Is it possible to divide by zero? Mathematician answers. Why can't you divide by zero? An illustrative example Any number multiplied by 0 equals

Very often, many people wonder why it is impossible to use division by zero? In this article, we will go into great detail about where this rule came from, as well as what actions can be performed with zero.

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Zero can be called one of the most interesting numbers. This number has no meaning, it means emptiness in the truest sense of the word. However, if you put zero next to any digit, then the value of this digit will become several times larger.

The number is very mysterious in itself. It was used by the ancient Mayan people. For the Maya, zero meant "beginning", and the countdown of calendar days also started from zero.

A very interesting fact is that the sign of zero and the sign of uncertainty were similar for them. By this, the Maya wanted to show that zero is the same identical sign as uncertainty. In Europe, the designation of zero appeared relatively recently.

Also, many people know the prohibition associated with zero. Any person will say that cannot be divided by zero. This is said by teachers at school, and children usually take their word for it. Usually, children are either simply not interested in knowing this, or they know what will happen if, upon hearing an important prohibition, they immediately ask “Why can’t you divide by zero?”. But when you get older, interest awakens, and you want to know more about the reasons for such a ban. However, there is reasonable evidence.

Actions with zero

First you need to determine what actions can be performed with zero. Exist several types of activities:

• Addition;
• Multiplication;
• Subtraction;
• Division (zero by number);
• Exponentiation.

Important! If zero is added to any number during addition, then this number will remain the same and will not change its numerical value. The same thing happens if you subtract zero from any number.

With multiplication and division, things are a little different. If a multiply any number by zero, then the product will also become zero.

Consider an example:

Let's write this as an addition:

There are five added zeros in total, so it turns out that

Let's try to multiply one by zero. The result will also be null.

Zero can also be divided by any other number not equal to it. In this case, it will turn out, the value of which will also be zero. The same rule applies to negative numbers. If you divide zero by a negative number, you get zero.

You can also raise any number to zero power. In this case, you get 1. It is important to remember that the expression "zero to the zero power" is absolutely meaningless. If you try to raise zero to any power, you get zero. Example:

We use the multiplication rule, we get 0.

Is it possible to divide by zero

So, here we come to the main question. Is it possible to divide by zero generally? And why is it impossible to divide a number by zero, given that all other operations with zero fully exist and apply? To answer this question, you need to turn to higher mathematics.

Let's start with the definition of the concept, what is zero? School teachers claim that zero is nothing. Emptiness. That is, when you say that you have 0 pens, it means that you have no pens at all.

In higher mathematics, the concept of "zero" is broader. It doesn't mean empty at all. Here, zero is called uncertainty, because if you do a little research, it turns out that by dividing zero by zero, we can get any other number as a result, which may not necessarily be zero.

Do you know that those simple arithmetic operations that you studied at school are not so equal among themselves? The most basic steps are addition and multiplication.

For mathematicians, the concepts of "" and "subtraction" do not exist. Suppose: if three are subtracted from five, then two will remain. This is what subtraction looks like. However, mathematicians would write it this way:

Thus, it turns out that the unknown difference is a certain number that needs to be added to 3 to get 5. That is, you don’t need to subtract anything, you just need to find a suitable number. This rule applies to addition.

Things are a little different with multiplication and division rules. It is known that multiplication by zero leads to zero result. For example, if 3:0=x, then if you flip the record, you get 3*x=0. And the number that is multiplied by 0 will give zero in the product. It turns out that a number that would give any value other than zero in the product with zero does not exist. This means that division by zero is meaningless, that is, it fits our rule.

But what happens if you try to divide zero by itself? Let's take x as some indefinite number. It turns out the equation 0 * x \u003d 0. It can be solved.

If we try to take zero instead of x, we get 0:0=0. It would seem logical? But if we try to take any other number instead of x, for example, 1, then we end up with 0:0=1. The same situation will be if you take any other number and plug it into the equation.

In this case, it turns out that we can take any other number as a factor. The result will be an infinite number of different numbers. Sometimes, nevertheless, division by 0 in higher mathematics makes sense, but then usually there is a certain condition due to which we can still choose one suitable number. This action is called "uncertainty disclosure". In ordinary arithmetic, division by zero will again lose its meaning, since we will not be able to choose any one number from the set.

