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 to use this number relatively recently, but the sages Ancient India were using zero a thousand years before the empty number came into regular use 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 are, in fact, 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're talking about 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 headache for school mathematics. 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 do not have a solution in school courses mathematics:
- 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 desired value division by zero is obtained and converted. 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 remarkable 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 is as follows. 
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 they divide integers. 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. A whole part is missing. 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.
Solution 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. Grades 5–9. – 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 meaning. 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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Evgeniy Shiryaev, teacher and head of the Mathematics Laboratory of the Polytechnic Museum, told AiF.ru about division by zero:
1. Jurisdiction of the issue
Agree, what makes the rule especially provocative is the ban. How can this not be done? Who banned? What about our civil rights?
Neither the Constitution of the Russian Federation, nor the Criminal Code, nor even the charter of your school objects to the intellectual action that interests us. This means that the ban has no legal force, and nothing prevents you from trying to divide something by zero right here, on the pages of AiF.ru. For example, a thousand.
2. Let's divide as taught
Remember, when you first learned how to divide, the first examples were solved by checking multiplication: the result multiplied by the divisor had to be the same as the divisible. If it didn’t match, they didn’t decide.
Example 1. 1000: 0 =...
Let's forget about the forbidden rule for a moment and make several attempts to guess the answer.
Incorrect ones will be cut off by the check. Try the following options: 100, 1, −23, 17, 0, 10,000. For each of them, the check will give the same result:
100 0 = 1 0 = − 23 0 = 17 0 = 0 0 = 10,000 0 = 0
By multiplying zero, everything turns into itself and never into a thousand. The conclusion is easy to formulate: no number will pass the test. That is, no number can be the result of dividing a non-zero number by zero. Such division is not prohibited, but simply has no result.
3. Nuance
We almost missed one opportunity to refute the ban. Yes, we admit that a non-zero number cannot be divided by 0. But maybe 0 itself can?
Example 2. 0: 0 = ...
What are your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor 0 is equal to the dividend 0.
More options! 1? Fits too. And −23, and 17, and that’s it. In this example, the 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 doesn’t take long to agree that Alice is not Alice, but Mary Ann, and both of them are a rabbit’s dream.
4. What about higher mathematics?
The problem has been resolved, the nuances have been taken into account, the dots have been placed, everything has become clear - the answer to the example with division by zero cannot be a single number. Solving such problems is hopeless and impossible. Which means... interesting! Take two.
Example 3. Figure out how to divide 1000 by 0.
But no way. But 1000 can be easily divided by other numbers. Well, let's at least do what we can, even if we change the task at hand. And then, you see, we get carried away, and the answer will appear by itself. Let’s forget about zero for a minute and divide by one hundred:
A hundred is far from zero. Let's take a step towards it by decreasing the divisor:
1000: 25 = 40,
1000: 20 = 50,
1000: 10 = 100,
1000: 8 = 125,
1000: 5 = 200,
1000: 4 = 250,
1000: 2 = 500,
1000: 1 = 1000.
The dynamics are obvious: the closer the divisor is to zero, the larger the quotient. The trend can be observed further by moving to fractions and continuing to reduce the numerator:
It remains to note that we can get as close to zero as we like, making the quotient as large as we like.
In this process there is no zero and there is no last quotient. We indicated the movement towards them by replacing the number with a sequence converging to the number we are interested in:
This implies a similar replacement for the dividend:
1000 ↔ { 1000, 1000, 1000,... }
It’s not for nothing that the arrows are double-sided: some sequences can converge to numbers. Then we can associate the sequence with its numerical limit.
Let's look at the sequence of quotients:
It grows unlimitedly, not striving for any number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:
Comparison with the numbers of sequences that have a limit allows us to propose a solution to the third example:
When elementwise dividing a sequence converging to 1000 by a sequence of positive numbers converging to 0, we obtain a sequence converging to ∞.
5. And here is the nuance with two zeros
What is the result of dividing two sequences of positive numbers that converge to zero? If they are the same, then the unit is identical. If the dividend sequence converges to zero faster, then in the quotient the sequence has a zero limit. And when the elements of the divisor decrease much faster than those of the dividend, the sequence of the quotient will grow greatly:
Uncertain situation. And that’s what it’s called: uncertainty of type 0/0 . When mathematicians see sequences that fit such uncertainty, they do not rush to divide two identical numbers by each other, but figure out which of the sequences runs faster to zero and how exactly. And each example will have its own specific answer!
