mathematics:
- infinity divided by infinity: ∞:∞;
- infinity minus infinity: ∞−∞;
- unit raised to an infinite power: 1 ∞ ;
- infinity multiplied by 0: ∞*0;
- some others.
It is impossible to solve such expressions by elementary methods. But higher mathematics, thanks to additional possibilities for a number of similar examples, gives final solutions. This is especially evident in the consideration of problems from the theory of limits.
Uncertainty Disclosure
In the theory of limits, the value 0 is replaced by a conditional infinitesimal variable. And expressions in which division by zero is obtained when substituting the desired value are converted. Below is a standard example of limit expansion using the usual algebraic transformations:
As you can see in the example, a simple reduction of a fraction brings its value to a completely rational answer.
When considering the limits trigonometric functions their expressions tend to be reduced to the first remarkable limit. When considering the limits in which the denominator goes to 0 when the limit is substituted, the second remarkable limit is used.
L'Hopital method
In some cases, the limits of expressions can be replaced by the limit of their derivatives. Guillaume Lopital - French mathematician, founder of the French school mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule is as follows.
Zero itself is a very interesting number. By itself, it means emptiness, the absence of meaning, and next to another number increases its significance by 10 times. Any numbers to the zero power always give 1. This sign was used back in the Mayan civilization, and they also denoted the concept of “beginning, reason”. Even the calendar started from day zero. And this figure is associated with a strict ban.
Ever since the primary school years, we all clearly learned the rule “you cannot divide by zero.” But if in childhood you take a lot on faith and the words of an adult rarely cause doubts, then over time, sometimes you still want to understand the reasons, to understand why certain rules were established.
Why can't you divide by zero? I would like to get a clear logical explanation for this question. In the first grade, teachers could not do this, because in mathematics the rules are explained with the help of equations, and at that age we had no idea what it was. And now it's time to figure it out and get a clear logical explanation of why you can't divide by zero.
The fact is that in mathematics only two of the four basic operations (+, -, x, /) with numbers are recognized as independent: multiplication and addition. The rest of the operations are considered to be derivatives. Let's consider a simple example.
Tell me, how much will it turn out if 18 is subtracted from 20? Naturally, the answer immediately arises in our head: it will be 2. And how did we come to such a result? To some, this question will seem strange - after all, everything is clear that it will turn out 2, someone will explain that he took 18 from 20 kopecks and he got two kopecks. Logically, all these answers are not in doubt, but from the point of view of mathematics, this problem should be solved differently. Let us recall once again that the main operations in mathematics are multiplication and addition, and therefore in our case the answer lies in solving the following equation: x + 18 = 20. From which it follows that x = 20 - 18, x = 2. It would seem, why paint everything in such detail? After all, everything is so simple. However, without this it is difficult to explain why it is impossible to divide by zero.
Now let's see what happens if we wish to divide 18 by zero. Let's make the equation again: 18: 0 = x. Since the division operation is a derivative of the multiplication procedure, then by transforming our equation we get x * 0 = 18. This is where the impasse begins. Any number in place of x when multiplied by zero will give 0 and we will not succeed in getting 18. Now it becomes extremely clear why you cannot divide by zero. Zero itself can be divided by any number, but vice versa - alas, it is impossible.
What happens when zero is divided by itself? This can be written in this form: 0: 0 = x, or x * 0 = 0. This equation has an infinite number of solutions. So the end result is infinity. Therefore, the operation in this case also does not make sense.
Dividing by 0 is at the root of many imaginary mathematical jokes, which, if desired, can puzzle any ignorant person. For example, consider the equation: 4 * x - 20 \u003d 7 * x - 35. We will take 4 out of brackets on the left side, and 7 on the right. We get: 4 * (x - 5) \u003d 7 * (x - 5). Now we multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will look like this: 4 * (x - 5) / (x - 5) \u003d 7 * (x - 5) / (x - 5). We reduce the fractions by (x - 5) and we get that 4 \u003d 7. From this we can conclude that 2 * 2 \u003d 7! Of course, the catch here is that it is equal to 5 and it was impossible to reduce fractions, since this led to division by zero. Therefore, when reducing fractions, you must always check that zero does not accidentally end up in the denominator, otherwise the result will turn out to be completely unpredictable.