Important! Zero cannot be divided by zero.

Zero and infinity

Infinity is very common in higher mathematics. Since it is simply not important for schoolchildren to know that there are still mathematical operations with infinity, teachers cannot properly explain to children why it is impossible to divide by zero.

Students begin to learn the basic mathematical secrets only in the first year of the institute. Higher mathematics provides a large set of problems that have no solution. The most famous problems are the problems with infinity. They can be solved with mathematical analysis.

You can also apply to infinity elementary mathematical operations: addition, multiplication by a number. Subtraction and division are also commonly used, but in the end they still come down to two simple operations.

But what will if you try:

• Multiply infinity by zero. In theory, if we try to multiply any number by zero, we will get zero. But infinity is an indefinite set of numbers. Since we cannot choose one number from this set, the expression ∞*0 has no solution and is absolutely meaningless.
• Zero divided by infinity. This is the same story as above. We can’t choose one number, which means we don’t know what to divide by. The expression doesn't make sense.

Important! Infinity is a little different from uncertainty! Infinity is a type of uncertainty.

Now let's try to divide infinity by zero. It would seem that there should be uncertainty. But if we try to replace division with multiplication, we get a very definite answer.

For example: ∞/0=∞*1/0= ∞*∞ = ∞.

It turns out like this mathematical paradox.

Why you can't divide by zero

Thought experiment, try to divide by zero

Conclusion

So, now we know that zero is subject to almost all operations that are performed with, except for one single one. You can't divide by zero just because the result is uncertainty. We also learned how to operate on zero and infinity. The result of such actions will be uncertainty.

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 inability to divide 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. The use of the number by Europeans is relatively recent, 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 mentioned, 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 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 is as follows.

At present, the L'Hopital method is successfully used in solving uncertainties of the type 0:0 or ?:?.

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 comma.

For example it was 56 , became 5,6 .

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

For example it was 56 , became 560 .

With the number 0.01, everything is the same, but you need to transfer it to 2 characters, and not to one.

In general, how many zeros, so much and transfer.

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 became 0.123456789.

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

It was 123456789 became 123456789000000000.

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

Dividing a number by 0.1 is equivalent to multiplying that number by 10

Dividing a number by 0.01 is equivalent 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, ignoring the comma, and multiply by the resulting number.

Example: 50: 0.0001. It's like multiplying 50 by (read from right to left without a comma - 10000) 10000. It turns out 500000.

The same 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.

Whoever finds it more convenient to transfer commas to the right when dividing and to the left when multiplying when multiplying and dividing by such numbers can do so.

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5.5.6. Decimal division

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

Primery.

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

Decision.

Example 1) 16,38: 0,7.

In the divider 0,7 there is one digit after the decimal point, therefore, we will move the commas in the dividend and divisor one digit to the right.

Then we will need to share 163,8 on the 7 .

Perform division according to the rule of dividing a decimal fraction by a natural number.

We divide as we divide natural numbers. How to take down the number 8 - the first digit after the decimal point (i.e. the digit in the tenth place), so immediately put a private comma and continue dividing.

Answer: 23.4.

Example 2) 15,6: 0,15.

Move commas in 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.

Remember that as many zeros as you like can be assigned to the decimal fraction on the right, and the decimal fraction will not change from this.

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

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 the 45 according to the rule of dividing a decimal fraction by a natural number.

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

In private, put a comma as soon as we demolish the figure 1 in the tenth place. Then we continue the division.

To complete the division we had to assign zero to the number 9 - difference of numbers 414 and 405 . (we know that zeros can be assigned to the decimal fraction on the right)

Answer: 0.692.

Example 4) 53,84: 0,1.

We transfer commas in the dividend and the divisor by 1 number to the right.

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

Let's analyze the equality: 53,84:0,1=538,4. We pay attention to the comma in the dividend in this example and to the comma in the resulting quotient. Note that the comma in the dividend has been moved to 1 digit to the right, as if we were multiplying 53,84 on the 10. (Watch 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 comma 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.

Decision.

Example 1) 617,35: 0,1.

According to the rule II division into 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 that we will transfer the comma in the dividend on the 2 digits to the right:

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

Example 3) 2,7845: 0,001.

