Whether divided by. Can you divide by zero? Mathematician answers. When zero appeared
"You cannot divide by zero!" - the majority of schoolchildren memorize this rule without asking questions. All children know what is "not allowed" and what will happen if in response to him ask: "Why?" But in fact it is very interesting and important to know why it is impossible.
The point is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as complete - 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 subtraction for example. What does 5 - 3 mean? The student will answer this simply: you need to take five objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem in a completely different way. There is no subtraction, there is only addition. Therefore, writing 5 - 3 means a number that, when added to the number 3, gives the number 5. That is, 5 - 3 is just an abbreviated notation of the equation: x + 3 \u003d 5. There is no subtraction in this equation. There is only a task - to find a suitable number.
The same is the case with multiplication and division. The 8: 4 notation can be understood as the result of dividing eight items into four equal piles. But in reality, this is just a shorthand form of the equation 4 x \u003d 8.
This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Notation 5: 0 is an abbreviation for 0 x \u003d 5. That is, this task is to find a number that, when multiplied by 0, gives 5. But we know that when multiplied by 0, you always get 0. This is an inherent property of zero, strictly speaking, part of its definition.
There is no such number that, when multiplied by 0, will give something other than zero. That is, our task has no solution. (Yes, this happens, not every problem has a solution.) This means that the 5: 0 notation does not correspond to any specific number, and it simply does not mean anything, and therefore does not make sense. The meaninglessness of this recording is briefly expressed, saying that you cannot divide by zero.
The most attentive readers in this place will certainly ask: can zero be divided by zero? Indeed, the equation 0 x \u003d 0 is successfully solved. For example, you can take x \u003d 0, and then we get 0 0 \u003d 0. So, 0: 0 \u003d 0? But let's not rush. Let's try to take x \u003d 1. We get 0 1 \u003d 0. Right? So 0: 0 \u003d 1? But you can take any number this way and get 0: 0 \u003d 5, 0: 0 \u003d 317, etc.
But if any number is suitable, then we have no reason to opt for any one of them. That is, we cannot say to which number the entry 0: 0 corresponds. And if this is so, then we have to admit that this entry does not make sense either. It turns out that even zero cannot be divided by zero. (In mathematical analysis, there are cases when, thanks to additional conditions of the problem, one of the possible solutions to the equation 0 x \u003d 0 can be preferred; in such cases, mathematicians speak of “disclosing uncertainty,” but in arithmetic such cases do not occur.)
This is the peculiarity of the division operation. More precisely, the multiplication operation and the associated number have zero.
Well, and the most meticulous, having read this far, may ask: why is it that it is impossible to divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. You can answer it only after getting acquainted with the formal mathematical definitions of number sets and operations on them. It's not that difficult, but for some reason it is not taught at school. But at the lectures on mathematics at the university, you, first of all, will be taught exactly this.
Back in school, the teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still there is a lot of controversy around him. Someone just remembered the rule and does not bother with the question “why?”. "You can't and that's it, because they said so at school, a rule is a rule!" Someone can write half a notebook with formulas, proving this rule or, conversely, its illogicality.
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Who is right in the end
During these disputes, both people who have opposite points of view look at each other like a ram and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams resting their horns against each other. The only difference between them is that one is slightly less educated than the other.
More often than not, those who believe this rule to be incorrect try to invoke logic in this way:
I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!
Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 \u003d 2. So we discard such a conclusion right away - it is illogical, although it has the opposite purpose - to call to logic.
What is multiplication
The original rule of multiplication was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies that the number is natural. Thus, any number with multiplication can be reduced to this equation:
- 25 × 3 \u003d 75
- 25 + 25 + 25 = 75
- 25 × 3 \u003d 25 + 25 + 25
The conclusion follows from this equation, that multiplication is a simplified addition.
What is zero
Any person from childhood knows: zero is emptiness, Despite the fact that this emptiness has a designation, it does not carry anything at all. The ancient oriental scholars thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the meaning of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty places in decimal fractions, this is done both before and after the decimal point.
Is it possible to multiply by emptiness
You can multiply by zero, but it's useless, because, whatever one may say, but even when multiplying negative numbers, you will still get zero. It is enough just to remember this simple rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as the ancient scientists believed. Below, the most logical explanation will be given that this multiplication is useless, because when a number is multiplied by it, it will still get the same thing - zero.
Going back to the very beginning, to the argument about two apples, 2 times 0 looks like this:
- If you eat two apples five times, then you eat 2 × 5 \u003d 2 + 2 + 2 + 2 + 2 \u003d 10 apples
- If you eat them two three times, then 2 × 3 \u003d 2 + 2 + 2 \u003d 6 apples are eaten
- If you eat two apples zero times, then nothing will be eaten - 2 × 0 \u003d 0 × 2 \u003d 0 + 0 \u003d 0
After all, eating an apple 0 times means not eating a single one. Even the smallest child will understand this. Whatever one may say, 0 will come out, a two or three can be replaced with absolutely any number and absolutely the same will come out. To put it simply, then zero is nothingand when you have there is nothing, no matter how much you multiply, it doesn't matter will be zero... There is no magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to dissipate, and everything falls into place.
