The mathematical rule regarding division by zero was taught to all people in the first grade of secondary school. “You can’t divide by zero,” we were all taught and were forbidden, on pain of a slap on the head, to divide by zero and generally discuss this topic. Although some elementary school teachers still tried to explain with simple examples why one should not divide by zero, these examples were so illogical that it was easier to just remember this rule and not ask unnecessary questions. But all these examples were illogical for the reason that the teachers could not logically explain this to us in the first grade, since in the first grade we did not even know what an equation was, and this mathematical rule can be logically explained only with the help of equations.

Everyone knows that dividing any number by zero results in a void. We will look at why it is emptiness later.

In general, in mathematics, only two procedures with numbers are recognized as independent. These are addition and multiplication. The remaining procedures are considered derivatives of these two procedures. Let's look at this with an example.

Tell me, how much will it be, for example, 11-10? We will all immediately answer that it will be 1. How did we find such an answer? Someone will say that it is already clear that there will be 1, someone will say that from 11 apples he took 10 and calculated that the result was one apple. From a logical point of view, everything is correct, but according to the laws of mathematics, this problem is solved differently. It is necessary to remember that addition and multiplication are considered the main procedures, so you need to create the following equation: x + 10 = 11, and only then x = 11-10, x = 1. Note that addition comes first, and only then, based on the equation, can we subtract. It would seem, why so many procedures? After all, the answer is already obvious. But only such procedures can explain the impossibility of division by zero.

For example, we are doing the following mathematical problem: we want to divide 20 by zero. So, 20:0=x. To find out how much it will be, you need to remember that the division procedure follows from multiplication. In other words, division is a derivative procedure from multiplication. Therefore, you need to create an equation from multiplication. So, 0*x=20. This is where the dead end comes in. No matter what number we multiply by zero, it will still be 0, but not 20. This is where the rule follows: you cannot divide by zero. You can divide zero by any number, but unfortunately, you cannot divide a number by zero.

This brings up another question: is it possible to divide zero by zero? So, 0:0=x, which means 0*x=0. This equation can be solved. Let's take, for example, x=4, which means 0*4=0. It turns out that if you divide zero by zero, you get 4. But here, too, everything is not so simple. If we take, for example, x=12 or x=13, then the same answer will come out (0*12=0). In general, no matter what number we substitute, it will still come out 0. Therefore, if 0:0, then the result will be infinity. This is some simple math. Unfortunately, the procedure of dividing zero by zero is also meaningless.

In general, the number zero in mathematics is the most interesting. For example, everyone knows that any number to the zero power gives one. Of course, we don’t come across such an example in real life, but life situations with division by zero come across very often. Therefore, remember that you cannot divide by zero.

Everyone remembers from school that you cannot divide by zero. Primary schoolchildren are never explained why this should not be done. They simply offer to take this as a given, along with other prohibitions like “you can’t put your fingers in sockets” or “you shouldn’t ask stupid questions to adults.” AiF.ru decided to find out whether the school teachers were right.

Algebraic explanation of the impossibility of division by zero

From an algebraic point of view, you cannot divide by zero, since it makes no sense. Let's take two arbitrary numbers, a and b, and multiply them by zero. a × 0 is equal to zero and b × 0 is equal to zero. It turns out that a × 0 and b × 0 are equal, because the product in both cases is equal to zero. Thus, we can create the equation: 0 × a = 0 × b. Now let's assume that we can divide by zero: we divide both sides of the equation by it and get that a = b. It turns out that if we allow the operation of division by zero, then all the numbers coincide. But 5 is not equal to 6, and 10 is not equal to ½. Uncertainty arises, which teachers prefer not to tell inquisitive junior high school students.

Explanation of the impossibility of dividing by zero from the point of view of mathematical analysis

In high school they study the theory of limits, which also talks about the impossibility of dividing by zero. This number is interpreted there as an “undefined infinitesimal quantity.” So if we consider the equation 0 × X = 0 within the framework of this theory, we will find that X cannot be found because to do this we would have to divide zero by zero. And this also does not make any sense, since both the dividend and the divisor in this case are indefinite quantities, therefore, it is impossible to draw a conclusion about their equality or inequality.

When can you divide by zero?

Unlike schoolchildren, students of technical universities can divide by zero. An operation that is impossible in algebra can be performed in other areas of mathematical knowledge. New additional conditions of the problem appear in them that allow this action. Dividing by zero will be possible for those who listen to a course of lectures on non-standard analysis, study the Dirac delta function and become familiar with the extended complex plane.

