Line UMK A. G. Merzlyak. Mathematics (5-6)

Mathematics

Why can't you divide by zero?

The information that we cannot divide by zero has been known to us since school. We learn this rule once and for all. However, only a few of us wonder why we actually can’t do this. But it is important to know and understand the reasons for the impossibility of this action, as it reveals the principles of “work” and other mathematical operations.

All math operations are equal, but some are more equal than others.

Let's start with the fact that the four arithmetic operations - addition, subtraction, multiplication and division - are not equal. And the conversation is not about the order of actions when solving some example or equation. No, we mean the very concept of number. And according to him, the most important are addition and multiplication. And subtraction and division “follow” from them in one way or another.

Addition and subtraction

For example, let's look at a simple operation: “3 - 1”. What does this mean? The student can easily explain this problem: this means that there were three objects (for example, three oranges), one was subtracted, the remaining number of objects is the correct answer. Is it described correctly? Right. We ourselves would explain it in exactly the same way. But mathematicians view the process of subtraction differently.

The operation “3 - 1” is considered not from the standpoint of subtraction, but only from the standpoint of addition. According to this, there is no “three minus one”, there is “some unknown number that, when added to one, gives three.” Thus, the simple “three minus one” becomes an equation with one unknown: “x + 1 = 3.” Moreover, the appearance of the equation changed its sign - subtraction changed to addition. There is only one task left - to find a suitable number.

The reference manual contains all the basic formulas of the school mathematics course: algebra, geometry and principles of analysis. For ease of use of the reference book, a subject index has been compiled. The manual is intended for schoolchildren in grades 5-11 and applicants.

Multiplication and division

Similar metamorphoses occur with such an action as division. Mathematicians refuse to perceive the “6:3” problem as some kind of six objects divided into three parts. “Six divided by three” is nothing more than “an unknown number multiplied by three, resulting in six”: “x · 3.”

Divide by zero

Having clarified the principle of mathematical operations in relation to problems with subtraction and division, let's consider our division by zero.

The problem "4:0" becomes "x · 0". It turns out that we need to find a number whose multiplication will give us 4. It is known that multiplication by zero always gives zero. This unique property zero and, in fact, its essence. There is no such thing as a number multiplied by zero that produces any number other than zero. We have reached a contradiction, which means the problem has no solution. Consequently, the entry “4:0” does not correspond to any specific number, and hence its meaninglessness follows. Therefore, in order to briefly emphasize the unproductiveness of such a process as division by zero, they say that “you cannot divide by zero.”

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What happens if zero is divided by zero?

Let’s imagine the following equation: “0 x = 0”. On the one hand, it looks quite fair. We imagine zero instead of an unknown number and get a ready-made solution: “0 · 0 = 0.” From this it is quite logical to deduce that “0: 0 = 0”.

However, now let’s substitute any other number, for example “x = 7”, into the same equation with an unknown instead of “x = 0”. The resulting expression now looks like “0 · 7 = 0”. It seems that everything is correct. We do the reverse operation and get “0: 0 = 7”. But then, it turns out that you can take absolutely any number and output 0: 0 = 1, 0: 0 = 2... 0: 0 = 145... - and so on ad infinitum.

If the equation is valid for any number x, then we do not have the right to choose only one, excluding the rest. This means that we still cannot answer what number the expression “0: 0” corresponds to. Finding ourselves at a dead end again, we admit that this operation is also pointless. It turns out that zero cannot be divided even by itself.

Let's agree that in mathematical analysis sometimes there are special conditions tasks - the so-called “uncertainty disclosure”. In such cases, it is allowed to give preference to one of the possible solutions equation "0 x = 0". However, in arithmetic such “tolerances” do not occur.

