"You can't divide by zero!" - most schoolchildren memorize this rule by heart, without asking questions. All children know what “no” is and what will happen if you ask in response to it: “Why?” But in fact, it is very interesting and important to know why it is impossible.
The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as full-fledged - 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, for example, subtraction. What means 5 – 3 ? The student will answer this simply: you need to take five items, 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, only addition. Therefore, the entry 5 – 3 means a number that, when added to a number 3 will give the number 5 . I.e 5 – 3 is just a shorthand for the equation: x + 3 = 5. There is no subtraction in this equation. There is only one task - to find suitable number.
The same is true with multiplication and division. Recording 8: 4 can be understood as the result of the division of eight objects into four equal piles. But it's really just a shortened form of the equation 4 x = 8.
This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Recording 5: 0 is an abbreviation for 0 x = 5. That is, this task is to find a number that, when multiplied by 0 will give 5 . But we know that when multiplied by 0 always turns out 0 . This is an inherent property of zero, strictly speaking, part of its definition.
A number that, when multiplied by 0 will give something other than null, just doesn't exist. That is, our problem has no solution. (Yes, it happens, not every problem has a solution.) 5: 0 does not correspond to any specific number, and it simply does not stand for anything and therefore does not make sense. The meaninglessness of this entry is briefly expressed by saying that you cannot divide by zero.
The most attentive readers at this point will certainly ask: is it possible to divide zero by zero? Indeed, since the equation 0 x = 0 successfully resolved. For example, you can take x=0, and then we get 0 0 = 0. It turns out 0: 0=0 ? But let's not rush. Let's try to take x=1. Get 0 1 = 0. Correctly? Means, 0: 0 = 1 ? But you can take any number and get 0: 0 = 5 , 0: 0 = 317 etc.
But if any number is suitable, then we have no reason to opt for any one of them. That is, we cannot tell which number corresponds to the entry 0: 0 . And if so, then we are forced to admit that this record also does not make sense. 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 can give preference to one of the options solution of the equation 0 x = 0; in such cases, mathematicians speak of "disclosure of indeterminacy", but in arithmetic such cases do not occur.)
This is the feature of the division operation. To be more precise, the multiplication operation and the number associated with it have zero.
Well, the most meticulous, having read up to this point, may ask: why is it so that you cannot 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 numerical sets and operations on them. It is not so difficult, but for some reason it is not studied at school. But in lectures on mathematics at the university, you will be taught this in the first place.
The mathematical rule regarding division by zero was told to all people in the first grade. secondary school. “You can’t divide by zero,” they taught all of us and forbade, under pain of a slap in the back, to divide by zero and generally discuss this topic. Although some elementary school teachers still tried to explain why it is impossible to divide by zero using simple examples, these examples were so illogical that it was easier to just remember this rule and not ask too many questions. But all these examples were illogical for the reason that the teachers could not logically explain this to us in the first grade, since in the first grade we did not even know close what an equation is, but logically it is mathematical rule can only be explained using equations.
Everyone knows that when dividing any number by zero, a void will come out. Why exactly emptiness, we will consider later.
In general, in mathematics, only two procedures with numbers are recognized as independent. This is addition and multiplication. The remaining procedures are considered derivatives of these two procedures. Let's look at this with an example.
Tell me, how much will it be, for example, 11-10? We will all instantly answer that it will be 1. And how did we find such an answer? Someone will say that it’s already clear that it will be 1, someone will say that he took 10 from 11 apples and calculated that it turned out to be one apple. From the point of view of logic, everything is correct, but according to the laws of mathematics, this problem is solved differently. It must be remembered that addition and multiplication are considered the main procedures, so you need to make the following equation: x + 10 \u003d 11, and only then x \u003d 11-10, x \u003d 1. Note that addition comes first, and only then, based on the equation, can we subtract. It would seem, why so many procedures? After all, the answer is so obvious. But only such procedures can explain the impossibility of dividing by zero.
For example, we do this math problem: want to divide 20 by zero. So 20:0=x. To find out how much it will be, you need to remember that the division procedure follows from multiplication. In other words, division is the derivative procedure of multiplication. Therefore, you need to make an equation from multiplication. So, 0*x=20. Here is the dead end. Whatever number we multiply by zero, it will still be 0, but not 20. This is where the rule follows: you cannot divide by zero. Zero can be divided by any number, but a number cannot be divided by zero.
This raises another question: is it possible to divide zero by zero? So 0:0=x means 0*x=0. This equation can be solved. Take, for example, x=4, which means 0*4=0. It turns out that if you divide zero by zero, you get 4. But even here everything is not so simple. If we take, for example, x=12 or x=13, then the same answer will come out (0*12=0). In general, no matter what number we substitute, 0 will still come out. Therefore, if 0: 0, then infinity will turn out. Here's some simple math. Unfortunately, the procedure for dividing zero by zero is also meaningless.
In general, the number zero in mathematics is the most interesting. For example, everyone knows that any number to the zero power gives one. Of course, with such an example in real life we do not meet, but with division by zero life situations come across very often. So remember that you can't divide by zero.
