Not divisible by zero. How to remove division by zero error in Excel using a formula. Is it possible to divide by zero

In mathematics, division by zero is impossible! One way to explain this rule is to analyze the process, which shows what happens when one number is divided by another.

Divide by zero error in Excel

In reality, division is essentially the same as subtraction. For example, dividing 10 by 2 is subtracting 2 from 10 multiple times. The multiplicity is repeated until the result is equal to 0. Thus, it is necessary to subtract the number 2 from ten exactly 5 times:

1. 10-2=8
2. 8-2=6
3. 6-2=4
4. 4-2=2
5. 2-2=0

If we try to divide the number 10 by 0, we will never get the result equal to 0, since when subtracting 10-0 there will always be 10. An infinite number of subtractions of zero from ten will not lead us to the result =0. There will always be the same result after the subtraction =10 operation:

• 10-0=10
• 10-0=10
• 10-0=10
• ∞ infinity.

In the lobby of mathematicians, they say that the result of dividing any number by zero is "unlimited." Any computer program that tries to divide by 0 simply returns an error. In Excel, this error is displayed by the value in the #DIV/0! cell.

But if necessary, you can work around the occurrence of a division by 0 error in Excel. You just need to skip the division operation if the denominator is 0. The solution is implemented by placing the operands in the arguments of the =IF() function:

Thus, the Excel formula allows us to "divide" the number by 0 without errors. When dividing any number by 0, the formula will return the value 0. That is, we get the following result after division: 10/0=0.



How does the formula for eliminating the divide-by-zero error work?

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

1. Boolean condition.
2. Actions or values ​​that will be performed if the resulting boolean condition evaluates to TRUE.
3. Actions or values ​​to be executed when the boolean condition evaluates to FALSE.

In this case, the conditional argument contains a value check. Whether the cell values ​​in the Sales column are 0. The first argument to the IF function must always have comparison operators between two values ​​to get the result of the condition as TRUE or FALSE. In most cases, the equals sign is used as the comparison operator, but others can be used, such as greater than > or less than >. Or their combinations - greater than or equal to >=, not equal to!=.

If the condition in the first argument returns TRUE, then the formula will fill the cell with the value from the second argument to the IF function. In this example, the second argument contains the number 0 as its value. This means that the cell in the "Performance" column will simply be filled with the number 0 if there are 0 sales in the cell opposite from the "Sales" column.

If the condition in the first argument evaluates to FALSE, then the value from the third argument to the IF function is used. In this case, this value is formed after the action of dividing the indicator from the "Sales" column by the indicator from the "Plan" column.

Formula for dividing by zero or zero by a number

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

Copy this formula to all cells in the Execution column:

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

Why can't you divide by zero? Who banned? The school stubbornly forbids us to divide by 0, but as soon as we cross the threshold of the university, an indulgence is received. What was considered taboo at school is now possible. You can divide by zero to get infinity. Higher mathematics… Well, almost. It can be explained even better.

History and philosophy of zero

In fact, the story of division by zero haunted its inventors (a). But Indians are philosophers accustomed to abstract problems. What does it mean to divide by nothing? For the Europeans of that time, such a question did not exist at all, since they did not know about zero or negative numbers (which are to the left of zero on the scale).

In India, subtracting a larger from a smaller one and getting a negative number was not a problem. After all, what does 3-5 \u003d -2 mean in ordinary life? This means that someone owes someone 2. Negative numbers were called debts.

Now let's just as simply deal with the issue of division by zero. Back in 598 AD (just think about how long ago, more than 1400 years ago!) In India, the mathematician Brahmagupta was born, who also wondered about dividing by zero.

He suggested that if we take a lemon and start cutting it into pieces, sooner or later we will come to the fact that the slices will be very small. In imagination, we can reach the point where the segments become equal to zero. So, the question is, if you divide a lemon not into 2, 4 or 10 parts, but into an infinite number of parts, what size are the slices? You will get an infinite number of "zero slices". Everything is quite simple, we cut the lemon very finely, we get a puddle with an infinite number of parts - lemon juice.

Just ask yourself the question:

If division by infinity gives zero, then division by zero must give infinity.

x/ ∞=0 means x/0=∞

What happens if you divide by zero?

But if you take up the math, it turns out somehow illogical:

a*0=0? What if b*0=0? So: a*0=b*0

And from here: a=b

That is, any number is equal to any number. The first incorrectness of division by zero, let's move on. In mathematics, division is considered the inverse of multiplication. This means that if we divide 4 by 2, we need to find the number that when multiplied by 2 will give 4.