As division into 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 decimal by 0,0001 is the same as multiplying it by 10000 (move a comma by 4 digits right). We get:

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Multiplication and division by numbers like 10, 100, 0.1, 0.01

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In this lesson, we will look at how to perform multiplication and division by numbers like 10, 100, 0.1, 0.001. Various examples on this topic will also be solved.

Multiplying numbers by 10, 100

An exercise. How to multiply the number 25.78 by 10?

The decimal notation for a given number is an abbreviated notation for the sum. You need 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 by 10 is very simple: you need to shift the comma to the right by one position.

An exercise. Multiply 25.486 by 100.

Multiplying by 100 is the same as multiplying twice by 10. In other words, you need to shift the comma to the right two times:

Division of numbers by 10, 100

An exercise. Divide 25.78 by 10.

As in the previous case, it is necessary to represent 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 comma to the left by one position. For example:

An exercise. Divide 124.478 by 100.

Dividing by 100 is the same as dividing by 10 twice, so the comma is shifted 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 shift the comma to the right as many positions as there are zeros in the multiplier.

And vice versa, if the decimal fraction needs to be divided by 10, 100, 1000, and so on, you need to shift the comma to the left as many positions as there are zeros in the multiplier.

Examples when you need to move a comma, but there are no more digits

Multiplying by 100 means shifting the decimal point to the right by two places.

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 the comma is not needed, the number turned out to be an integer.

You need to move 4 positions to the right. But there are only two digits after the decimal point. It is 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 Entries

Entry 52 means the following:

If we put 0 in front, we get record 052. These records are equivalent.

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

Now let's look at the decimal:

If we assign zero, then we get:

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

Thus, any number can be assigned 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 shift the comma 2 positions to the left. There are no digits to the left of the decimal point. The whole part is missing. This notation is often used by programmers. In mathematics, if there is no integer part, then put zero instead of it.

You need to shift to the left by three positions, but there are only two positions. If you write several zeros before the number, then this will be an equivalent notation.

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

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

Multiplication and division by 0.1, 0.01, 0.001

Multiplication and division by the numbers 10, 100, 1000 is a very simple procedure. The same is true with the numbers 0.1, 0.01, 0.001.

Example. Multiply 25.34 by 0.1.

Let's write the decimal fraction 0.1 in the form of an ordinary. But multiplying by is the same as dividing by 10. Therefore, you need to move the comma 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 an ordinary fraction, and dividing by is the same as multiplying by 10:

That is, to divide by 0.1, you need to shift the comma to the right by one position, which is equivalent to multiplying by 10.

Rule for multiplying and dividing by 0.1, 0.01, 0.001

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

Divide by 10 and multiply by 0.1 is the same thing. The comma needs to be shifted to the right by 1 position:

Solution of examples

Conclusion

In this lesson, the rules for dividing and multiplying by 10, 100, and 1000 were studied. In addition, the rules for multiplying and dividing by 0.1, 0.01, 0.001 were considered.

Examples on the application of these rules were considered and decided.

Bibliography

1. Vilenkin N. Ya. Mathematics: textbook. for 5 cells. general const. 17th ed. – M.: Mnemosyne, 2005.

2. Shevkin A.V. Word Problems in Mathematics: 5–6. – M.: Ileksa, 2011.

3. Ershova A.P., Goloborodko V.V. All school mathematics in independent and control works. Mathematics 5–6. – M.: Ileksa, 2006.

4. Khlevnyuk N.N., Ivanova M.V. Formation of computational skills in mathematics lessons. 5th-9th grades. – M.: Ileksa, 2011 .

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2. Internet portal "Matematika-na.ru" (Source)

3. Internet portal "School.xvatit.com" (Source)

Homework

3. Compare expression values:

Actions with zero

In mathematics, the number zero occupies a special place. The fact is that it, in fact, 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 countdown of the coordinates of the point position in any coordinate system begins.

Zero widely used in decimals to determine the values ​​of "blank" digits, both before and after the decimal point. In addition, one of the fundamental rules of arithmetic is associated with it, which says that on zero cannot be divided. His logic, in fact, stems from the very essence of this number: indeed, it is impossible to imagine that some value different from it (and it itself, too) was divided into “nothing”.

With zero all arithmetic operations are carried out, and integers, ordinary and decimal fractions can be used as its "partners", and all of them can have both positive and negative values. We give examples of their implementation and some explanations for them.

