In mathematics, division by zero is impossible! One way to explain this rule is by analyzing 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 10 by 2 is multiple subtraction of 2 from 10. The repetition is repeated until the result equals 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, because 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 \u003d 0. There will always be one and the same result after the subtraction operation \u003d 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, when trying to divide by 0, it just returns an error. In Excel, this error is displayed by the value in the # DIV / 0! Cell.

However, if necessary, you can work around the division by 0 error in Excel. You just need to skip the division operation if the denominator contains the number 0. The solution is implemented by placing operands in the arguments of the function \u003d IF ():

In this way excel formula allows us to "divide" a number by 0 without error. When dividing any number by 0, the formula will return the value 0. That is, we get the following result after division: 10/0 \u003d 0.



How the formula works to eliminate division by zero error

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

1. Boolean condition.
2. Actions or values \u200b\u200bto be performed if the result of the Boolean condition returns TRUE.
3. Actions or values \u200b\u200bto be performed when the boolean condition returns FALSE.

In this case, the conditional argument contains validation of values. Whether the cell values \u200b\u200bin the Sales column are 0. The first argument of an IF function must always have comparison operators between two values \u200b\u200bin order to get the result of the condition as TRUE or FALSE. In most cases, the equal sign is used as the comparison operator, but others can be used, such as greater than\u003e or less than\u003e. Or combinations thereof - greater than or equal to\u003e \u003d, not equal to! \u003d.

If the condition in the first argument returns TRUE, then the formula will fill the cell with the value from the second argument to the IF function. In this example, the second argument contains the number 0 as its value. This means that the cell in the "Progress" column will simply be filled with the number 0 if there will be 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 division by zero or zero by a number

Let's complicate our formula with the function \u003d OR (). Let's add another salesperson with zero sales. Now the formula should be changed to:

Copy this formula to all cells in the Progress column:

Now, regardless of where the zero is in the denominator or in the numerator, the formula will work as the user needs.

Textbook: "Mathematics" by M. I. Moro

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

Lesson Objectives:

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

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

The structure of the lesson included:

1. Org. moment, the purpose of which was to positively set children up for educational activities.
2. Motivation allowed to update knowledge, form the goals and objectives of the lesson. For this, tasks were proposed for finding an extra number, classifying examples into groups, adding missing numbers... In the course of solving these tasks, the children were faced with problem: an example was found, for the solution of which there is not enough available knowledge. In this regard, children independently formulated a goal and set ourselves the learning objectives of the lesson.
3. Search and discovery of new knowledge made it possible for children offer different options solving the problem. Based on previously studied material, they were able to find the right solution and come to conclusion, in which a new rule was formulated.
4. During primary anchoring students commented on their actions, working by rule, were additionally selected their examples to this rule.
5. For automation of actions and the ability to use the rules in non-standard assignments children solved equations, expressions in several actions.
6. Independent work and carried out mutual check showed that most of the children learned the topic.
7. During reflectionsthe children concluded that the set goal of the lesson was achieved and assessed themselves with the help of cards.