The number 0 can be imagined as a certain boundary 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. Impossibility of dividing by zero - bright that example. 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 using this number relatively recently, but the sages of ancient India used zero a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Mayan numerical system. These American people used the duodecimal number system, and the first day of each month began with a zero. It is interesting that among the Mayans the sign denoting “zero” completely coincided with the sign denoting “infinity”. Thus, the ancient Mayans concluded that these quantities are identical and unknowable.

Mathematical operations with zero

Standard mathematical operations with zero can be reduced to a few rules.

Addition: if you add zero to an arbitrary number, it will not change its value (0+x=x).

Subtraction: When subtracting zero from any number, the value of the subtrahend remains unchanged (x-0=x).

Multiplication: Any number multiplied by 0 produces 0 (a*0=0).

Division: zero can be divided by any number, not equal to zero. 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 zero power will give 1 (x 0 =1).

Zero to any power is equal to 0 (0 a = 0).

In this case, a contradiction immediately arises: the expression 0 0 does not make sense.

Paradoxes of mathematics

Many people know from school that division by zero is impossible. But for some reason it is impossible to explain the reason for such a ban. In fact, why does the formula for dividing by zero 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 learn in primary school, in fact, are not nearly as equal as we think. All simple number operations can be reduced to two: addition and multiplication. These actions constitute the essence of the very concept of number, and other operations are built on the use of these two.

Addition and Multiplication

Let's take standard example for subtraction: 10-2=8. At school they consider it simply: if you subtract two from ten subjects, eight remain. But mathematicians look at this operation completely differently. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x+2=10. To mathematicians, the unknown difference is simply the number that needs to be added to two to make eight. And no subtraction is required here, you just need to find the appropriate numerical value.

Multiplication and division are treated the same. In the example 12:4=3 you can understand that we are talking about dividing eight objects into two equal piles. But in reality, this is just an inverted formula for writing 3x4 = 12. Such examples of division can be given endlessly.

Examples for division by 0

This is where it becomes a little clear why you can’t divide by zero. Multiplication and division by zero follow their own rules. All examples of dividing this quantity can be formulated as 6:0 = x. But this is an inverted notation 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 zero value.

It turns out that there is no such number that, when multiplied by 0, gives any tangible value, that is, this problem has no solution. You should not be afraid of this answer; it is a natural answer for problems of this type. It's just that the 6:0 record doesn't make any sense and it can't explain anything. In short, this expression can be explained by the immortal “division by zero is impossible.”

Is there a 0:0 operation? Indeed, if the operation of multiplication by 0 is legal, can zero be divided by zero? After all, an equation of the form 0x 5=0 is quite legal. Instead of the number 5 you can put 0, the product will not change.

Indeed, 0x0=0. But you still can't divide by 0. As stated, division is simply 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 just one from an infinite number of numbers. And if so, this 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 school math. Mathematical analysis studied in technical universities slightly expands the concept of problems that have no solution. For example, to already famous expression 0:0 new ones are added that do not have solutions 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 using elementary methods. But higher mathematics thanks additional features for a number of similar examples gives finite solutions. This is especially evident in the consideration of problems from the theory of limits.

Unlocking Uncertainty

In the theory of limits, the value 0 is replaced by a conditional infinitesimal variable. And expressions in which, when substituting the desired value, division by zero is obtained, are transformed. Below is a standard example of expanding a limit using ordinary algebraic transformations:

As you can see in the example, simply reducing a fraction leads its value to a completely rational answer.

When considering the limits trigonometric functions their expressions tend to be reduced to the first wonderful limit. When considering limits in which the denominator becomes 0 when a limit is substituted, a second remarkable limit is used.

L'Hopital method

In some cases, the limits of expressions can be replaced by the limits of their derivatives. Guillaume L'Hopital is a French mathematician, the 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 looks like this.

Currently, L'Hopital's method is successfully used to solve uncertainties of the 0:0 or?:? type.

How to divide and multiply by 0.1; 0.01; 0.001, etc.?

Write the rules for division and multiplication.

To multiply a number by 0.1, you just need to move the decimal point.