When adding zero to some number (both whole and fractional, both positive and negative), its value remains absolutely unchanged.

twenty four plus zero equals twenty-four.

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

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In this lesson, we will look at how to perform multiplication and division by numbers like 10, 100, 0.1, 0.001. Various examples on this topic will also be solved.

An exercise. How to multiply the number 25.78 by 10?

The decimal notation for a given number is an abbreviated notation for the sum. You need 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 by 10 is very simple: you need to shift the comma to the right by one position.

An exercise. Multiply 25.486 by 100.

Multiplying by 100 is the same as multiplying twice by 10. In other words, you need to shift the comma to the right two times:

An exercise. Divide 25.78 by 10.

As in the previous case, it is necessary to represent 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 comma to the left by one position. For example:

An exercise. Divide 124.478 by 100.

Dividing by 100 is the same as dividing by 10 twice, so the comma is shifted to the left by 2 places:

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

And vice versa, if the decimal fraction needs to be divided by 10, 100, 1000, and so on, you need to shift the comma to the left as many positions as there are zeros in the multiplier.

Example 1

Multiplying by 100 means shifting the decimal point to the right by two places.

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 the comma is not needed, the number turned out to be an integer.

Example 2

You need to move 4 positions to the right. But there are only two digits after the decimal point. It is 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 Entries

Entry 52 means the following:

If we put 0 in front, we get record 052. These records are equivalent.

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

Now let's look at the decimal:

If we assign zero, then we get:

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

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

Example 3

Since division by 100 occurs, it is necessary to shift the comma 2 positions to the left. There are no digits to the left of the decimal point. The whole part is missing. This notation is often used by programmers. In mathematics, if there is no integer part, then put zero instead of it.

Example 4

You need to shift to the left by three positions, but there are only two positions. If you write several zeros before the number, then this will be an equivalent notation.

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

Example 5

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

Multiplication and division by numbers 10, 100, 1000 is a very simple procedure. The same is true with the numbers 0.1, 0.01, 0.001.

Example. Multiply 25.34 by 0.1.

Let's write the decimal fraction 0.1 in the form of an ordinary. But multiplying by is the same as dividing by 10. Therefore, you need to move the comma 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 an ordinary fraction, and dividing by is the same as multiplying by 10:

That is, to divide by 0.1, you need to shift the comma to the right by one position, which is equivalent to multiplying by 10.

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

Divide by 10 and multiply by 0.1 is the same thing. The comma needs to be shifted to the right by 1 position:

Division by zero in mathematics, a division at which the divisor is zero. Such a division can be formally written as ⁄ 0, where is the dividend.

In ordinary arithmetic (with real numbers), this expression does not make sense, because:

• at ≠ 0, there is no number that, when multiplied by 0, gives, therefore, no number can be taken as a quotient ⁄ 0;
• at = 0, division by zero is also undefined, since any number, when multiplied by 0, gives 0 and can be taken as a quotient 0 ⁄ 0.

Historically, one of the first references to the mathematical impossibility of assigning the value ⁄ 0 is in George Berkeley's criticism of infinitesimal calculus.

Logic errors

Since when multiplying any number by zero, we always get zero as a result, when dividing both parts of the expression × 0 = × 0, which is true regardless of the value of and, by 0, we get the expression = , which is incorrect in the case of arbitrarily given variables. Since zero can be given implicitly, but in the form of a rather complex mathematical expression, for example, in the form of the difference between two values ​​reduced to each other by algebraic transformations, such a division can be a rather unobvious mistake. The imperceptible introduction of such a division into the proof process in order to show the identity of obviously different quantities, thereby proving any absurd statement, is one of the varieties of mathematical sophism.