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

During the classes

Stage goal	Stage content	Student activities
1. Org. moment
Preparation of students for work, a positive attitude towards learning activities.	Incentives for learning activities. Check your readiness for the lesson, sit up straight, lean your elbows on the back of the chair. Rub your ears to help blood flow to your brain. Today you will have a lot of interesting work, which I am sure you will cope with perfectly well.	Organization of the workplace, check of landing.
2. Motivation.
Stimulating cognitive activity, activation of the thought process	Actualization of knowledge sufficient to acquire new knowledge. Verbal counting. Testing knowledge of table multiplication:	Solving tasks based on knowledge of table multiplication.
	A) find an extra number: 2 4 6 7 10 12 14 6 18 24 29 36 42 Explain why it is superfluous and with what number it should be replaced.	Finding an extra number.
	B) insert the missing numbers: … 16 24 32 … 48 …	Adding the missing number.
	Creating a problem situation Tasks in pairs: C) arrange examples in 2 groups: Why was it so distributed? (with answer 4 and 5).	Classification of examples into groups.
	Cards: 8 7-6 + 30: 6 \u003d 28: (16: 4) 6 \u003d 30-(20-10:2):5= 30- (20-10 2): 5 \u003d	Strong students work on individual cards.
	What have you noticed? Is there an extra example here? Have you been able to solve all the examples? Who is having difficulty? How is this example different from the rest? If someone has decided, then well done. But why not everyone was able to cope with this example?	Finding the embarrassment. Identification of missing knowledge, the reasons for the difficulty.
	Statement of the educational problem. Here is an example with 0. And from 0 you can expect different tricks. This is an unusual number. Remember what you know about 0? (a 0 \u003d 0, 0 a \u003d 0, 0 + a \u003d a) Give examples. Look how cunning he is: when he is added, he does not change the number, and when he is multiplied, he turns it into 0. Do these rules fit our example? How will he behave when eating?	Observation of known techniques from 0 and relation to the original example.
	So what is our goal? Solving this example is correct. Table on the board. What is needed for that? Learn the rule for dividing 0 by a number.	Making a hypothesis
How do you find the right solution? What action is multiplication associated with? (with division) Give an example 2 3 \u003d 6 6: 2 = 3 Can we now 0: 5? This means you need to find a number that, when multiplied by 5, you get 0. x 5 \u003d 0 This number is 0. So 0: 5 \u003d 0. Give your examples.	finding a solution based on what was previously learned,
Formulation of the rule. What rule can be formulated now? When you divide 0 by a number, you get 0. 0: a \u003d 0.	Decision typical assignments with commenting. Work according to the scheme (0: a \u003d 0)
5. Physical minutes.
Prevention of posture disorders, removal of fatigue from the eyes, general fatigue.
6. Automation of knowledge.
Revealing the boundaries of applicability of new knowledge.	In what other tasks might you need to know this rule? (in solving examples, equations)	Using the knowledge gained in different tasks. Working in groups.
	What is unknown in these equations? Remember how to find out the unknown factor. Solve the equations. What's the solution in 1 equation? (0) At 2? (no solution, you cannot divide by 0)	Referring to previously learned skills.
	** Make an equation with the solution x \u003d 0 (x 5 \u003d 0)	For strong learners creative task
7. Independent work.
Development of independence, cognitive abilities	Independent work with subsequent mutual check. №6	Active mental actions of students associated with finding a solution, based on their knowledge. Self-control and mutual control. Strong learners test and help weaker ones.
8. Work on previously covered material. Practicing problem solving skills.
Formation of problem solving skills.	Do you think the number 0 is often used in tasks? (No, not often, because 0 is nothing, and tasks should have some amount of something.) Then we will solve problems where there are other numbers. Read the problem. What will help solve the problem? (table) What columns should be recorded in the table? Fill the table. Make a plan for the solution: what do you need to learn in 1, 2 steps?	Work on a task using a table. Planning the solution to the problem. Self-recording of the decision. Self-control by sample.
9. Reflection. Lesson summary.
Organization of self-assessment activities. Increasing the motivation of the child.	What topic did you work on today? What did you not know at the beginning of the lesson? What was your goal? Have you reached it? What rule did you meet? Rate your work by displaying the appropriate icon: sun - I am pleased with myself, I did it white cloud - everything is fine, but I could work better; gray cloud - the lesson is ordinary, nothing interesting; droplet - nothing succeeded	Awareness of their activities, introspection of their work. Fixing the correspondence of the results of activities and the set goal
10. Homework.

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

Maths

Why can't you divide by zero?

The information that one cannot divide by zero is known to us from school. We learn this rule once and for all. However, only a few of us are wondering why it is actually impossible to 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 is 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 performing actions when solving an example or equation. No, I mean the very concept of number. And according to him, the most important are addition and multiplication. And already subtraction and division "follow" from them in one way or another.

Addition and subtraction

For example, let's analyze a simple operation: "3 - 1". What does this mean? The student will 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. Described correctly? Right. We ourselves would explain exactly the same. But mathematicians view the process of subtraction differently.

The operation "3 - 1" is considered not from the position of subtraction, but only from the side of addition. According to this there is no "three minus one", there is "some unknown number, which, when one adds one, gives three." Thus, a simple "three minus one" turns into an equation with one unknown: "x + 1 \u003d 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 book contains all the basic formulas of the school mathematics course: algebra, geometry and the beginnings of analysis. For the convenience of using the reference book, a subject index has been compiled. The manual is intended for schoolchildren of 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 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 figured out the principle of mathematical operations in relation to problems with subtraction and division, consider our division by zero.