For example it was 56 , it became 5,6 .

To divide by the same number, you need to move the comma in the opposite direction:

For example it was 56 , it became 560 .

With the number 0.01 everything is the same, but you need to move it to 2 digits, not one.

In general, transfer as many zeros as you need.

For example, there is a number 123456789.

You need to multiply it by 0.000000001

There are nine zeros in the number 0.000000001 (we also count the zero to the left of the decimal point), which means we shift the number 123456789 by 9 digits:

It was 123456789 and now it is 0.123456789.

In order not to multiply, but to divide by the same number, we shift in the other direction:

It was 123456789 and now it is 123456789000000000.

To shift an integer this way, we simply add a zero to it. And in the fractional we move the comma.

Dividing a number by 0.1 corresponds to multiplying that number by 10

Dividing a number by 0.01 corresponds to multiplying that number by 100

Dividing by 0.001 is multiplying by 1000.

To make it easier to remember, we read the number by which we need to divide from right to left, not paying attention to the comma, and multiply by the resulting number.

Example: 50: 0.0001. This is the same as 50 multiplied by (read from right to left without a comma - 10000) 10000. It turns out 500000.

The same thing with multiplication, only in reverse:

400 x 0.01 is the same as dividing 400 by (read from right to left without a comma - 100) 100: 400: 100 = 4.

For those who find it more convenient to move commas to the right when dividing and to the left when multiplying when multiplying and dividing by such numbers, you can do this.

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5.5.6. Division by decimal

I. To divide a number by a decimal fraction, you need to move the commas in the dividend and divisor as many digits to the right as there are after the decimal point in the divisor, and then divide by the natural number.

Primary.

Perform division: 1) 16,38: 0,7; 2) 15,6: 0,15; 3) 3,114: 4,5; 4) 53,84: 0,1.

Solution.

Example 1) 16,38: 0,7.

In the divider 0,7 there is one digit after the decimal point, so let’s move the commas in the dividend and divisor one digit to the right.

Then we will need to divide 163,8 on 7 .

Let's perform the division according to the rule for dividing a decimal fraction by a natural number.

We divide as natural numbers are divided. How to remove the number 8 - the first digit after the decimal point (i.e. the digit in the tenths place), so immediately put a comma in the quotient and continue dividing.

Answer: 23.4.

Example 2) 15,6: 0,15.

We move commas in the dividend ( 15,6 ) and divisor ( 0,15 ) two digits to the right, since in the divisor 0,15 there are two digits after the decimal point.

We remember that you can add as many zeros as you like to the decimal fraction on the right, and this will not change the decimal fraction.

15,6:0,15=1560:15.

We perform division of natural numbers.

Answer: 104.

Example 3) 3,114: 4,5.

Move the commas in the dividend and divisor one digit to the right and divide 31,14 on 45 according to the rule for dividing a decimal fraction by a natural number.

3,114:4,5=31,14:45.

In the quotient we put a comma as soon as we remove the number 1 in the tenth place. Then we continue dividing.

To complete the division we had to assign zero to the number 9 - differences between numbers 414 And 405 . (we know that zeros can be added to the right side of a decimal fraction)

Answer: 0.692.

Example 4) 53,84: 0,1.

Move the commas in the dividend and divisor to 1 number to the right.

We get: 538,4:1=538,4.

Let's analyze the equality: 53,84:0,1=538,4. Pay attention to the comma in the dividend in this example and the comma in the resulting quotient. We notice that the comma in the dividend has been moved to 1 number to the right, as if we were multiplying 53,84 on 10. (See the video “Multiplying a decimal by 10, 100, 1000, etc.”) Hence the rule for dividing a decimal by 0,1; 0,01; 0,001 etc.

II. To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1, 0.01, 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)

Examples.

Perform division: 1) 617,35: 0,1; 2) 0,235: 0,01; 3) 2,7845: 0,001; 4) 26,397: 0,0001.