In computer science

In programming, depending on the programming language, data type, and value of the dividend, an attempt to divide by zero can lead to different consequences. The consequences of division by zero in integer and real arithmetic are fundamentally different:

• Attempt integer division by zero is always a critical error that makes it impossible to continue executing the program. It leads either to throwing an exception (which the program can handle itself, thereby avoiding an emergency stop), or to immediately stop the program with a fatal error message and, possibly, the contents of the call stack. In some programming languages, such as Go, an integer division by a zero constant is considered a syntax error and will cause the program to compile abort.
• AT real arithmetic consequences can be different in different languages:

• throwing an exception or stopping the program, as with integer division;
• obtaining a special non-numeric value as a result of the operation. In this case, the calculations are not interrupted, and their result can subsequently be interpreted by the program itself or by the user as a meaningful value or as evidence of incorrect calculations. The principle is widely used, according to which, when dividing the form ⁄ 0, where ≠ 0 is a floating point number, the result is equal to positive or negative (depending on the sign of the dividend) infinity - or, and when = 0, the result is a special value NaN (abbreviated from English not a number - “not a number”). This approach is adopted in the IEEE 754 standard, which is supported by many modern programming languages.

Random division by zero in a computer program can sometimes cause costly or dangerous failures in the equipment controlled by the program. For example, on September 21, 1997, a division by zero in the computerized control system of the USS Yorktown (CG-48) US Navy cruiser shut down all electronic equipment in the system, causing the ship's power plant to stop working.

see also

Notes

Function = 1 ⁄ . When tends to zero from the right, tends to infinity; when tends to zero from the left, tends to minus infinity

If you divide any number by zero on a conventional calculator, then it will give you the letter E or the word Error, that is, “error”.

The computer calculator in a similar case writes (in Windows XP): "Division by zero is prohibited."

Everything is consistent with the rule known from school that you cannot divide by zero.

Let's see why.

Division is the mathematical operation that is the inverse of multiplication. Division is defined through multiplication.

Divide a number a(dividend, for example 8) by a number b(divisor, for example, the number 2) - means to find such a number x(quotient), when multiplied by a divisor b it turns out divisible a(4 2 = 8), i.e. a divide by b means to solve the equation x · b = a.

The equation a: b = x is equivalent to the equation x · b = a.

We replace division with multiplication: instead of 8: 2 = x we ​​write x 2 = 8.

8: 2 = 4 is equivalent to 4 2 = 8

18: 3 = 6 is equivalent to 6 3 = 18

20: 2 = 10 is equivalent to 10 2 = 20

The result of division can always be checked by multiplication. The result of multiplying a divisor by a quotient must be the dividend.

Similarly, let's try to divide by zero.

For example, 6: 0 = ... We need to find a number that, when multiplied by 0, will give 6. But we know that when multiplied by zero, zero is always obtained. There is no number that, when multiplied by zero, would give something other than zero.

When they say that it is impossible or forbidden to divide by zero, it means that there is no number corresponding to the result of such a division (it is possible to divide by zero, but not to divide :)).

Why do they say in school that you can't divide by zero?

Therefore, in definition operations of dividing a by b, it is immediately emphasized that b ≠ 0.

If everything written above seemed too complicated for you, then it’s completely on your fingers: Dividing 8 by 2 means finding out how many twos you need to take to get 8 (answer: 4). Dividing 18 by 3 means to find out how many triples you need to take to get 18 (answer: 6).

Dividing 6 by zero means finding out how many zeros you need to take to get 6. No matter how many zeros you take, you still get zero, but you never get 6, i.e. division by zero is not defined.

An interesting result is obtained if you try to divide the number by zero on the android calculator. The screen will display ∞ (infinity) (or - ∞ if you divide by a negative number). This result is incorrect, since there is no number ∞. Apparently, programmers have confused completely different operations - dividing numbers and finding the limit of a numerical sequence n / x, where x → 0. When dividing zero by zero, NaN (Not a Number - Not a number) will be written.

"You can't divide by zero!" - Most students memorize this rule by heart, without asking questions. All children know what “no” is and what will happen if you ask in response to it: “Why?” But in fact, it is very interesting and important to know why it is impossible.

The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as full-fledged - 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 means 5 - 3 ? The student will answer this simply: you need to take five items, take away (remove) three of them and see how many remain. But mathematicians look at this problem in a completely different way. There is no subtraction, only addition. Therefore, the entry 5 - 3 means a number that, when added to a number 3 will give the number 5 . I.e 5 - 3 is just a shorthand for the equation: x + 3 = 5. There is no subtraction in this equation.

Division by zero

There is only a task - to find a suitable number.

The same is true with multiplication and division. Recording 8: 4 can be understood as the result of the division of eight objects 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 always turns out 0 . This is an inherent property of zero, strictly speaking, part of its definition.