Divide 4 by zero - you need to find a number that, when multiplied by zero, will give 4. That is, x * 0 \u003d 4? But x*0=0! Again bad luck. So we are asking: "How many zeros do you need to take to get 4?" Infinity? An infinite number of zeros will still add up to zero.

And dividing 0 by 0 generally gives uncertainty, because 0 * x \u003d 0, where x is anything at all. That is, an infinite number of solutions. So what will happen in the end?

A simple explanation from life

Here's a problem from physics and real life. Let's say we want to calculate how long it will take to walk 10 kilometers. So Speed ​​* time = distance (S=Vt). To find out the time, divide the distance by the speed (t=S/V). What happens if we have 0 speed? t=10/0. There will be infinity!

We stand still, the speed is zero, and at this speed we will forever get to the 10 km mark. So the time will be… t=∞. Here we have infinity!

And in this example, you can divide by zero, life experience allows. It's a pity that teachers at school can't explain such things in such a simple way.

Another explanation

Let's define what division is. For example, 8/4 - means the question "how many fours can fit in an eight?" Answer: "two fours", that is, mathematically 8/4=2.

And if you ask yourself the question 5/0=? How many zeros will fit inside a five? Yes, as much as you want. Infinite amount.

But if instead of abstract figures we take material things, for example, an apple. 6/3 - "if you put 6 apples into boxes of 3, how many boxes do you need?" Answer: 2 boxes. We go further 4/0 - “if we put 4 apples into boxes by zero (!) Pieces, then how many ...” It turns out that the boxes are not needed, we don’t put anything anywhere!

A very simple explanation

10/2 =5 10/4 =2,5 10/8 \u003d 1.25 .... The larger the number in the denominator, the smaller the result

10/2 =5 10/1 =10 10/1,5 \u003d 20 .... The smaller the number in the denominator, the greater the result, but if you take a very small number? For example, 0.0000001 would be 1,00,000,000. And if you go further in your thinking and reduce the denominator to zero? As a result, we get something so huge that it will be called "infinity".

So is it possible to divide by zero?

It all depends on why you need it and under what rules you decide to “separate”. If this is algebra, then everything is simply “you can’t divide by zero” because there is no such thing as “infinity” (it’s actually not a number at all), and it’s not clear what should happen in the end.

Is it possible to divide by zero in higher mathematics - yes please. After all, zero can be represented by the number zero (the number means a number with the value "0", that is, nothing at all), or maybe by some infinitesimal (that is, it tends to zero, almost nothing, but still - not nothing). Then nothing prevents you from quietly dividing by "infinitely small".

The illogicality and abstractness of operations with zero is not allowed in the narrow framework of algebra, more precisely, this is an indefinite operation. It needs a more serious apparatus - higher mathematics. So, in a way, you can’t divide by zero, but if you really want to, then you can divide by zero ... But you need to be ready to understand such things as the Dirac delta function and other things that are difficult to comprehend. Share for health.

Zero itself is a very interesting number. By itself, it means emptiness, the absence of meaning, and next to another number increases its significance by 10 times. Any numbers to the zero degree always give 1. This sign was used back in the Mayan civilization, and they also denoted the concept of “beginning, reason”. Even the calendar started from day zero. And this figure is associated with a strict ban.

Ever since the beginning school years we all clearly learned the rule "you cannot divide by zero." But if in childhood you take a lot on faith and the words of an adult rarely cause doubts, then over time, sometimes you still want to understand the reasons, to understand why certain rules were established.

Why can't you divide by zero? I would like to get a clear logical explanation for this question. In the first grade, teachers could not do this, because in mathematics the rules are explained with the help of equations, and at that age we had no idea what it was. And now it's time to figure it out and get a clear logical explanation of why you can't divide by zero.

The fact is that in mathematics only two of the four basic operations (+, -, x, /) with numbers are recognized as independent: multiplication and addition. The rest of the operations are considered to be derivatives. Let's consider a simple example.

Tell me, how much will it turn out if 18 is subtracted from 20? Naturally, the answer immediately arises in our head: it will be 2. And how did we come to such a result? To some, this question will seem strange - after all, everything is clear that it will turn out 2, someone will explain that he took 18 from 20 kopecks and he got two kopecks. Logically, all these answers are not in doubt, but from the point of view of mathematics, this problem should be solved differently. Let us recall once again that the main operations in mathematics are multiplication and addition, and therefore, in our case, the answer lies in solving the following equation: x + 18 = 20. From which it follows that x = 20 - 18, x = 2. It would seem, why paint everything in such detail? After all, everything is so simple. However, without this it is difficult to explain why it is impossible to divide by zero.