The problem "4: 0" turns into "x · 0". It turns out that we need to find such a number, multiplication with which will give us 4. It is known that multiplication by zero always gives zero. it unique property zero and, in fact, its essence. There is no number multiplied by zero to produce any other number other than zero. We have come to a contradiction, so the problem has no solution. Consequently, the record "4: 0" does not correspond to any definite number, and hence its meaninglessness already follows. Therefore, in order to briefly emphasize the unproductiveness of such a process as dividing by zero, they say that “you cannot divide by zero”.

More interesting materials:

• Typical mistakes teachers make when teaching mathematics in elementary school
• Extracurricular activities in mathematics in elementary school
• Formation of mathematical literacy in primary school

What happens if zero is divided by zero?

Let's imagine the following equation: "0 · x \u003d 0". On the one hand, it looks quite fair. We represent zero instead of an unknown number and we get a ready-made solution: "0 · 0 \u003d 0". From this it is quite logical to deduce that "0: 0 \u003d 0".

However, now let's substitute any other number in the same equation with unknown instead of "x \u003d 0", for example "x \u003d 7". The resulting expression now looks like "0 · 7 \u003d 0". It seems that everything is correct. We do the opposite and get "0: 0 \u003d 7". But then, it turns out that you can take absolutely any number and print 0: 0 \u003d 1, 0: 0 \u003d 2 ... 0: 0 \u003d 145 ... - and so on ad infinitum.

If for any number x the equation is valid, then we have no right to choose only one, excluding the rest. This means that we still cannot answer what number the expression "0: 0" corresponds to. Once again stumped, we admit that this operation is also pointless. It turns out that zero cannot be divided even by itself.

Let us make a reservation that in mathematical analysis sometimes there are special conditions of the problem - the so-called "disclosure of uncertainty." In such cases, it is allowed to give preference to one of the possible solutions of the equation "0 · x \u003d 0". However, such "tolerances" do not occur in arithmetic.

The number 0 can be thought of 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 division by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.

Zero story

Zero is the reference point in all standard systems of calculation. Europeans began to use 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 number system. This American people used the duodecimal system of number, and they began with a zero on the first day of each month. Interestingly, the Maya sign for "zero" was exactly the same as the sign for "infinity." Thus, the ancient Maya concluded that these values \u200b\u200bwere identical and unknowable.

Math operations with zero

Standard math operations with zero can be boiled down to a few rules.

Addition: if you add zero to an arbitrary number, then it will not change its value (0 + x \u003d x).

Subtraction: when subtracting zero from any number, the value of the subtracted remains unchanged (x-0 \u003d x).

Multiplication: Any number multiplied by 0 gives 0 in the product (a * 0 \u003d 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 zero power will give 1 (x 0 \u003d 1).

Zero to any power is 0 (0 a \u003d 0).

In this case, a contradiction immediately arises: the expression 0 0 has no meaning.

Paradoxes of mathematics

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

Addition and multiplication

Let's take a standard example of subtraction: 10-2 \u003d 8. At school, it is considered simply: if two are taken away from ten subjects, eight remain. But mathematicians look at this operation in a completely different way. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x + 2 \u003d 10. For mathematicians, the unknown difference is simply a number that needs to be added to two to make eight. And no subtraction is required here, you just need to find a suitable numeric value.

Multiplication and division are treated the same way. In example 12: 4 \u003d 3, you can understand that we are talking about dividing eight objects into two equal piles. But in reality it is just an inverted formula for writing 3x4 \u003d 12. There are endless examples of division.

Division by 0 examples

This is where it becomes a little clear why you can't divide by zero. Multiplication and division by zero obey their own rules. All examples of the division of this quantity can be formulated as 6: 0 \u003d x. But this is an inverted notation of the expression 6 * x \u003d 0. But, as you know, any number multiplied by 0 gives in the product only 0. This property is inherent in the very concept of a zero value.

It turns out that such a number that, when multiplied by 0, gives some tangible value, does not exist, that is, this problem has no solution. You should not be afraid of such an answer, it is a natural answer for problems of this type. It's just that 6-0 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 multiplying by 0 is legal, can zero be divided by zero? After all, an equation of the form 0x 5 \u003d 0 is completely legal. Instead of the number 5, you can put 0, the product will not change from this.