MKOU Sarybalyk secondary school
Teacher primary school: Makoveeva Marina Valentinovna
Math lesson in 4th grade. (textbook for special (correctional) educational institutionsVIIIspecies, author M. N. Perova)
Topic: “Multiplying the number zero and zero. Division of zero.
Target: introduce the rule of multiplying the number 0 and by 0, dividing 0; consolidate knowledge of the multiplication table, the ability to solve problems of the types studied; learn to reason and draw conclusions.
Planned results: students will learn how to multiply 0 by a number, a number by 0, divide 0; use the multiplication and division tables; solve problems of the studied species; evaluate the correctness of the actions.
Equipment: cards for the game "Postman"; table with geometric shapes, handout,Personal Computer, media projector, textbook "Mathematics" by M. N. Perov(4th grade).
Lesson type: new topic.
Type of lesson: game lesson.
During the classes
I . Org. moment:
Checking homework.
II . Verbal counting.
Teacher: remember tabular multiplication and division. Now we will play the game "Postmen". Sveta, you will be a postman. Houses with numbers on the board. Your task is to take an example letter, solve it correctly and determine which house we need to take the letter to.
3x4 2x2 9x2 3x1 3x8 25:5
6x2 16:4 3x6 9:3 6x4 5:1
4:1 3:1
Teacher: insert a missing action character.
4…0=4 1…3=4 5…1=6
4…4=0 1…3=3 5…1=5
3…3=0 1…0=1 9…0=0
III . Introduction to new material
PRO ZERO
In vain they think that zero
Plays a small role
At one time, many believed
That zero means nothing
And, oddly enough, they thought
That he is not a number at all.
But about its special properties
We will now tell the story
If you add zero to the number
Or you take away from him
In response you will immediately receive
Again the same number
Hitting as a multiplier among numbers
He instantly brings everything to naught
And therefore in the work
One for all bears the answer
As for dividing
We must firmly remember that
What has long been in the scientific world
Dividing by zero is not allowed
And indeed: which of the famous
We take the number for the quotient
When with zero in the product
All numbers zero can only give
Teacher: Let's check if everything in the poem is correct:
7+0=7 7-0=7 7 0=0 7:0
Teacher: apply the commutative property of multiplication and replace multiplication with addition: 7 0=0 7=0+0+0+0+0+0+0=0
What happened?
Teacher: we know that division is checked by multiplication: then we multiply the quotient by 0 - it should turn out 7, but this is not possible! Whatever number we multiply by 0, the product will always be 0.
IV . Fizminutka
V . Consolidation of the studied material
1. Solving the problem (p. 143 No. 7)
Teacher: what is the task about?
Student: about repair, foundation, bricks.
Teacher: What do you need to know?
Student: how many bricks are left to lay.
Teacher: can we immediately answer this question?
Student: no.
Teacher: Why?
Student: Because we don't know how many bricks the worker used.
Teacher: Can we find out?
Student: yes.
Teacher: what action?
Student: division.
Teacher: can we now answer the question of the problem?
Student: yes.
Teacher: what action?
Student: subtraction.
Teacher: how many bricks are left for the worker to lay?
Student: (40:5=8, 40-8=32) 32 bricks.
2. Independent work (p. 144 No. 18)
7*0 7:1 3*0 8:1
7*1 0*7 0*3 0:8
1*6 0*1 3*1 0*8
0*6 0:1 1*3 0*1
3. Work at the blackboard (p. 144 No. 11)
7*0 0*8 0:5 1*3 5+0
7+1 0:8 6*0 1+3 5*0
7-1 8+0 8-0 4-1 5-1
VI. Repetition
1.Circular examples
Teacher: We will be foresters. We need to determine the height of some trees, for this we need to solve circular examples.
2. Arithmetic dictation
Teacher: And now we will be stenographers. I dictate, and you write down - take shorthand with the help of cards.
The sum of the numbers 45 and 18 (45+18=63)
Product of numbers 8 and 3 (8*3=24)
The difference between the numbers 35 and 7 (35-7=22)
The quotient of numbers 20 and 4 (20:4=5)
3.Geometric material.
Teacher: last task. Which geometric figures you see?
Count and say how many times each figure occurs.
(Circle - 12, square - 6, triangle - 6, rectangle - 5.)
VII . Reflection
Do it yourself with. 144 No. 17 (1.2 st.). The answers are written on the board: 0,0,0; 5,5,5.
Appreciate your work in the lesson with an emoticon.
VIII. Homework
S. 144 No. 12.