A number that, when multiplied by 0 will give something other than null, just doesn't exist. That is, our problem has no solution. (Yes, it happens, not every problem has a solution.) 5: 0 does not correspond to any specific number, and it simply does not stand for anything and therefore does not make sense. The meaninglessness of this entry is briefly expressed by saying that you cannot divide by zero.

The most attentive readers at this point will certainly ask: is it possible to divide zero by zero?

Indeed, since the equation 0 x = 0 successfully resolved. For example, you can take x=0, and then we get 0 0 = 0. It turns out 0: 0=0 ? But let's not rush. Let's try to take x=1. Get 0 1 = 0. Correctly? Means, 0: 0 = 1 ? But 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 opt for any one of them. That is, we cannot tell which number corresponds to the entry 0: 0 . And if so, then we are forced to admit that this record also does not make 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 the possible options for solving the equation 0 x = 0; in such cases, mathematicians speak of "disclosure of indeterminacy", but in arithmetic such cases do not occur.)

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

Well, the most meticulous, having read up to this point, may ask: why is it so that you cannot divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. It can be answered only by getting acquainted with the formal mathematical definitions of numerical sets and operations on them. It is not so difficult, but for some reason it is not studied at school. But in lectures on mathematics at the university, you will be taught this in the first place.

The division function is not defined for a range where the divisor is zero. You can divide, but the result is not defined

You can't delt by zero. Mathematics 2 classes of high school.

If my memory serves me right, then zero can be represented as an infinitesimal value, so there will be infinity. And the school "zero - nothing" is just a simplification, there are so many of them in school mathematics. But without them in any way, everything in due time.

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Division by zero

Private from division by zero there is no number other than zero.

The reasoning here is as follows: since in this case no number can satisfy the definition of a quotient.

Let's write, for example,

whatever number you take for testing (say, 2, 3, 7), it is not good because:

\[ 2 0 = 0 \]

\[ 3 0 = 0 \]

\[ 7 0 = 0 \]

What happens if you divide by 0?

etc., but you need to get in the product 2,3,7.

We can say that the problem of dividing by zero a number other than zero has no solution. However, a number other than zero can be divided by a number arbitrarily close to zero, and the closer the divisor is to zero, the larger the quotient will be. So if we divide 7 by

\[ \frac(1)(10), \frac(1)(100), \frac(1)(1000), \frac(1)(10000) \]

then we get private 70, 700, 7000, 70,000, etc., which increase indefinitely.

Therefore, it is often said that the quotient of dividing 7 by 0 is "infinitely large", or "equal to infinity", and they write

\[7:0 = \infin\]

The meaning of this expression is that if the divisor approaches zero, and the dividend remains equal to 7 (or approaches 7), then the quotient increases indefinitely.

In mathematics, the number zero occupies a special place. The fact is that it, in fact, 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 countdown of the coordinates of the point position in any coordinate system begins.

Zero widely used in decimals to determine the values ​​of "blank" digits, both before and after the decimal point. In addition, one of the fundamental rules of arithmetic is associated with it, which says that on zero cannot be divided. His logic, in fact, stems from the very essence of this number: indeed, it is impossible to imagine that some value different from it (and it itself, too) was divided into “nothing”.

Calculation examples

With zero all arithmetic operations are carried out, and integers, ordinary and decimal fractions can be used as its "partners", and all of them can have both positive and negative values. We give examples of their implementation and some explanations for them.

ADDITION

When adding zero to some number (both whole 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 eighth plus zero equals seventeen point three eighths.

MULTIPLICATION

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

Example 1

five hundred and eighty six times zero equals zero.

Example 2

Zero times one hundred thirty-five point six equals zero.

Example 3

Zero multiply by zero equals zero.

DIVISION

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

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

Example 1

Zero divided by two hundred and sixty five equals zero.

Example 2

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

0:

= 0

Share zero to zero according to the rules of mathematics is impossible. This means that when such a procedure is performed, the quotient is indeterminate. Thus, theoretically, it can be absolutely any number.

0: 0 = 8 because 8 × 0 = 0

In mathematics, a problem like divide 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 indicated that may affect the final result.

Those, if any, should be to indicate 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 it is defined, then an expression like zero divide by zero, in the vast majority of cases, some meaning can be given.