Solution.

Example 1) 617,35: 0,1.

According to the rule II division by 0,1 is equivalent to multiplying by 10 , and move the comma in the dividend 1 digit to the right:

1) 617,35:0,1=6173,5.

Example 2) 0,235: 0,01.

Division by 0,01 is equivalent to multiplying by 100 , which means we move the comma in the dividend on 2 digits to the right:

2) 0,235:0,01=23,5.

Example 3) 2,7845: 0,001.

Because division by 0,001 is equivalent to multiplying by 1000 , then move the comma 3 digits to the right:

3) 2,7845:0,001=2784,5.

Example 4) 26,397: 0,0001.

Divide a decimal by 0,0001 - it's the same as multiplying it by 10000 (move the comma by 4 digits right). We get:

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Multiplication and division by numbers of the form 10, 100, 0.1, 0.01

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This lesson will look at how to perform multiplication and division by numbers of the form 10, 100, 0.1, 0.001. Will also be decided various examples on this topic.

Multiplying numbers by 10, 100

Exercise. How to multiply the number 25.78 by 10?

The decimal notation of a given number is a shorthand notation for the amount. It is necessary to describe it in more detail:

Thus, you need to multiply the amount. To do this, you can simply multiply each term:

It turns out that...

We can conclude that multiplying a decimal fraction by 10 is very simple: you need to move the decimal point to the right one position.

Exercise. Multiply 25.486 by 100.

Multiplying by 100 is the same as multiplying by 10 twice. In other words, you need to move the decimal point to the right twice:

Dividing numbers by 10, 100

Exercise. Divide 25.78 by 10.

As in the previous case, you need to present the number 25.78 as a sum:

Since you need to divide the sum, this is equivalent to dividing each term:

It turns out that to divide by 10, you need to move the decimal point to the left one position. For example:

Exercise. Divide 124.478 by 100.

Dividing by 100 is the same as dividing by 10 twice, so the decimal point is moved to the left by 2 places:

Rule of multiplication and division by 10, 100, 1000

If a decimal fraction needs to be multiplied by 10, 100, 1000, and so on, you need to move the decimal point to the right by as many positions as there are zeros in the multiplier.

Conversely, if a decimal fraction needs to be divided by 10, 100, 1000, and so on, you need to move the decimal point to the left by as many positions as there are zeros in the multiplier.

Examples when it is necessary to move a comma, but there are no more numbers left

Multiplying by 100 means moving the decimal place two places to the right.

After the shift, you can find that there are no more digits after the decimal point, which means that the fractional part is missing. Then there is no need for a comma, the number is an integer.

You need to move 4 positions to the right. But there are only two digits after the decimal point. It's worth remembering that there is an equivalent notation for the fraction 56.14.

Now multiplying by 10,000 is easy:

If it is not very clear why you can add two zeros to the fraction in the previous example, then the additional video at the link can help with this.

Equivalent decimal notations

Entry 52 means the following:

If we put 0 in front, we get entry 052. These entries are equivalent.

Is it possible to put two zeros in front? Yes, these entries are equivalent.

Now let's look at the decimal fraction:

If you assign zero, you get:

These entries are equivalent. Similarly, you can assign multiple zeros.

Thus, any number can have several zeros after the fractional part and several zeros before the integer part. These will be equivalent entries of the same number.

Since division by 100 occurs, it is necessary to move the decimal point 2 positions to the left. There are no numbers left to the left of the decimal point. Whole part absent. This notation is often used by programmers. In mathematics, if there is no whole part, then they put a zero in its place.

You need to move it to the left by three positions, but there are only two positions. If you write several zeros in front of a number, it will be an equivalent notation.

That is, when shifting to the left, if the numbers run out, you need to fill them with zeros.

In this case, it is worth remembering that a comma always comes after the whole part. Then:

Multiplying and dividing by 0.1, 0.01, 0.001

Multiplying and dividing by numbers 10, 100, 1000 is a very simple procedure. The situation is exactly the same with the numbers 0.1, 0.01, 0.001.