Evgeniy SHIRYAEV, teacher and head of the Mathematics Laboratory of the Polytechnic Museum, told AiF about division by zero:

1. Jurisdiction of the issue

Agree, what makes the rule especially provocative is the ban. How can this not be done? Who banned? What about our civil rights?

Neither the Constitution, nor the Criminal Code, nor even the charter of your school objects to the intellectual action that interests us. This means that the ban has no legal force, and nothing prevents you from trying to divide something by zero right here, on the pages of AiF. For example, a thousand.

2. Let's divide as taught

Remember, when you first learned how to divide, the first examples were solved with a multiplication check: the result multiplied by the divisor had to coincide with the dividend. It didn’t match - they didn’t decide.

Example 1. 1000: 0 =...

Let's forget about the forbidden rule for a moment and make several attempts to guess the answer.

Incorrect ones will be cut off by the check. Try the following options: 100, 1, −23, 17, 0, 10,000. For each of them, the check will give the same result:

100 0 = 1 0 = − 23 0 = 17 0 = 0 0 = 10,000 0 = 0

By multiplying zero, everything turns into itself and never into a thousand. The conclusion is easy to formulate: no number will pass the test. That is, no number can be the result of dividing a non-zero number by zero. Such division is not prohibited, but simply has no result.

3. Nuance

We almost missed one opportunity to refute the ban. Yes, we admit that a non-zero number cannot be divided by 0. But maybe 0 itself can?

Example 2. 0: 0 = ...

What are your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor 0 is equal to the dividend 0.

More options! 1? Fits too. And −23, and 17, and that’s it. In this example, the result check will be positive for any number. And, to be honest, the solution in this example should be called not a number, but a set of numbers. Everyone. And it doesn’t take long to agree that Alice is not Alice, but Mary Ann, and both of them are a rabbit’s dream.

4. What about higher mathematics?

The problem has been resolved, the nuances have been taken into account, the dots have been placed, everything has become clear - the answer to the example with division by zero cannot be a single number. Solving such problems is hopeless and impossible. Which means... interesting! Take two.

Example 3. Figure out how to divide 1000 by 0.

But no way. But 1000 can be easily divided by other numbers. Well, let's at least do what works, even if we change the task. And then, you see, we get carried away, and the answer will appear by itself. Let’s forget about zero for a minute and divide by one hundred:

A hundred is far from zero. Let's take a step towards it by decreasing the divisor:

1000: 25 = 40,
1000: 20 = 50,
1000: 10 = 100,
1000: 8 = 125,
1000: 5 = 200,
1000: 4 = 250,
1000: 2 = 500,
1000: 1 = 1000.

The dynamics are obvious: the closer the divisor is to zero, the larger the quotient. The trend can be observed further by moving to fractions and continuing to reduce the numerator:

It remains to note that we can get as close to zero as we like, making the quotient as large as we like.

In this process there is no zero and there is no last quotient. We indicated the movement towards them by replacing the number with a sequence converging to the number we are interested in:

This implies a similar replacement for the dividend:

1000 ↔ { 1000, 1000, 1000,... }

It’s not for nothing that the arrows are double-sided: some sequences can converge to numbers. Then we can associate the sequence with its numerical limit.

Let's look at the sequence of quotients:

It grows unlimitedly, not striving for any number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to place a double-sided arrow next to such a sequence:

Comparison with the numbers of sequences that have a limit allows us to propose a solution to the third example:

When elementwise dividing a sequence converging to 1000 by a sequence of positive numbers converging to 0, we obtain a sequence converging to ∞.

5. And here is the nuance with two zeros

What will be the result of dividing two sequences of positive numbers that converge to zero? If they are the same, then the unit is identical. If a dividend sequence converges to zero faster, then in particular it is a sequence with a zero limit. And when the elements of the divisor decrease much faster than those of the dividend, the sequence of the quotient will grow greatly:

Uncertain situation. And that’s what it’s called: uncertainty of type 0/0 . When mathematicians see sequences that fit such uncertainty, they do not rush to divide two identical numbers by each other, but figure out which of the sequences runs faster to zero and how exactly. And each example will have its own specific answer!