Indeed, 0x0 \u003d 0. But you still can't divide by 0. As said, division is simply the inverse of multiplication. Thus, if in the example 0x5 \u003d 0, you need to determine the second factor, we get 0x0 \u003d 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 the infinite set of numbers. And if so, it means 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 mathematics. Studied in technical universities mathematical analysis 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: ???;
• one raised to an infinite power: 1? ;
• infinity times 0:? * 0;
• some others.

It is impossible to solve such expressions by elementary methods. But higher mathematics thanks to the additional possibilities for a number of similar examples, it provides final solutions. This is especially evident in the consideration of problems from the theory of limits.

Disclosure of uncertainty

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

As you can see in the example, a simple reduction of the fraction leads its value to a completely rational answer.

When considering the limits 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, a second remarkable limit is used.

Lopital's 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.

Currently, L'Hôpital's method is successfully used to solve uncertainties such as 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 by 2 characters, not one.

In general, as many zeros, transfer as much.

For example, there is a number 123456789.

You need to multiply it by 0.000000001

There are nine zeros in the number 0.000000001 (zero to the left of the comma is also counted), so we shift the number 123456789 by 9 digits:

It was 123456789 now 0.123456789.

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

It was 123456789 now 123456789000000000.

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

Dividing a number by 0.1 is the same as multiplying that number by 10

Dividing a number by 0.01 is the same as multiplying that number by 100

Division by 0.001 is multiplied 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 50 times (read from right to left without comma - 10000) 10000. That's 500000.

It's the same with multiplication, just the opposite:

400 x 0.01 is the same as dividing 400 by (read from right to left without comma - 100) 100: 400: 100 \u003d 4.

Who is more convenient to transfer commas to the right when dividing and to the left when multiplying when multiplying and dividing by such numbers, you can do so.

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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 by as many digits to the right as there are after the decimal point in the divisor, and then divide by a natural number.

Let's takery.

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 divider 0,7 there is one digit after the comma, therefore, move the commas in the dividend and divisor by one digit to the right.

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

Let's divide by the rule of dividing a decimal fraction by a natural number.

Divide as Divide integers... How to demolish a digit 8 - the first digit after the decimal point (i.e. the digit in the tenth place), so immediately put in a private comma and continue dividing.

Answer: 23.4.

Example 2) 15,6: 0,15.

We carry 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.

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

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 by one digit to the right and divide 31,14 on 45 according to the rule of dividing a decimal fraction by a natural number.

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

In the private, we put a comma as soon as we demolish a digit 1 in the tenth place. Then we continue to divide.

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 right to the decimal fraction)

Answer: 0.692.

Example 4) 53,84: 0,1.

Move commas in dividend and divisor by 1 digit 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. Note that the comma in the dividend has been moved to 1 digit to the right, as if we were multiplying 53,84 on 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 equivalent to 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 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 that the comma in the dividend is transferred 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 decimal by 0,0001 - it's like multiplying it by 10000 (carry the comma 4 digits to the 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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On this lesson will consider how to perform multiplication and division by numbers of the form 10, 100, 0.1, 0.001. Various examples on this topic will also be solved.

Multiply numbers by 10, 100

An exercise. How to multiply 25.78 by 10?

Decimal notation this number Is an abbreviated notation for the amount. It is necessary to paint 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 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 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 comma to the left one position. For instance:

An exercise. Divide 124.478 by 100.

Divide by 100 is the same as divide by 10 twice, so the comma is shifted 2 positions to the left:

The rule of multiplication and division by 10, 100, 1000

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

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

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

Multiplying by 100 is to shift the comma two places to the right.

After the shift, you can find that there are no numbers after the decimal point, which means that the fractional part is missing. Then the comma is not needed, the number is an integer.

You need to shift 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 on the link can help with this.

Equivalent decimal notation

Entry 52 means the following:

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

Can you put two zeros in front? Yes, these entries are equivalent.

Now let's look at the decimal fraction:

If you assign zero, it turns out:

These entries are equivalent. Similarly, you can assign multiple 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 for the same number.

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

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

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

In this case, remember that the comma always comes after the whole part. Then:

Multiplication and division by 0.1, 0.01, 0.001

Multiplication and division 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 one. But multiplying by is the same as dividing by 10. Therefore, you need to shift the comma 1 position to the left:

Similarly, multiplying by 0.01 is divided by 100:

Example. 5.235 divided by 0.1.

Decision 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 shift the comma to the right one position, which is equivalent to multiplying by 10.

