Evgeniy Shiryaev, teacher and head of the Mathematics Laboratory of the Polytechnic Museum, told AiF.ru about division by zero:
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.
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.
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.
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 ∞.
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!
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 If it doesn’t work out, physics throws up an interesting problem, behind which, obviously, there is a 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!
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. The impossibility of dividing by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.
Zero is the reference point in all standard number systems. Europeans began using this number relatively recently, but the sages of ancient India used zero a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Mayan numerical system. These American people used the duodecimal number system, and the first day of each month began with a zero. It is interesting that among the Mayans the sign denoting “zero” completely coincided with the sign denoting “infinity”. Thus, the ancient Mayans concluded that these quantities are identical and unknowable.
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.
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.
Let's take a standard subtraction example: 10-2=8. At school they consider it simply: if you subtract two from ten subjects, eight remain. But mathematicians look at this operation completely differently. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x+2=10. To mathematicians, the unknown difference is simply the number that needs to be added to two to make eight. And no subtraction is required here, you just need to find the appropriate numerical value.
Multiplication and division are treated the same. In the example 12:4=3 you can understand that we are talking about dividing eight objects into two equal piles. But in reality, this is just an inverted formula for writing 3x4 = 12. Such examples of division can be given endlessly.
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.
Division by zero is a headache for high school math. Mathematical analysis studied in technical universities slightly expands the concept of problems that have no solution. For example, new ones are added to the already known expression 0:0, which do not have solutions in school mathematics courses:
It is impossible to solve such expressions using elementary methods. But higher mathematics, thanks to additional possibilities for a number of similar examples, provides final solutions. This is especially evident in the consideration of problems from the theory of limits.
In the theory of limits, the value 0 is replaced by a conditional infinitesimal variable. And expressions in which, when substituting the desired value, division by zero is obtained, are 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 of 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.
In some cases, the limits of expressions can be replaced by the limits of their derivatives. Guillaume L'Hopital - 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 looks like this.
Very often, many people wonder why division by zero cannot be used? In this article we will talk in great detail about where this rule came from, as well as what actions can be performed with a zero.
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Zero can be called one of the most interesting numbers. This number has no meaning, it means emptiness in the truest sense of the word. However, if a zero is placed next to any number, then the value of this number will become several times greater.
The number itself is very mysterious. It was used by the ancient Mayan people. For the Mayans, zero meant “beginning,” and calendar days also began from zero.
A very interesting fact is that the zero sign and the uncertainty sign were similar. By this, the Mayans wanted to show that zero is the same identical sign as uncertainty. In Europe, the designation zero appeared relatively recently.
Many people also know the prohibition associated with zero. Anyone will say that you can't divide by zero. Teachers at school say this, and children usually take their word for it. Usually, children are either simply not interested in knowing this, or they know what will happen if, having heard an important prohibition, they immediately ask, “Why can’t you divide by zero?” But when you get older, your interest awakens, and you want to know more about the reasons for this ban. However, there is reasonable evidence.
First you need to determine what actions can be performed with zero. Exists several types of actions:
Important! If you add zero to any number during addition, then this number will remain the same and will not change its numerical value. The same thing happens if you subtract zero from any number.
When multiplying and dividing things are a little different. If multiply any number by zero, then the product will also become zero.
Let's look at an example:
Let's write this as an addition:
There are five zeros in total, so it turns out that
Let's try to multiply one by zero. The result will also be zero.
Zero can also be divided by any other number that is not equal to it. In this case, the result will be , the value of which will also be zero. The same rule applies to negative numbers. If zero is divided by a negative number, the result is zero.
You can also construct any number to the zero degree. In this case, the result will be 1. It is important to remember that the expression “zero to the power of zero” is absolutely meaningless. If you try to raise zero to any power, you get zero. Example:
We use the multiplication rule and get 0.
So, here we come to the main question. Is it possible to divide by zero? at all? And why can’t we divide a number by zero, given that all other actions with zero exist and are applied? To answer this question it is necessary to turn to higher mathematics.
Let's start with the definition of the concept, what is zero? School teachers say that zero is nothing. Emptiness. That is, when you say that you have 0 handles, it means that you have no handles at all.
In higher mathematics, the concept of “zero” is broader. It does not mean emptiness at all. Here zero is called uncertainty because if we do a little research, it turns out that when we divide zero by zero, we can end up with any other number, which may not necessarily be zero.
Did you know that those simple arithmetic operations that you studied at school are not so equal to each other? The most basic actions are addition and multiplication.
For mathematicians, the concepts of “” and “subtraction” do not exist. Let's say: if you subtract three from five, you will be left with two. This is what subtraction looks like. However, mathematicians would write it this way:
Thus, it turns out that the unknown difference is a certain number that needs to be added to 3 to get 5. That is, you don’t need to subtract anything, you just need to find the appropriate number. This rule applies to addition.