6. In life

Ohm's law relates current, voltage and resistance in a circuit. It is often written in this form:

Let's allow ourselves to neglect the neat physical understanding and formally look at the right-hand side as the quotient of two numbers. Let's imagine that we are solving a school problem on electricity. The condition gives the voltage in volts and resistance in ohms. The question is obvious, the solution is in one action.

Now let's look at the definition of superconductivity: this is the property of some metals to have zero electrical resistance.

Well, let's solve the problem for a superconducting circuit? Just set it up like that R= 0 If it doesn’t work out, physics throws up an interesting problem, behind which, obviously, there is a scientific discovery. And the people who managed to divide by zero in this situation received the Nobel Prize. It’s useful to be able to bypass any prohibitions!

Textbook:“Mathematics” by M.I. Moreau

Lesson objectives: create conditions for developing the ability to divide 0 by a number.

Lesson objectives:

• reveal the meaning of dividing 0 by a number through the connection between multiplication and division;
• develop independence, attention, thinking;
• develop skills in solving examples of table multiplication and division.

To achieve the goal, the lesson was designed taking into account activity approach.

The structure of the lesson included:

1. Org. moment, the goal of which was to positively motivate children to learn.
2. Motivation allowed us to update knowledge, formulate the goals and objectives of the lesson. For this purpose, tasks were proposed for finding an extra number, classifying examples into groups, adding missing numbers. While solving these tasks, children were faced with problem: an example was found for which the existing knowledge is not enough to solve. In this regard, children independently formulated a goal and set themselves the learning objectives of the lesson.
3. Search and discovery of new knowledge gave the children an opportunity offer various options task solutions. Based on previously studied material, they were able to find the right solution and come to conclusion, in which a new rule was formulated.
4. During primary consolidation students commented your actions, working according to the rule, were additionally selected your examples to this rule.
5. For automation of actions And ability to use rules in non-standard In the tasks, children solved equations and expressions in several steps.
6. Independent work and carried out mutual verification showed that most children understood the topic.
7. During reflections The children concluded that the goal of the lesson had been achieved and assessed themselves using the cards.

The lesson was based on independent actions of students at each stage, complete immersion in the learning task. This was facilitated by such techniques as working in groups, self- and mutual testing, creating a situation of success, differentiated tasks, and self-reflection.

During the classes

Purpose of the stage	Contents of the stage	Student activity
1. Org. moment
Preparing students for work, a positive attitude towards learning activities.	Incentives for educational activities. Check your readiness for the lesson, sit upright, lean on the back of the chair. Rub your ears so that blood flows more actively to the brain. Today you will have a lot of interesting work, which, I am sure, you will cope with perfectly.	Organization of the workplace, checking the fit.
2. Motivation.
Stimulation of cognitive activity, activation of the thought process	Updating knowledge sufficient to acquire new knowledge. Verbal counting. Testing your knowledge of table multiplication:	Solving problems based on knowledge of table multiplication.
	A) find the extra number: 2 4 6 7 10 12 14 6 18 24 29 36 42 Explain why it is redundant and what number should be used to replace it.	Finding the extra number.
	B) insert the missing numbers: … 16 24 32 … 48 …	Adding the missing number.
	Creating a problem situation Tasks in pairs: C) arrange the examples into 2 groups: Why was it distributed this way? (with answer 4 and 5).	Classification of examples into groups.
	Cards: 8·7-6+30:6= 28:(16:4) 6= 30-(20-10:2):5= 30-(20-10 2):5=	Strong students work on individual cards.
	What did you notice? Is there another example here? Were you able to solve all the examples? Who's having trouble? How is this example different from the others? If someone has decided, then well done. But why couldn’t everyone cope with this example?	Finding the problem. Identifying missing knowledge and causes of difficulty.
	Setting a learning task. Here is an example with 0. And from 0 you can expect different tricks. This is an unusual number. Remember what you know about 0? (a 0=0, 0 a=0, 0+a=a) Give examples. Look how insidious it is: when it is added, it does not change the number, but when it is multiplied, it turns it into 0. Do these rules apply to our example? How will he behave when eating?	Observation of known techniques for operating with 0 and correlation with the original example.
	So what is our goal? Solve this example correctly. Table on the board. What is needed for that? Learn the rule for dividing 0 by a number.	Proposing a hypothesis
How to find the right solution? What action is involved in multiplication? (with division) Give an example 2 3 = 6 6: 2 = 3 Can we now 0:5? This means you need to find a number that, when multiplied by 5, equals 0. x 5=0 This number is 0. So 0:5=0. Give your own examples.	searching for a solution based on what has been previously studied,
Formulation of the rule. What rule can now be formulated? When you divide 0 by a number, you get 0. 0: a = 0.	Solving typical tasks with commenting. Work according to the scheme (0:a=0)
5. Physical exercise.
Prevention of poor posture, relieving eye fatigue and general fatigue.
6. Automation of knowledge.
Identifying the limits of applicability of new knowledge.	What other tasks might require knowledge of this rule? (in solving examples, equations)	Using the acquired knowledge in various tasks. Work in groups.
	What is unknown in these equations? Remember how to find out an unknown multiplier. Solve the equations. What is the solution to equation 1? (0) At 2? (no solution, cannot divide by 0)	Recalling previously learned skills.
	** Create an equation with the solution x=0 (x 5=0)	For strong students a creative task
7. Independent work.
Development of independence and cognitive abilities	Independent work followed by mutual verification. №6	Active mental actions of students associated with searching for solutions based on their knowledge. Self-control and mutual control. Strong students check and help weaker ones.
8. Work on previously covered material. Practicing problem solving skills.
Formation of problem solving skills.	Do you think the number 0 is often used in problems? (No, not often, because 0 is nothing, and tasks must contain some amount of something.) Then we will solve problems where there are other numbers. Read the problem. What will help solve the problem? (table) What columns in the table should be written down? Fill the table. Make a solution plan: what needs to be learned in steps 1 and 2?	Working on a problem using a table. Planning to solve a problem. Self-recording of the solution. Self-control according to the model.
9. Reflection. Lesson summary.
Organization of self-assessment of activities. Increasing the child's motivation.	What topic did you work on today? What didn't you know at the beginning of the lesson? What goal did you set for yourself? Have you achieved it? What rule did you come across? Rate your work by checking the appropriate icon: Sun – I’m pleased with myself, I did it all White cloud – everything is fine, but I could have worked better; gray cloud – the lesson is ordinary, nothing interesting; droplet - nothing succeeded	Awareness of your activities, self-analysis of your work. Recording the correspondence of performance results and the set goal.
10. Homework.