Division
Another important rule follows from all of the above:
You cannot divide by zero!
This rule has also been stubbornly hammered into our heads since childhood. We just know that it is impossible and everything without stuffing our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many controversies and contradictions around this rule.
Everyone just memorized the rule and did not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are complete from the above, and all other manipulations with numbers are built from them. That is, writing 10: 2 is an abbreviation of the equation 2 * x \u003d 10. So, writing 10: 0 is the same abbreviation from 0 * x \u003d 10. It turns out that division by zero is the task of finding a number, multiplying it by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be incorrect a priori.
Let me tell you
To not divide by 0!
Cut 1 as you want, lengthwise,
Just don't divide by 0!
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 of interest to 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.
Invalid checks will cut off. 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 nonzero number cannot be divisible by 0. But maybe 0 itself can?
Example 2. 0: 0 = ...
Your suggestions for a private? one hundred? Please: quotient 100 times the divisor 0 equals the divisible 0.
More options! one? 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, without 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 with a limit allows us to propose 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 they are the same 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 such an 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 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. 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 the Nobel Prize. It is useful to be able to bypass any prohibitions!
And here's another interesting statement. "You cannot divide by zero!" - the majority of schoolchildren memorize this rule without asking questions. All children know what is "not allowed" and what will happen if in response to him ask: "Why?". This is what will happen if
But in fact it is very interesting and important to know why it is impossible.
The point is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as complete - 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 subtraction for example. What does 5 - 3 mean? The student will answer this simply: you need to take five objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem in a completely different way. There is no subtraction, there is only addition. Therefore, writing 5 - 3 means a number that, when added to the number 3, gives the number 5. That is, 5 - 3 is just an abbreviated notation of the equation: x + 3 \u003d 5. There is no subtraction in this equation. There is only a task - to find a suitable number.
The same is the case with multiplication and division. The 8: 4 notation can be understood as the result of dividing eight items into four equal piles. But in reality it is just an abbreviated form of the equation 4 x \u003d 8.
This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Notation 5: 0 is an abbreviation for 0 x \u003d 5. That is, this task is to find such a number that, when multiplied by 0, will give 5. But we know that when multiplied by 0, you always get 0. This is an inherent property of zero, strictly speaking , part of its definition.
There is no such number that, when multiplied by 0, will give something other than zero. That is, our task has no solution. (Yes, this happens, not every problem has a solution.) This means that the 5: 0 notation does not correspond to any specific number, and it simply does not mean anything and therefore does not make sense. The meaninglessness of this recording is briefly expressed, saying that you cannot divide by zero.
The most attentive readers in this place will certainly ask: can zero be divided by zero? Indeed, the equation 0 x \u003d 0 is successfully solved. For example, we can take x \u003d 0, and then we get 0 · 0 \u003d 0. It turns out that 0: 0 \u003d 0? But let's not rush. Let's try to take x \u003d 1. We get 0 · 1 \u003d 0. Right? So 0: 0 \u003d 1? But you can take any number this way and get 0: 0 \u003d 5, 0: 0 \u003d 317, etc.
But if any number is suitable, then we have no reason to opt for any one of them. That is, we cannot say to which number the entry 0: 0 corresponds. And if this is so, then we have to admit that this entry does not make sense either. 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 of the possible solutions to the equation 0 · x \u003d 0 can be preferred; in such cases, mathematicians speak of "disclosing uncertainty", but in arithmetic such cases do not occur.)
This is the peculiarity of the division operation. More precisely, the multiplication operation and the associated number have zero.
Well, and the most meticulous, having read this far, may ask: why is it that it is impossible to divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. You can answer it only after getting acquainted with the formal mathematical definitions of number sets and operations on them.
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 school are actually far from being 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's just an inverted formula for writing 3x4 \u003d 12, and 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. The 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: ???;
- 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, gives 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 a 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 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, 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 comma 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 decimal point, 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 according to the rule of dividing a decimal fraction by a natural number.
Divide as natural numbers divide. 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 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 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 , so we will transfer the comma in the dividend on 2 digits to the right:
2) 0,235:0,01=23,5.
Example 3) 2,7845: 0,001.
As 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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This lesson will show you how to multiply and divide by numbers like 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?
The decimal notation for 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 example:
An exercise. Divide 124.478 by 100.
Divide by 100 is the same as divide by 10 twice, so the comma is shifted left 2 positions:
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.
The solution to this example is built 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 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
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 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. Word problems in mathematics: 5-6. - M .: Ileksa, 2011.
3. Ershova A.P., Goloborodko V.V. All school mathematics in independent and test papers. 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 a certain 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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