How much will it be when multiplied by 0. Why can't you divide by zero? illustrative example

If we can rely on other laws of arithmetic, then this particular fact can be proven.

Suppose there is a number x for which x * 0 = x", and x" is not zero (for simplicity, we will assume that x" > 0)

Then, on the one hand, x * 0 = x", on the other hand, x * 0 = x * (1 - 1) = x - x

It turns out that x - x = x", whence x = x + x", i.e. x > x, which cannot be true.

This means that our assumption leads to a contradiction and there is no such number x for which x * 0 would not be equal to zero.

the assumption cannot be true because it is just an assumption! no one can explain in simple language or find it difficult! if 0 * x = 0 then 0 * x = (0 + 0) * x \u003d 0 * x + 0 * x and as a result they reduced the right to the left 0 \u003d 0 * x this is supposedly a mathematical proof! but such nonsense with this zero terribly contradicts and in my opinion 0 should not be a number, but only an abstract concept! So that mere mortals would not be burned in the brain by the fact that the physical presence of objects, when miraculously multiplied by nothing, gave rise to nothing!

P / s it’s not entirely clear to me, not a mathematician, but to a mere mortal where did you get units in the reasoning equation (like 0 is the same as 1-1)

I'm crazy about reasoning like there is some kind of X and let it be any number

is in the equation 0 and when multiplied by it, we set all numerical values to zero

therefore X is a numeric value, and 0 is the number of actions performed on the number X (and the actions, in turn, are also displayed in a numeric format)

EXAMPLE on apples)) :

Kolya had 5 apples, he took these apples and went to the market in order to increase capital, but the day turned out to be rainy, cloudy trade did not work out and Kalek returned home with nothing. In mathematical language, the story about Kolya and apples looks like this

5 apples * 0 sales = made 0 profits 5*0=0

Before going to the bazaar, Kolya went and picked 5 apples from a tree, and tomorrow he went to pick but did not reach for some reason of his own ...

Apples 5, tree 1, 5*1=5 (Kolya picked 5 apples on the 1st day)

Apples 0, tree 1, 0*1=0 (actually the result of Kolya's work on the second day)

The scourge of mathematics is the word "Suppose"

And if in another way, 5 apples for 0 apples \u003d how many apples, in mathematics it should be zero, and so

In fact, any numbers make sense only when they are associated with material objects, such as 1 cow, 2 cows, or whatever, and an account has appeared in order to count objects and not just like that, and there is a paradox if I don’t have a cow , and the neighbor has a cow, and we multiply my absence by the neighbor’s cow, then his cow should disappear, multiplication is generally invented to facilitate the addition of large quantities of identical objects, when it is difficult to calculate them using the addition method, for example, money was stacked in columns of 10 coins, and then the number of columns was multiplied by the number of coins in the column, much easier than adding up. but if the number of columns is multiplied by zero coins, then it will naturally turn out to be zero, but if there are both columns and coins, then how not to multiply them by zero, the coins will not go anywhere because they are, and even if it is one coin, then the column is consisting of one coin, so you can’t get anywhere, so zero when multiplied by zero is obtained only under certain conditions, that is, in the absence of a material component, and if I have 2 socks, since you don’t multiply them by zero, they won’t go anywhere .

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 do not obey 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 elementary grades are in fact far from being 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 a headache for high school math. 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 mathematics courses:

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 of 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 of 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.

Which of these sums do you think can be replaced by the product?

Let's argue like this. In the first sum, the terms are the same, the number five is repeated four times. So we can replace addition with multiplication. The first factor shows which term is repeated, the second factor shows how many times this term is repeated. We replace the sum with the product.

Let's write down the solution.

In the second sum, the terms are different, so it cannot be replaced by a product. We add the terms and get the answer 17.

Let's write down the solution.

Can the product be replaced by the sum of the same terms?

Consider works.

Let's take action and draw a conclusion.

1*2=1+1=2

1*4=1+1+1+1=4

1*5=1+1+1+1+1=5

We can conclude: always the number of unit terms is equal to the number by which the unit is multiplied.

Means, multiplying the number one by any number gives the same number.

1 * a = a

Consider works.

These products cannot be replaced by a sum, since the sum cannot have one term.

The products in the second column differ from the products in the first column only in the order of the factors.

This means that in order not to violate the commutative property of multiplication, their values must also be equal, respectively, to the first factor.

Let's conclude: When any number is multiplied by the number one, the number that was multiplied is obtained.

We write this conclusion as an equality.

a * 1= a

Solve examples.

Hint: do not forget the conclusions that we made in the lesson.

Test yourself.

Now let's observe the products, where one of the factors is zero.

Consider products where the first factor is zero.

Let us replace the products with the sum of identical terms. Let's take action and draw a conclusion.

0*3=0+0+0=0

0*6=0+0+0+0+0+0=0

0*4=0+0+0+0=0

The number of zero terms is always equal to the number by which zero is multiplied.

Means, When you multiply zero by a number, you get zero.

We write this conclusion as an equality.

0 * a = 0

Consider products where the second factor is zero.

These products cannot be replaced by a sum, since the sum cannot have zero terms.

Let's compare the works and their meanings.

0*4=0

The products of the second column differ from the products of the first column only in the order of the factors.

This means that in order not to violate the commutative property of multiplication, their values must also be equal to zero.

Let's conclude: Multiplying any number by zero results in zero.

We write this conclusion as an equality.

a * 0 = 0

But you can't divide by zero.

Solve examples.

Hint: don't forget the conclusions drawn in the lesson. When calculating the values of the second column, be careful when determining the order of operations.

Test yourself.

Today in the lesson we got acquainted with special cases of multiplication by 0 and 1, practiced multiplying by 0 and 1.