Now let's see what happens if we wish to divide 18 by zero. Let's make the equation again: 18: 0 = x. Since the division operation is a derivative of the multiplication procedure, then by transforming our equation we get x * 0 = 18. This is where the impasse begins. Any number in place of x when multiplied by zero will give 0 and we will not succeed in getting 18. Now it becomes extremely clear why you cannot divide by zero. Zero itself can be divided by any number, but vice versa - alas, it is impossible.

What happens when zero is divided by itself? This can be written in this form: 0: 0 = x, or x * 0 = 0. This equation has an infinite number of solutions. So the end result is infinity. Therefore, the operation in this case also does not make sense.

Dividing by 0 is at the root of many imaginary mathematical jokes, which, if desired, can puzzle any ignorant person. For example, consider the equation: 4 * x - 20 \u003d 7 * x - 35. We will take 4 out of brackets on the left side, and 7 on the right. We get: 4 * (x - 5) \u003d 7 * (x - 5). Now we multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will take the following form: 4 * (x - 5) / (x - 5) \u003d 7 * (x - 5) / (x - 5). We reduce the fractions by (x - 5) and we get that 4 \u003d 7. From this we can conclude that 2 * 2 \u003d 7! Of course, the catch here is that it is equal to 5 and it was impossible to reduce fractions, since this led to division by zero. Therefore, when reducing fractions, you must always check that zero does not accidentally end up in the denominator, otherwise the result will turn out to be completely unpredictable.

Evgeny Shiryaev, lecturer and head of the Laboratory of Mathematics of the Polytechnic Museum, told AiF.ru about division by zero:

1. Jurisdiction of the issue

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

Neither the constitution of the Russian Federation, nor the Criminal Code, nor even the charter of your school object to the intellectual action that interests us. This means that the ban has no legal force, and nothing prevents right here, on the pages of AiF.ru, from trying 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 by checking by multiplication: the result multiplied by the divisor had to match the divisible. Did not match - did not decide.

Example 1 1000: 0 =...

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

Incorrect will cut off the check. Iterate over the options: 100, 1, −23, 17, 0, 10,000. For each of them, the test will give the same result:

100 0 = 1 0 = − 23 0 = 17 0 = 0 0 = 10,000 0 = 0

Zero by multiplication turns everything 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 a division is not prohibited, but simply has no result.

3. Nuance

Almost missed one opportunity to refute the ban. Yes, we recognize that a non-zero number will not be divisible by 0. But maybe 0 itself can?

Example 2 0: 0 = ...

Your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor of 0 is equal to the divisible of 0.

More options! one? Also suitable. And -23, and 17, and all-all-all. In this example, the result check will be positive for any number. And to be honest, the solution in this example should not be called a number, but a set of numbers. Everyone. And it won’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 is solved, the nuances are taken into account, the dots are placed, everything is clear - no number can be the answer for the example with division by zero. Solving such problems is hopeless and impossible. So... interesting! Double 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 works, even if we change the task. And there, you see, we will get carried away, and the answer will appear by itself. 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 is to zero, the greater the quotient. The trend can be observed further, moving to fractions and continuing to reduce the numerator:

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

There is no zero in this process and no last quotient. We indicated the movement towards them by 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 double-sided for a reason: some sequences can converge to numbers. Then we can associate a sequence with its numerical limit.

Let's look at the sequence of quotients:

It grows indefinitely, striving for no number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:

Comparing the numbers of sequences with a limit allows us to propose a solution to the third example:

Dividing a sequence converging to 1000 element-wise by a sequence of positive numbers converging to 0, we get a sequence converging to ∞.

5. And here is the nuance with two zeros

What will be the result of dividing two sequences of positive numbers that converge to zero? If they are the same, then the identical unit. If a sequence-dividend converges to zero faster, then in a particular sequence with 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 strongly:

Uncertain situation. And so it is called: the uncertainty of the form 0/0 . When mathematicians see sequences that fit such uncertainty, they don't rush to divide two identical numbers by each other, but figure out which of the sequences runs to zero faster and how. 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 us neglect accurate physical understanding and formally look at the right side as a quotient of two numbers. Imagine that we are solving a school problem on electricity. The condition is given voltage in volts and resistance in ohms. The question is obvious, the decision 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 put it like that R= 0 does not 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 got Nobel Prize. It is useful to be able to bypass any prohibitions!

The number 0 can be represented as a kind of border separating the world of real numbers from imaginary or negative ones. Due to the ambiguous position, many operations with this numerical value do not obey mathematical logic. The inability to divide by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.