Number in mathematics zero occupies a special place. The fact is that it, in essence, means “nothing”, “emptiness”, but its significance is really difficult to overestimate. To do this, it is enough to remember at least what exactly with zero mark and the counting of the coordinates of the point’s position in any coordinate system begins.

Zero widely used in decimal fractions to determine the values ​​of the “empty” places, both before and after the decimal point. In addition, one of the fundamental rules of arithmetic is associated with it, which states that zero cannot be divided. Its logic, strictly speaking, stems from the very essence of this number: indeed, it is impossible to imagine that some value different from it (and it itself too) would be divided into “nothing”.

Calculation examples

WITH zero all arithmetic operations are carried out, and integer numbers, ordinary and decimal fractions can be used as its “partners”, and all of them can have both positive and negative values. Let us give examples of their implementation and some explanations for them.

Addition

When adding zero to a certain number (both integer and fractional, both positive and negative), its value remains absolutely unchanged.

Example 1

twenty four plus zero equals twenty-four.

Example 2

Seventeen point three eighths plus zero equals seventeen point three eighths.

Multiplication

When multiplying any number (integer, fraction, positive or negative) by zero it turns out zero.

Example 1

Five hundred eighty six times zero equals zero.

Example 2

Zero multiplied by one hundred thirty-five point six seven equals zero.

Example 3

Zero multiply by zero equals zero.

Division

The rules for dividing numbers by each other in cases where one of them is a zero differ depending on what role the zero itself plays: a dividend or a divisor?

In cases where zero represents the dividend, the result is always equal to it, regardless of the value of the divisor.

Example 1

Zero divided by two hundred sixty five equals zero.

Example 2

Zero divided by seventeen five hundred ninety-six equals zero.

0:

= 0

Divide zero to zero According to the rules of mathematics, it is impossible. This means that when performing such a procedure, the quotient is uncertain. Thus, in theory, it can represent absolutely any number.

0: 0 = 8 because 8 × 0 = 0

In mathematics there is a problem like division of zero by zero, does not make any sense, since its result is an infinite set. This statement, however, is true if no additional data is provided that could affect the final result.

These, if present, should consist of indicating the degree of change in the magnitude of both the dividend and the divisor, and even before the moment when they turned into zero. If this is defined, then an expression such as zero divide by zero, in the vast majority of cases some meaning can be attached.