6. In life
Ohm's law relates current, voltage and resistance in a circuit. It is often written in this form:
Let's allow ourselves to ignore the neat physical understanding and formally look at the right-hand side as the quotient of two numbers. Let's imagine that we are solving a school problem on electricity. The condition gives the voltage in volts and resistance in ohms. The question is obvious, the solution is in one action.
Now let's look at the definition of superconductivity: this is the property of some metals to have zero electrical resistance.
Well, let's solve the problem for a superconducting circuit? Just set it up R= 0 will not work, physics throws up an interesting problem, behind which, obviously, there is scientific discovery. And the people who managed to divide by zero in this situation received the Nobel Prize. It’s useful to be able to bypass any prohibitions!
In this method, the decision values are taken equally, and the likelihood ratio takes the form
The solution is similar to the minimum risk method.
Here the ratio of a priori probabilities of a serviceable ( R 1) and faulty (R 2) states accepted equal to one, and the condition for finding K 0 looks like that:
Example
Define parameter limit value K 0 , above which the facility is subject to decommissioning.
The object is a gas turbine engine.
Parameter - iron content in oil K , (g/t). The parameter has a normal distribution if ( D 1 ) and faulty ( D 2 ) states. Known:
Solution
Minimum Risk Method
According to expression (2.4) 
After substituting the expression
and taking logarithms we get
Transforming and solving the given quadratic equation, we get:
K01=2,24; K 02=0.47. Required limit value K 0 =2,24.
Minimum number method wrong decisions
Condition of receipt K 0
:
Substituting and expanding the corresponding probability densities, we obtain
the equation:
The appropriate root for this equation is 2.57.
So, K 0 = 2,57.
Maximum likelihood method
Condition of receipt K 0 :
F(K 0 /D 1) = F(K 0 /D 2).
The final quadratic equation will look like this:
What you are looking for K 0 = 2,31.
Let's determine the probability of a false alarm P(H 21 ) , probability of missing a defect P(H 12), as well as the average risk R for boundary values K 0, found by various methods.
If under initial conditions K 1
And 
If under initial conditions K 1 > K 2, That
And
For the minimum risk method at K 0=2.29 we get the following
For the method of the minimum number of erroneous decisions with K 0 =2,57:
For the maximum likelihood method at K 0 =2,37:
Let us summarize the calculation results in the final table.
Assignments for task No. 2.
The assignment option is selected based on the last two digits of the grade book number. All tasks require defining a limit value K 0 , dividing objects into two classes: serviceable and faulty. The results of the decisions are plotted on a graph (Fig. 9.1), which is drawn on graph paper and pasted into the work.
So, technical diagnostics of an object is carried out according to the parameter K. For a serviceable object, the average value of the parameter is given K 1 and standard deviation σ 1 . For the faulty one, respectively K2 And σ 2 . The source data also shows the price ratio for each option C 12 / C 21. Distribution K is accepted as normal. In all variants P 1=0,9; P2=0,1.
Options for tasks are given in table. 2.1-2.10.
Initial data for options 00÷09 (Table 2.1):
An object- gas turbine engine.
Parameter- vibration velocity (mm/s).
Faulty condition- violation normal conditions operation of the engine rotor supports.
Table 2.1
| Designation of quantities | Options | |||||||||
| K 1 | ||||||||||
| K2 | ||||||||||
| σ 1 | ||||||||||
| σ 2 | ||||||||||
| C 12 / C 21 |
Initial data for options 10÷19 (Table 2.2):
An object- gas turbine engine.
Parameter Cu ) in oil (g/t).
Faulty condition - increased concentration Cu
Table 2.2
| Designation of quantities | Options | |||||||||
| K 1 | 1,0 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 | 1,7 | 1,8 | 1,9 |
| K2 | ||||||||||
| σ 1 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 |
| σ 2 | ||||||||||
| C 12 / C 21 |
Initial data for options 20÷29 (Table 2.3):
An object- pumpable fuel pump fuel system.
Parameter- fuel pressure at the outlet (kg/cm2).
Faulty condition- deformation of the impeller.
Table 2.3
| Designation of quantities | Options | |||||||||
| K 1 | 2,50 | 2,55 | 2,60 | 2,65 | 2,70 | 2,75 | 2,80 | 2,85 | 2,90 | 2,95 |
| K2 | 1,80 | 1,85 | 1,90 | 1,95 | 2,00 | 2,05 | 2,10 | 2,15 | 2,20 | 2,25 |
| σ 1 | 0,20 | 0,20 | 0,20 | 0,20 | 0,20 | 0,20 | 0,20 | 0,20 | 0,20 | 0,20 |
| σ 2 | 0,30 | 0,30 | 0,30 | 0,30 | 0,30 | 0,30 | 0,30 | 0,30 | 0,30 | 0,30 |
| C 12 / C 21 |
Initial data for options 30÷39 (Table 2.4):
An object- gas turbine engine.