History of Zero

Zero is the reference point in all standard number systems. Europeans began to use this number relatively recently, but the wise men ancient india used zero for a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Maya numerical system. This American people used the duodecimal system, and they began the first day of each month with a zero. Interestingly, among the Maya, the sign for "zero" completely coincided with the sign for "infinity". Thus, the ancient Maya concluded that these quantities were identical and unknowable.

Math operations with zero

Standard mathematical operations with zero can be reduced to a few rules.

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

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

Multiplication: any number multiplied by 0 gives 0 in the product (a*0=0).

Division: Zero can be divided by any non-zero number. In this case, the value of such a fraction will be 0. And division by zero is prohibited.

Exponentiation. This action can be performed with any number. An arbitrary number raised to the power of zero will give 1 (x 0 =1).

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

In this case, a contradiction immediately arises: the expression 0 0 does not make sense.

Paradoxes of mathematics

The fact that division by zero is impossible, many people know from school. But for some reason it is not possible to explain the reason for such a ban. Indeed, why does the division-by-zero formula not exist, but other actions with this number are quite reasonable and possible? The answer to this question is given by mathematicians.

The thing is that the usual arithmetic operations that schoolchildren study in elementary grades are in fact far from being as equal as we think. All simple operations with numbers can be reduced to two: addition and multiplication. These operations are the essence of the very concept of a number, and the rest of the operations are based on the use of these two.

Addition and multiplication

Let's take a standard subtraction example: 10-2=8. At school, it is considered simply: if two are taken away from ten objects, eight remain. But mathematicians look at this operation quite differently. After all, there is no such operation as subtraction for them. This example can be written in another way: x+2=10. For mathematicians, the unknown difference is simply the number that must be added to two to make eight. And no subtraction is required here, you just need to find a suitable numerical value.

Multiplication and division are treated in the same way. In the example of 12:4=3 it can be understood that we are talking about the division of eight objects into two equal piles. But in reality, this is just an inverted formula for writing 3x4 \u003d 12. Such examples for division can be given endlessly.

Examples for dividing by 0

This is where it becomes a little clear why it is impossible to divide by zero. Multiplication and division by zero have their own rules. All examples per division of this quantity can be formulated as 6:0=x. But this is an inverted expression of the expression 6 * x = 0. But, as you know, any number multiplied by 0 gives only 0 in the product. This property is inherent in the very concept of a zero value.

It turns out that such a number, which, when multiplied by 0, gives any tangible value, does not exist, that is, this problem has no solution. One should not be afraid of such an answer, it is a natural answer for problems of this type. Just writing 6:0 doesn't make any sense, and it can't explain anything. In short, this expression can be explained by the immortal "no division by zero".

Is there a 0:0 operation? Indeed, if the operation of multiplying by 0 is legal, can zero be divided by zero? After all, an equation of the form 0x5=0 is quite legal. Instead of the number 5, you can put 0, the product will not change from this.

Indeed, 0x0=0. But you still can't divide by 0. As mentioned, division is just the inverse of multiplication. Thus, if in the example 0x5=0, you need to determine the second factor, we get 0x0=5. Or 10. Or infinity. Dividing infinity by zero - how do you like it?

But if any number fits into the expression, then it does not make sense, we cannot choose one from an infinite set of numbers. And if so, it means that the expression 0:0 does not make sense. It turns out that even zero itself cannot be divided by zero.

higher mathematics

Division by zero is a headache for high school math. Mathematical analysis studied in technical universities slightly expands the concept of problems that have no solution. For example, to the already known expression 0:0, new ones are added that have no solution in school mathematics courses:

• infinity divided by infinity: ?:?;
• infinity minus infinity: ???;
• unit raised to an infinite power: 1? ;
• infinity multiplied by 0: ?*0;
• some others.

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

Uncertainty Disclosure

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

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

When considering the limits of trigonometric functions, their expressions tend to be reduced to the first remarkable limit. When considering the limits in which the denominator goes to 0 when the limit is substituted, the second remarkable limit is used.

L'Hopital Method

In some cases, the limits of expressions can be replaced by the limit of their derivatives. Guillaume Lopital is a French mathematician, the founder of the French school of mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule is as follows.

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

How to divide and multiply by 0.1; 0.01; 0.001 etc.?

Write the rules for division and multiplication.

To multiply a number by 0.1, you just need to move the comma.

For example it was 56 , became 5,6 .

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

For example it was 56 , became 560 .

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

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

For example, there is a number 123456789.

You need to multiply it by 0.000000001

There are nine zeros in the number 0.000000001 (we also count the zero to the left of the decimal point), which means we shift the number 123456789 by 9 digits:

It was 123456789 became 0.123456789.