Example. Multiply 25.34 by 0.1.

Let's write the decimal fraction 0.1 as an ordinary fraction. But multiplying by is the same as dividing by 10. Therefore, you need to move the decimal point 1 position to the left:

Similarly, multiplying by 0.01 is dividing by 100:

Example. 5.235 divided by 0.1.

The solution to this example is constructed in a similar way: 0.1 is expressed as common fraction, and dividing by is the same as multiplying by 10:

That is, to divide by 0.1, you need to move the decimal point to the right one position, which is equivalent to multiplying by 10.

Rule of multiplication and division by 0.1, 0.01, 0.001

Multiplying by 10 and dividing by 0.1 is the same thing. The comma must be moved to the right by 1 position.

Dividing by 10 and multiplying by 0.1 are the same thing. The comma needs to be moved to the right by 1 position:

Solving Examples

Conclusion

In this lesson, the rules of division and multiplication by 10, 100 and 1000 were studied. In addition, the rules of multiplication and division by 0.1, 0.01, 0.001 were examined.

Examples of the application of these rules were reviewed and resolved.

Bibliography

1. Vilenkin N.Ya. Mathematics: textbook. for 5th grade. general education uchr. 17th ed. – M.: Mnemosyne, 2005.

2. Shevkin A.V. Math word problems: 5–6. – M.: Ilexa, 2011.

3. Ershova A.P., Goloborodko V.V. All school mathematics in independent and tests. Math 5–6. – M.: Ilexa, 2006.

4. Khlevnyuk N.N., Ivanova M.V. Formation of computing skills in mathematics lessons. 5–9 grades. – M.: Ilexa, 2011 .

1. Internet portal “Festival of Pedagogical Ideas” (Source)

2. Internet portal “Matematika-na.ru” (Source)

3. Internet portal “School.xvatit.com” (Source)

Homework

3. Compare the meanings of the expressions:

Actions with zero

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”.

WITH zero all arithmetic operations are carried out, and integers, ordinary and decimals, and all of them can have both positive and negative meanings. Let us give examples of their implementation and some explanations for them.

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

twenty four plus zero equals twenty-four.

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

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In mathematics, division by zero is impossible! One way to explain this rule is to analyze the process, which shows what happens when one number is divided by another.

Division by zero error in Excel

In reality, division is essentially the same as subtraction. For example, dividing the number 10 by 2 is repeatedly subtracting 2 from 10. The repetition is repeated until the result is equal to 0. Thus, it is necessary to subtract the number 2 from ten exactly 5 times:

1. 10-2=8
2. 8-2=6
3. 6-2=4
4. 4-2=2
5. 2-2=0

If we try to divide the number 10 by 0, we will never get the result equal to 0, since when subtracting 10-0 there will always be 10. An infinite number of times subtracting zero from ten will not lead us to the result =0. There will always be the same result after the subtraction operation =10:

• 10-0=10
• 10-0=10
• 10-0=10
• ∞ infinity.

On the sidelines of mathematicians they say that the result of dividing any number by zero is “unlimited.” Any computer program that tries to divide by 0 simply returns an error. In Excel, this error is indicated by the value in the cell #DIV/0!.

But if necessary, you can work around the division by 0 error in Excel. You should simply skip the division operation if the denominator contains the number 0. The solution is implemented by placing the operands in the arguments of the =IF() function:

Thus, the Excel formula allows us to “divide” a number by 0 without errors. When dividing any number by 0, the formula will return the value 0. That is, we get the following result after division: 10/0=0.



How does the formula for eliminating division by zero error work?

To work correctly, the IF function requires filling in 3 of its arguments:

1. Logical condition.
2. Actions or values ​​that will be performed if the Boolean condition returns TRUE.
3. Actions or values ​​that will be performed when a Boolean condition returns FALSE.