The rule of multiplication and division by 0.1, 0.01, 0.001

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

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

Solution examples

Output

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 considered.

Examples on the application of these rules have been reviewed and resolved.

Bibliography

1. Vilenkin N.Ya. Mathematics: textbook. for 5 cl. general uchr. 17th ed. - M .: Mnemosina, 2005.

2. Shevkin A.V. Text tasks 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 NN, Ivanova MV. Formation of computing skills in mathematics lessons. 5-9 grades. - M .: Ileksa, 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 values \u200b\u200bof the expressions:

Actions with zero

In mathematics, the number zero occupies a special place. The fact is that it, in fact, means "nothing", "emptiness", but its meaning is really difficult to overestimate. To do this, it is enough to remember at least what exactly with zero markand starts counting the coordinates of the position of the point in any coordinate system.

Zero It is widely used in decimal fractions to define the values \u200b\u200bof "empty" digits located both before and after the decimal point. In addition, it is with him that one of the fundamental rules of arithmetic is associated, which states that on zero cannot be divided. Its logic, as a matter of 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".

FROM zero all arithmetic operations are carried out, and as its "partners" in them can be used whole numbers, ordinary and decimal fractions, and all of them can have both positive and negative values. Here are examples of their implementation and some explanations for them.

When adding scratch 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 eighths plus zero equals seventeen point three eighths.

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Evgeny SHIRYAEV, Lecturer and Head of the Mathematics Laboratory of the Polytechnic Museum, told "AiF" about division by zero:

1. Jurisdiction of the issue

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

Neither the constitution, nor the Criminal Code, nor even the statutes 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", to try 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 with the test of multiplication: the result multiplied by the divisor had to coincide with the dividend. Didn't match - didn't decide.

Example 1. 1000: 0 =...

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

Check will cut off incorrect ones. Go through the options: 100, 1, −23, 17, 0, 10,000. For each of them, the check will give the same result:

100 0 \u003d 1 0 \u003d - 23 0 \u003d 17 0 \u003d 0 0 \u003d 10 000 0 \u003d 0

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

3. Nuance

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

Example 2. 0: 0 = ...

Your suggestions for a private? 100? Please: quotient 100 times the divisor 0 equals the dividend 0.

More options! 1? Also fits. And -23, and 17, and all-all-all. In this example, the test will be positive for any number. And, to be honest, the solution in this example should be called not a number, but a set of numbers. Everyone. And it’s not long to come to an agreement to the point that Alice is not Alice, but Mary Ann, and both of them are a rabbit's dream.

4. What about higher mathematics?

The problem was resolved, the nuances were taken into account, the dots were placed, everything became clear - the answer for the example with division by zero cannot be a single number. To solve such problems is a hopeless and impossible task. Which means ... interesting! Take two.

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

No way. But 1000 can be easily divided by other numbers. Well, let's at least do what we get, even if we change the task. And there, you see, we will get carried away, and the answer will appear by itself. We 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 to zero, the larger the quotient. The trend can be observed further, moving to fractions and continuing to decrease the numerator:

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

In this process, there is no zero and no last quotient. We designated the movement towards them, 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 not in vain put double-sided: some sequences can converge to numbers. Then we can assign the sequence to its numerical limit.

Let's look at the sequence of quotients:

It grows indefinitely, not striving for any number and surpassing any. Mathematicians add the symbol to numbers ∞ to be able to put a double-headed arrow next to such a sequence:

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

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

5. And here is a 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 the dividend sequence converges to zero faster, then in the quotient it is a sequence with a zero limit. And when the elements of the divisor decrease much faster than that of the dividend, the sequence of quotients will grow strongly:

Uncertain situation. And so it is called: the uncertainty of the species 0/0 ... When mathematicians see sequences that fit this uncertainty, they don't rush to divide two. the same numbers against each other, but they figure out which of the sequences runs faster to zero and how exactly. And each example will have its own specific answer!

6. In life

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

Let us neglect the accurate physical understanding and formally look at the right side as a quotient of two numbers. Let's imagine we are solving a school electricity problem. The condition gives voltage in volts and resistance in ohms. The question is obvious, a one-step solution.

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

Well, let's solve the problem for the superconducting circuit? Just substitute R \u003d0 will not work, physics throws up an interesting problem, behind which, obviously, there is a scientific discovery. And people who managed to divide by zero in this situation received Nobel Prize... It is useful to be able to bypass any prohibitions!