Parameter- level of vibration overload ( g ).
Faulty condition- rolling out the outer race of bearings.
Table 2.4
| Designation of quantities | Options | |||||||||
| K 1 | 4,5 | 4,6 | 4,7 | 4,8 | 4,9 | 5,0 | 5,1 | 5,2 | 5,3 | 5,4 |
| K2 | 6,0 | 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6 | 6,7 | 6,8 | 6,9 |
| σ 1 | 0,5 | 0,5 | 0,5 | 0,5 | 0,5 | 0,5 | 0,5 | 0,5 | 0,5 | 0,5 |
| σ 2 | 0,7 | 0,7 | 0,7 | 0,7 | 0,7 | 0,7 | 0,7 | 0,7 | 0,7 | 0,7 |
| C 12 / C 21 |
Initial data for options 40÷49 (Table 2.5):
An object- intershaft bearing of a gas turbine engine.
Parameter- readings of a vibroacoustic device for monitoring the condition of the bearing (µa).
Faulty condition- appearance of traces of chipping on the bearing raceways.
Table 2.5
| Designation of quantities | Options | |||||||||
| K 1 | ||||||||||
| K2 | ||||||||||
| σ 1 | ||||||||||
| σ 2 | ||||||||||
| C 12 / C 21 |
Initial data for options 50÷59 (Table 2.6)
An object- gas turbine engine.
Parameter- iron content ( Fe ) in oil (g/t).
Faulty condition- increased concentration Fe in oil due to accelerated wear of gear connections in the drive box.
Table 2.6
| Designation of quantities | Options | |||||||||
| K 1 | 1,95 | 2,02 | 1,76 | 1,82 | 1,71 | 1,68 | 1,73 | 1,81 | 1,83 | 1,86 |
| K2 | 4,38 | 4,61 | 4,18 | 4,32 | 4,44 | 4,10 | 4,15 | 4,29 | 4,39 | 4,82 |
| σ 1 | 0,3 | 0,3 | 0,3 | 0.3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 | 0,3 |
| σ 2 | ||||||||||
| C 12 / C 21 |
Initial data for options 60÷69 (Table 2.7):
An object- oil for lubricating gas turbine engines.
Parameter- optical density of the oil, %.
Faulty condition- reduced operational properties oil having optical density.
Table 2.7
| Designation of quantities | Options | |||||||||
| K 1 | ||||||||||
| K2 | ||||||||||
| σ 1 | ||||||||||
| σ 2 | ||||||||||
| C 12 / C 21 |
Initial data for options 70÷79 (Table 2.8):
An object- fuel filter elements.
Parameter- concentration of copper impurities ( Cu ) in oil (g/t).
Faulty condition- increased concentration Cu in oil due to intensified wear processes of copper-plated splined joints of drive shafts.
Table 2.8
| Designation of quantities | Options | |||||||||
| K 1 | ||||||||||
| K2 | ||||||||||
| σ 1 | ||||||||||
| σ 2 | ||||||||||
| C 12 / C 21 |
Initial data for options 80÷89 (Table 2.9)
An object- axial piston pump.
Parameter- the value of pump performance, expressed by volumetric
Efficiency (in fractions of 1.0).
Faulty condition- low volumetric efficiency associated with pump failure.
Table 2.9
| Designation of quantities | Options | |||||||||
| K 1 | 0,92 | 0,91 | 0,90 | 0,89 | 0,88 | 0,07 | 0,86 | 0,85 | 0,84 | 0,83 |
| K2 | 0,63 | 0,62 | 0,51 | 0,50 | 0,49 | 0,48 | 0,47 | 0,46 | 0,45 | 0,44 |
| σ 1 | 0,11 | 0,11 | 0,11 | 0,11 | 0,11 | 0,11 | 0,11 | 0,11 | 0,11 | 0,11 |
| σ 2 | 0,14 | 0,14 | 0,14 | 0,14 | 0,14 | 0,14 | 0,14 | 0,14 | 0,14 | 0,14 |
| C 12 / C 21 |
Initial data for options 90÷99 (Table 2.10)
An object- an aircraft control system consisting of rigid rods.
Parameter- total axial play of joints, microns.
Faulty condition- increased total axial play due to wear of mating pairs.
Table 2.10
| Designation of quantities | Options | |||||||||
| K 1 | ||||||||||
| K2 | ||||||||||
| σ 1 | ||||||||||
| σ 2 | ||||||||||
| C 12 / C 21 |