Things are a little different with rules of multiplication and division. It is known that multiplication by zero leads to a zero result. For example, if 3:0=x, then if you reverse the entry, you get 3*x=0. And a number that was multiplied by 0 will give zero in the product. It turns out that there is no number that would give any value other than zero in the product with zero. This means that division by zero is meaningless, that is, it fits our rule.
But what happens if you try to divide zero itself by itself? Let's take some indefinite number as x. The resulting equation is 0*x=0. It can be solved.
If we try to take zero instead of x, we will get 0:0=0. It would seem logical? But if we try to take any other number, for example, 1, instead of x, we will end up with 0:0=1. The same situation will happen if we take any other number and plug it into the equation.
In this case, it turns out that we can take any other number as a factor. The result will be an infinite number of different numbers. Sometimes division by 0 in higher mathematics still makes sense, but then usually a certain condition appears, thanks to which we can still choose one suitable number. This action is called "uncertainty disclosure." In ordinary arithmetic, division by zero will again lose its meaning, since we will not be able to choose one number from the set.
Important! You cannot divide zero by zero.
Infinity can be found very often in higher mathematics. Since it is simply not important for schoolchildren to know that there are also mathematical operations with infinity, teachers cannot properly explain to children why it is impossible to divide by zero.
Students begin to learn basic mathematical secrets only in the first year of institute. Higher mathematics provides a large complex of problems that have no solution. The most famous problems are problems with infinity. They can be solved using mathematical analysis.
Can also be applied to infinity elementary mathematical operations: addition, multiplication by number. Usually they also use subtraction and division, but in the end they still come down to two simple operations.
But what will happen if you try:
Important! Infinity is a little different from uncertainty! Infinity is one of the types of uncertainty.
Now let's try dividing infinity by zero. It would seem that there should be uncertainty. But if we try to replace division with multiplication, we get a very definite answer.
For example: ∞/0=∞*1/0= ∞*∞ = ∞.
It turns out like this mathematical paradox.
The answer to why you can't divide by zero
Thought experiment, trying to divide by zero
So, now we know that zero is subject to almost all operations that are performed with, except for one single one. You can't divide by zero just because the result is uncertainty. We also learned how to perform operations with zero and infinity. The result of such actions will be uncertainty.
Even at school, teachers tried to hammer into our heads the simplest rule: “Any number multiplied by zero equals zero!”, - but still a lot of controversy constantly arises around him. Some people just remember the rule and don’t bother themselves with the question “why?” “You can’t and that’s it, because they said so at school, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.
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During these disputes, both people with opposing 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 on each other. The only difference between them is that one is slightly less educated than the other.
Most often, those who consider this rule to be incorrect try to appeal to logic in this way:
I have two apples on my table, if I put zero apples on them, that is, I don’t put a single one, then my two apples will not disappear! 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 = 2. So let’s discard this conclusion right away - it is illogical, although it has the opposite goal - to call to logic.
Originally the multiplication rule 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:
From this equation it follows that that multiplication is a simplified addition.
Any person knows from childhood: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient Eastern scientists 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 about multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimal fractions, this is done both before and after the decimal point.
You can multiply by zero, but it is useless, because, whatever one may say, even when multiplying negative numbers, you will still get zero. It’s 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 ancient scientists believed. Below we will give the most logical explanation that this multiplication is useless, because when you multiply a number by it, you will still get the same thing - zero.
Returning to the very beginning, to the argument about two apples, 2 times 0 looks like this:
After all, eating an apple 0 times means not eating a single one. This will be clear to even the smallest child. Whatever one may say, the result will be 0, two or three can be replaced with absolutely any number and the result will be absolutely the same. And to put it simply, then zero is nothing, and when do you have there is nothing, then no matter how much you multiply, it’s still the same will be zero. There is no such thing as 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 who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.
From all of the above, another important rule follows:
You can't divide by zero!
This rule has also been persistently hammered into our heads since childhood. We just know that it’s impossible to do everything without filling our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then most will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions surrounding this rule.
Everyone simply memorized the rule and did not divide by zero, not suspecting that the answer was hidden on the surface. Addition, multiplication, division and subtraction are unequal; of the above, only multiplication and addition are valid, and all other manipulations with numbers are built from them. That is, the notation 10: 2 is an abbreviation of the equation 2 * x = 10. This means that the notation 10: 0 is the same abbreviation for 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying 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 a priori incorrect.
Let me tell you,
So as not to divide by 0!
Cut 1 as you want, lengthwise,
Just don't divide by 0!
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. The impossibility of dividing by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.
Zero is the reference point in all standard number systems. Europeans began using this number relatively recently, but the sages of ancient India used zero a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Mayan numerical system. These American people used the duodecimal number system, and the first day of each month began with a zero. It is interesting that among the Mayans the sign denoting “zero” completely coincided with the sign denoting “infinity”. Thus, the ancient Mayans concluded that these quantities are identical and unknowable.
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.
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.