In this case, the conditional argument contains a value check. Are the cell values ​​in the Sales column equal to 0? The first argument of the IF function must always have comparison operators between two values ​​to produce the result of the condition as TRUE or FALSE. In most cases, the equal sign is used as a comparison operator, but others can be used, such as greater than > or less than >. Or their combinations – greater than or equal to >=, not equal!=.

If the condition in the first argument returns TRUE, then the formula will fill the cell with the value from the second argument of the IF function. In this example, the second argument contains the number 0 as its value. This means that the cell in the “Execution” column will simply be filled with the number 0 if there are 0 sales in the cell opposite from the “Sales” column.

If the condition in the first argument returns FALSE, then the value from the third argument of the IF function is used. In this case, this value is formed after dividing the indicator from the “Sales” column by the indicator from the “Plan” column.

Formula for dividing by zero or zero by a number

Let's complicate our formula with the =OR() function. Let's add another sales agent with zero sales. Now the formula should be changed to:

Copy this formula to all cells in the Progress column:

Now, no matter where the zero is in the denominator or in the numerator, the formula will work as the user needs.

Here's another interesting statement. “You can’t divide by zero!” - Most schoolchildren learn this rule by heart, without asking questions. All children know what “you can’t” is and what will happen if you ask in response to it: “Why?” This is what will happen if

But in fact, it is very interesting and important to know why it is not possible.

The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as valid - addition and multiplication. These operations and their properties are included in the very definition of the concept of number. All other actions are built in one way or another from these two.

Consider, for example, subtraction. What does 5 – 3 mean? The student will answer this simply: you need to take five objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem completely differently. There is no subtraction, there is only addition. Therefore, the notation 5 – 3 means a number that, when added to the number 3, will give the number 5. That is, 5 – 3 is simply an abbreviated notation of the equation: x + 3 = 5. There is no subtraction in this equation. There is only a task - to find a suitable number.

The same is true with multiplication and division. Entry 8:4 can be understood as the result of dividing eight items into four equal piles. But it's really just a shortened form of the equation 4 x = 8.

This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Recording 5: 0 is an abbreviation for 0 x = 5. That is, this task is to find a number that, when multiplied by 0, will give 5. But we know that when multiplied by 0, the result is always 0. This is an inherent property of zero, strictly speaking , part of its definition.

There is no such number that when multiplied by 0 will give something other than zero. That is, our problem has no solution. (Yes, this happens; not every problem has a solution.) This means that the entry 5:0 does not correspond to any specific number, and it simply does not mean anything and therefore has no meaning. The meaninglessness of this entry is briefly expressed by saying that you cannot divide by zero.

The most attentive readers in this place will certainly ask: is it possible to divide zero by zero? Indeed, the equation 0 x = 0 can be solved safely. For example, we can take x = 0, and then we get 0 · 0 = 0. So, 0: 0=0? But let's not rush. Let's try to take x = 1. We get 0 · 1 = 0. Correct? So 0:0 = 1? But this way you can take any number and get 0: 0 = 5, 0: 0 = 317, etc.

But if any number is suitable, then we have no reason to choose any one of them. That is, we cannot say to which number the entry 0:0 corresponds. And if so, then we are forced to admit that this entry also makes no sense. It turns out that even zero cannot be divided by zero. (In mathematical analysis there are cases when, due to additional conditions of the problem, one can give preference to one of possible options solutions to the equation 0 x = 0; In such cases, mathematicians talk about “unfolding uncertainty,” but such cases do not occur in arithmetic.)

This is the peculiarity of the division operation. More precisely, the operation of multiplication and the number associated with it have zero.

Well, the most meticulous ones, having read this far, may ask: why does it happen that you can’t divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. You can answer it only by becoming familiar with the formal mathematical definitions of numerical sets and operations on them.

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 can't divide by zero because it doesn't make any 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 technical universities You 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.