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

It was 123456789 became 123456789000000000.

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

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

Dividing a number by 0.01 is equivalent to multiplying that number by 100

Dividing by 0.001 is multiplying by 1000.

To make it easier to remember, we read the number by which we need to divide from right to left, ignoring the comma, and multiply by the resulting number.

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

The same with multiplication, only in reverse:

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

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

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5.5.6. Decimal division

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

Primery.

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

Decision.

Example 1) 16,38: 0,7.

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

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

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

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

Answer: 23.4.

Example 2) 15,6: 0,15.

Move commas in dividend ( 15,6 ) and divisor ( 0,15 ) two digits to the right, since in the divisor 0,15 there are two digits after the decimal point.

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

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

Perform division of natural numbers.

Answer: 104.

Example 3) 3,114: 4,5.

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

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

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

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

Answer: 0.692.

Example 4) 53,84: 0,1.

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

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

Let's analyze the equality: 53,84:0,1=538,4. We pay attention to the comma in the dividend in this example and to the comma in the resulting quotient. Note that the comma in the dividend has been moved to 1 digit to the right, as if we were multiplying 53,84 on the 10. (Watch the video “Multiplying a decimal by 10, 100, 1000, etc.”) Hence the rule for dividing a decimal by 0,1; 0,01; 0,001 etc.

II. To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the comma to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1; 0.01; 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)

Examples.

Perform division: 1) 617,35: 0,1; 2) 0,235: 0,01; 3) 2,7845: 0,001; 4) 26,397: 0,0001.

Decision.

Example 1) 617,35: 0,1.

According to the rule II division into 0,1 is equivalent to multiplying by 10 , and move the comma in the dividend 1 digit to the right:

1) 617,35:0,1=6173,5.

Example 2) 0,235: 0,01.

Division by 0,01 is equivalent to multiplying by 100 , which means that we will transfer the comma in the dividend on the 2 digits to the right:

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

Example 3) 2,7845: 0,001.

As division into 0,001 is equivalent to multiplying by 1000 , then move the comma 3 digits to the right:

3) 2,7845:0,001=2784,5.

Example 4) 26,397: 0,0001.

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

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

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

Multiplying numbers by 10, 100

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

The decimal notation for a given number is an abbreviated notation for the sum. You need to describe it in more detail:

Thus, you need to multiply the amount. To do this, you can simply multiply each term:

It turns out that.

We can conclude that multiplying a decimal by 10 is very simple: you need to shift the comma to the right by one position.

An exercise. Multiply 25.486 by 100.

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

Division of numbers by 10, 100

An exercise. Divide 25.78 by 10.

As in the previous case, it is necessary to represent the number 25.78 as a sum:

Since you need to divide the sum, this is equivalent to dividing each term:

It turns out that to divide by 10, you need to move the comma to the left by one position. For example:

An exercise. Divide 124.478 by 100.

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

Rule of multiplication and division by 10, 100, 1000

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

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

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

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

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

You need to move 4 positions to the right. But there are only two digits after the decimal point. It is worth remembering that there is an equivalent notation for the fraction 56.14.

Now multiplying by 10,000 is easy:

If it is not very clear why you can add two zeros to the fraction in the previous example, then the additional video at the link can help with this.

Equivalent Decimal Entries

Entry 52 means the following:

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

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

Now let's look at the decimal:

If we assign zero, then we get:

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

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

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

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

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

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

Multiplication and division by 0.1, 0.01, 0.001

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

Example. Multiply 25.34 by 0.1.

Let's write the decimal fraction 0.1 in the form of an ordinary. But multiplying by is the same as dividing by 10. Therefore, you need to move the comma 1 position to the left:

Similarly, multiplying by 0.01 is dividing by 100:

Example. 5.235 divided by 0.1.

The solution of this example is built in a similar way: 0.1 is expressed as common fraction, and dividing by is the same as multiplying by 10:

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

Rule for multiplying and dividing by 0.1, 0.01, 0.001

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

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

Solution of examples

Conclusion

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

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

Bibliography

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

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

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

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

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 expression values:

Actions with zero

In mathematics, the number zero occupies a special place. The fact is that it, in fact, means “nothing”, “emptiness”, but its significance is really difficult to overestimate. To do this, it is enough to remember at least what exactly with zero mark and the countdown of the coordinates of the point position in any coordinate system begins.

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

With zero all arithmetic operations are carried out, and integers, ordinary and decimals, and all of them can have both positive and negative meaning. We give examples of their implementation and some explanations for them.

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

twenty four plus zero equals twenty-four.

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

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