Let's take a standard subtraction example: 10-2=8. At school they consider it simply: if you subtract two from ten subjects, eight remain. But mathematicians look at this operation completely differently. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x+2=10. To mathematicians, the unknown difference is simply the number that needs to be added to two to make eight. And no subtraction is required here, you just need to find the appropriate numerical value.
Multiplication and division are treated the same. In the example 12:4=3 you can understand that we are talking about dividing eight objects into two equal piles. But in reality, this is just an inverted formula for writing 3x4 = 12. Such examples of division can be given endlessly. 
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.
Division by zero is a headache for school math. Mathematical analysis studied in technical universities slightly expands the concept of problems that have no solution. For example, new ones are added to the already known expression 0:0, which do not have solutions in school mathematics courses:
It is impossible to solve such expressions using elementary methods. But higher mathematics, thanks to additional possibilities for a number of similar examples, provides final solutions. This is especially evident in the consideration of problems from the theory of limits.
In the theory of limits, the value 0 is replaced by a conditional infinitesimal variable. And expressions in which, when substituting the desired value, division by zero is obtained, are 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 of 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.
In some cases, the limits of expressions can be replaced by the limits of their derivatives. Guillaume L'Hopital is a French mathematician, the founder of the French school of mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule looks like this. 
Currently, L'Hopital's method is successfully used to solve uncertainties of the 0:0 or?:? type.
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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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.
Primery.
Perform division: 1) 16,38: 0,7; 2) 15,6: 0,15; 3) 3,114: 4,5; 4) 53,84: 0,1.
Solution.
Example 1) 16,38: 0,7.
In the divider 0,7 there is one digit after the decimal point, so let’s move the commas in the dividend and divisor one digit to the right.
Then we will need to divide 163,8 on 7 .
Let's perform the division according to the rule for dividing a decimal fraction by a natural number.
We divide as natural numbers are divided. How to remove the number 8 - the first digit after the decimal point (i.e. the digit in the tenths place), so immediately put a comma in the quotient and continue dividing.
Answer: 23.4.
Example 2) 15,6: 0,15.
We move commas in the dividend ( 15,6 ) and divisor ( 0,15 ) two digits to the right, since in the divisor 0,15 there are two digits after the decimal point.
We remember that you can add as many zeros as you like to the decimal fraction on the right, and this will not change the decimal fraction.
15,6:0,15=1560:15.
We perform division of natural numbers.
Answer: 104.
Example 3) 3,114: 4,5.
Move the commas in the dividend and divisor one digit to the right and divide 31,14 on 45 according to the rule for dividing a decimal fraction by a natural number.
3,114:4,5=31,14:45.
In the quotient we put a comma as soon as we remove the number 1 in the tenth place. Then we continue dividing.
To complete the division we had to assign zero to the number 9 - differences between numbers 414 And 405 . (we know that zeros can be added to the right side of a decimal fraction)
Answer: 0.692.
Example 4) 53,84: 0,1.
Move the commas in the dividend and divisor to 1 number to the right.
We get: 538,4:1=538,4.
Let's analyze the equality: 53,84:0,1=538,4. Pay attention to the comma in the dividend in this example and the comma in the resulting quotient. We notice that the comma in the dividend has been moved to 1 number to the right, as if we were multiplying 53,84 on 10. (See the video “Multiplying a decimal by 10, 100, 1000, etc.”) Hence the rule for dividing a decimal by 0,1; 0,01; 0,001 etc.
II. To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1, 0.01, 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)
Examples.
Perform division: 1) 617,35: 0,1; 2) 0,235: 0,01; 3) 2,7845: 0,001; 4) 26,397: 0,0001.
Solution.
Example 1) 617,35: 0,1.
According to the rule II division by 0,1 is equivalent to multiplying by 10 , and move the comma in the dividend 1 digit to the right:
1) 617,35:0,1=6173,5.
Example 2) 0,235: 0,01.
Division by 0,01 is equivalent to multiplying by 100 , which means we move the comma in the dividend on 2 digits to the right:
2) 0,235:0,01=23,5.
Example 3) 2,7845: 0,001.
Because division by 0,001 is equivalent to multiplying by 1000 , then move the comma 3 digits to the right:
3) 2,7845:0,001=2784,5.
Example 4) 26,397: 0,0001.
Divide a decimal by 0,0001 - it's the same as multiplying it by 10000 (move the comma by 4 digits right). We get:
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This lesson will look at 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.
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:
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:
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.
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 numbers 10, 100, 1000 is a very simple procedure. The situation is exactly the same with the numbers 0.1, 0.01, 0.001.
Example. Multiply 25.34 by 0.1.
Let's write the decimal fraction 0.1 as an ordinary fraction. But multiplying by is the same as dividing by 10. Therefore, you need to move the decimal point 1 position to the left:
Similarly, multiplying by 0.01 is dividing by 100:
Example. 5.235 divided by 0.1.
The solution to this example is constructed in a similar way: 0.1 is expressed as a 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.
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:
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 test work. 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:
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 as its “partners” they can use integers, ordinary and decimal fractions, and all of them can have both positive and negative values. 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.
