If you divide by zero you get it. Division by zero

Each of us learned at least two unshakable rules from school: “zhi and shi - write with the letter I” and “ You can't divide by zero". And if the first rule can be explained by the peculiarity of the Russian language, then the second raises a completely logical question: “Why?”

Why can't you divide by zero?

It’s not entirely clear why they don’t talk about this in school, but from an arithmetic point of view, the answer is very simple.

Let's take a number 10 and divide it by 2 . This implies that we took 10 any objects and arranged them according to 2 equal groups, that is 10: 2 = 5 (By 5 items in the group). The same example can be written using the equation x * 2 = 10(And X here will be equal 5 ).

Now, let’s imagine for a second that you can divide by zero, and let’s try 10 divide by 0 .

You will get the following: 10: 0 = x, hence x * 0 = 10. But our calculations cannot be correct, since when multiplying any number by 0 it always works out 0 . In mathematics there is no such number that, when multiplied by 0 would give something other than 0 . Therefore, the equations 10: 0 = x And x * 0 = 10 don't have a solution. In view of this, they say that you cannot divide by zero.

When can you divide by zero?

There is an option in which division by zero still makes some sense. If we divide zero itself, we get the following 0: 0 = x, which means x * 0 = 0.

Let's assume that x=0, then the equation does not raise any questions, everything fits perfectly 0: 0 = 0 , and therefore 0 * 0 = 0 .

But what if X≠ 0 ? Let's assume that x = 9? Then 9 * 0 = 0 And 0: 0 = 9 ? What if x=45, That 0: 0 = 45 .

We can really share 0 on 0 . But this equation will have an infinite number of solutions, since 0: 0 = anything.

Why 0: 0 = NaN

Have you ever tried to divide 0 on 0 on a smartphone? Since zero divided by zero gives absolutely any number, programmers had to look for a way out of this situation, because the calculator cannot ignore your requests. And they found a unique way out: when you divide zero by zero, you get NaN (not a number).

Why x: 0 =∞ A x: -0 = — ∞

If you try to divide any number by zero on your smartphone, the answer will be equal to infinity. The thing is that in mathematics 0 sometimes considered not as “nothing”, but as an “infinitesimal quantity”. Therefore, if any number is divided by an infinitesimal value, the result is an infinitely large value (∞) .

So is it possible to divide by zero?

The answer, as is often the case, is ambiguous. At school, it’s best to note on your nose that You can't divide by zero- this will save you from unnecessary difficulties. But if you enroll in the mathematics department at a university, you will still have to divide by zero.

Zero itself is a very interesting number. By itself it means emptiness, lack of meaning, and next to another number it increases its significance 10 times. Any numbers to the zero power always give 1. This sign was used in the Mayan civilization, and it also denoted the concept of “beginning, cause.” Even the calendar began with day zero. This figure is also associated with a strict ban.

Since the beginning school years We have all clearly learned the rule “you can’t divide by zero.” But if in childhood you take a lot of things on faith and the words of an adult rarely raise 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 using 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 remaining operations are considered to be derivatives. Let's look at a simple example.

Tell me, how much do you get if you subtract 18 from 20? Naturally, the answer immediately arises in our head: it will be 2. How did we come to this result? This question will seem strange to some - after all, everything is clear that the result will be 2, someone will explain that he took 18 from 20 kopecks and got two kopecks. Logically, all these answers are not in doubt, but from a mathematical point of view, 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 describe everything in such detail? After all, everything is so simple. However, without this it is difficult to explain why you cannot divide by zero.

Now let's see what happens if we want to divide 18 by zero. Let's create the equation again: 18: 0 = x. Since the division operation is a derivative of the multiplication procedure, transforming our equation we get x * 0 = 18. This is where the dead end begins. Any number in place of X when multiplied by zero will give 0 and we will not be able to get 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’s impossible.

What happens if you divide zero by itself? This can be written as follows: 0: 0 = x, or x * 0 = 0. This equation has an infinite number of solutions. Therefore, the end result is infinity. Therefore, the operation in this case also does not make sense.

Division by 0 is at the root of many imaginary mathematical jokes that can be used to puzzle any ignorant person if desired. For example, consider the equation: 4*x - 20 = 7*x - 35. Let's take 4 out of brackets on the left side and 7 on the right. We get: 4*(x - 5) = 7*(x - 5). Now let's multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will take the following form: 4*(x - 5)/(x - 5) = 7*(x - 5)/ (x - 5). Let's reduce the fractions by (x - 5) and it turns out that 4 = 7. From this we can conclude that 2*2 = 7! Of course, the catch here is that it is equal to 5 and it was impossible to cancel fractions, since this led to division by zero. Therefore, when reducing fractions, you must always check that a zero does not accidentally end up in the denominator, otherwise the result will be completely unpredictable.

Everyone was taught the mathematical rule regarding division by zero in first grade. 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 he took 10 away from 11 apples and calculated that it turned out to be 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 the main procedures are addition and multiplication, 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 do this math 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, with such an example in real life We don’t meet, but life situations involving division by zero come across very often. Therefore, remember that you cannot divide by zero.

Number in mathematics zero occupies a special place. The fact is that it essentially 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.

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 put 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 is 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 R= 0 will not work, physics throws up an interesting problem, behind which, obviously, there is scientific discovery. And the people who managed to divide by zero in this situation received Nobel Prize. It’s useful to be able to bypass any prohibitions!