Bibliography

M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
"School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.

Nsportal.ru ().
Prosv.ru ().
Do.gendocs.ru ().

Homework

1. Find the meaning of expressions.

2. Find the meaning of expressions.

3. Compare expression values.

(56-54)*1 … (78-70)*1

4. Make a task on the topic of the lesson for your comrades.

Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still there is a lot of controversy around him. Someone just memorized the rule and does not bother with the question “why?”. “You can’t do everything here, because at school they said so, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.

In contact with

Who is right in the end

During these disputes, both people, having opposite points of view, look at each other like a ram, and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams resting against each other with their horns. The only difference between them is that one is slightly less educated than the other.

Most often, those who consider this rule to be wrong try to call for logic in this way:

I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!

Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 \u003d 2. So we will immediately discard such a conclusion - it is illogical, although it has the opposite goal - to call to logic.

What is multiplication

The original multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:

25x3=75
25 + 25 + 25 = 75
25x3 = 25 + 25 + 25

From this equation follows the conclusion, that multiplication is a simplified addition.

What is zero

Any person knows from childhood: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. The ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the value of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to determine empty digits in decimal fractions, this is done both before and after the decimal point.

Is it possible to multiply by emptiness

It is possible to multiply by zero, but it is useless, because, whatever one may say, but even when multiplying negative numbers, zero will still be obtained. It is enough just to remember this simplest rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as ancient scientists believed. The most logical explanation will be given below that this multiplication is useless, because when multiplying a number by it, the same thing will still be obtained - zero.

Going back to the very beginning, the argument about two apples, 2 times 0 looks like this:

If you eat two apples five times, then eaten 2×5 = 2+2+2+2+2 = 10 apples
If you eat two of them three times, then eaten 2 × 3 = 2 + 2 + 2 = 6 apples
If you eat two apples zero times, then nothing will be eaten - 2x0 = 0x2 = 0+0 = 0

After all, eating an apple 0 times means not eating a single one. This will be clear even to the smallest child. Like it or not, 0 will come out, two or three can be replaced with absolutely any number and absolutely the same thing will come out. And to put it simply, zero is nothing and when you have there is nothing, then no matter how much you multiply - it's all the same will be zero. There is no magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.

Division

From all of the above follows another important rule:

You can't divide by zero!

This rule, too, has been stubbornly hammered into our heads since childhood. We just know that it is impossible and that's it, without stuffing our heads with unnecessary information. If you are suddenly asked the question, for what reason it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions around this rule.

Everyone just memorized the rule and does not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are full of the above, and all other manipulations with numbers are built from them. That is, the entry 10: 2 is an abbreviation of the equation 2 * x = 10. Therefore, the entry 10: 0 is the same abbreviation for 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be a priori incorrect.

Let me tell you

To not divide by 0!

Cut 1 as you like, along,

Just don't divide by 0!

Evgeny Shiryaev, lecturer and head of the Laboratory of Mathematics of the Polytechnic Museum, told AiF.ru about division by zero:

1. Jurisdiction of the issue

Agree, the ban gives a special provocativeness to the rule. How is it impossible? Who banned? But what about our civil rights?

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

2. Divide as taught

Remember, when you first learned how to divide, the first examples were solved by checking by multiplication: the result multiplied by the divisor had to match the divisible. Did not match - did not decide.

Example 1 1000: 0 =...

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

Incorrect will cut off the check. Iterate over the options: 100, 1, −23, 17, 0, 10,000. For each of them, the test will give the same result:

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

Zero by multiplication turns everything 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 a division is not forbidden, but simply has no result.

3. Nuance

Almost missed one opportunity to refute the ban. Yes, we recognize that a non-zero number will not be divisible by 0. But maybe 0 itself can?

Example 2 0: 0 = ...

Your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor of 0 is equal to the divisible of 0.

More options! one? Also suitable. And -23, and 17, and all-all-all. In this example, the result check will be positive for any number. And to be honest, the solution in this example should not be called a number, but a set of numbers. Everyone. And it won’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 is solved, the nuances are taken into account, the dots are placed, everything is clear - no number can be the answer for the example with division by zero. Solving such problems is hopeless and impossible. So... interesting! Double 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 there, you see, we will get carried away, and the answer will appear by itself. 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.

Obvious dynamics: the closer the divisor is to zero, the greater the quotient. The trend can be observed further, moving to fractions and continuing to reduce the numerator:

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

There is no zero in this process and no last quotient. We indicated the movement towards them by replacing the number with a sequence converging to the number of interest to us:

This implies a similar replacement for the dividend:

1000 ↔ { 1000, 1000, 1000,... }

The arrows are double-sided for a reason: some sequences can converge to numbers. Then we can associate a sequence with its numerical limit.

Let's look at the sequence of quotients:

It grows indefinitely, striving for no number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:

Comparing the numbers of sequences with a limit allows us to propose a solution to the third example:

Dividing a sequence converging to 1000 element-wise by a sequence of positive numbers converging to 0, we get 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 identical unit. If a sequence-dividend converges to zero faster, then in a particular sequence with a zero limit. And when the elements of the divisor decrease much faster than the dividend, the quotient sequence will grow strongly:

Uncertain situation. And so it is called: the uncertainty of the form 0/0 . When mathematicians see sequences that fall under such uncertainty, they do not rush to divide two identical numbers by each other, but figure out which of the sequences runs to zero faster and how. 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 us neglect accurate physical understanding and formally look at the right side as a quotient of two numbers. Imagine that we are solving a school problem on electricity. The condition is given voltage in volts and resistance in ohms. The question is obvious, the decision in one action.

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

Well, let's solve the problem for a superconducting circuit? Just put it like that R= 0 does not 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 is useful to be able to bypass any prohibitions!