Very often, many people wonder why it is impossible to use division by zero? In this article, we will go into great detail about where this rule came from, as well as what actions can be performed with zero.
In contact with
Zero can be called one of the most interesting numbers. This number has no meaning, it means emptiness in literally the words. However, if you put zero next to any digit, then the value of this digit will become several times larger.
The number is very mysterious in itself. It has also been used ancient people Mayan. For the Maya, zero meant "beginning", and the countdown calendar days also started from scratch.
Highly interesting fact is that the zero sign and the uncertainty sign were similar. With this, the Maya wanted to show that zero is the same identical sign as well as uncertainty. In Europe, the designation of zero appeared relatively recently.
Also, many people know the prohibition associated with zero. Any person will say that cannot be divided by zero. This is said by teachers at school, 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, interest awakens, and you want to know more about the reasons for such a ban. However, there is reasonable evidence.
Actions with zero
First you need to determine what actions can be performed with zero. Exist several types of activities:
- Addition;
- Multiplication;
- Subtraction;
- Division (zero by number);
- Exponentiation.
Important! If zero is added to any number when adding, 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.
With multiplication and division, things are a little different. If a multiply any number by zero, then the product will also become zero.
Consider an example:
Let's write this as an addition:
There are five added zeros in total, so it turns out that
Let's try to multiply one by zero. The result will also be null.
Zero can also be divided by any other number not equal to it. In this case, it will turn out, the value of which will also be zero. The same rule applies for negative numbers. If you divide zero by a negative number, you get zero.
You can also raise any number in zero degree . In this case, you get 1. It is important to remember that the expression "zero to the zero power" is absolutely meaningless. If you try to raise zero to any power, you get zero. Example:
We use the multiplication rule, we get 0.
Is it possible to divide by zero
So, here we come to the main question. Is it possible to divide by zero generally? And why is it impossible to divide a number by zero, given that all other operations with zero fully exist and apply? To answer this question, you need 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 pens, it means that you have no pens at all.
In higher mathematics, the concept of "zero" is broader. It doesn't mean empty at all. Here zero is called uncertainty, since if we draw a little research, it turns out that when dividing zero by zero, we can get any other number as a result, which may not necessarily be zero.
Did you know that those simple arithmetic operations that you studied at school are not so equal among themselves? The most basic steps are addition and multiplication.
For mathematicians, the concepts of "" and "subtraction" do not exist. Suppose: if three are subtracted from five, then two will remain. 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 a suitable number. This rule applies to addition.
Things are a little different with multiplication and division rules. It is known that multiplication by zero leads to zero result. For example, if 3:0=x, then if you flip the record, you get 3*x=0. And the number that is multiplied by 0 will give zero in the product. It turns out that a number that would give any value other than zero in the product with zero does not exist. This means that division by zero is meaningless, that is, it fits our rule.
But what happens if you try to divide zero by itself? Let's take x as some indefinite number. It turns out the equation 0 * x \u003d 0. It can be solved.
If we try to take zero instead of x, we get 0:0=0. It would seem logical? But if we try to take any other number instead of x, for example, 1, then we end up with 0:0=1. The same situation will be if you 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 different numbers. Sometimes, nevertheless, division by 0 in higher mathematics makes sense, but then usually there is a certain condition due 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 any one number from the set.
Important! Zero cannot be divided by zero.
Zero and infinity
Infinity is very common in higher mathematics. Since it is simply not important for schoolchildren to know that there are still mathematical operations with infinity, teachers cannot properly explain to children why it is impossible to divide by zero.
Main mathematical secrets students begin to learn only in the first year of the institute. Higher mathematics provides a large set of problems that have no solution. The most famous problems are the problems with infinity. They can be solved with mathematical analysis.
You can also apply to infinity elementary mathematical operations: addition, multiplication by a number. Subtraction and division are also commonly used, but in the end they still come down to two simple operations.
Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still a lot of controversy constantly arises around him. Someone just memorized the rule and does not bother with the question “why?”. “You can’t do everything here, because at school they said so, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.
Who is right in the end
During these disputes, both people, having opposite points vision, 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 against each other with their horns. The only difference between them is that one is slightly less educated than the other. Most often, those who consider this rule to be wrong try to call for 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 let's discard this conclusion right away - it is illogical, although it has the opposite goal - to call to logic.
This is interesting: How to find the difference of numbers in mathematics?
What is multiplication
The original multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:
From this equation follows the conclusion, that multiplication is a simplified addition.
What is zero
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 otherwise - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw deep meaning in this number. After all, zero, which has the value 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 identify empty bits in decimal fractions, this is done both before and after the comma.
Is it possible to multiply by emptiness
It is possible to multiply by zero, but it is useless, because, whatever one may say, but even when multiplying negative numbers, zero will still be obtained. It is enough just to remember this simplest rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and mysteries, as ancient scholars believed. The most logical explanation will be given below that this multiplication is useless, because when multiplying a number by it, the same thing will still be obtained - zero.
This is interesting: what is the modulus of a number?
Going back to the very beginning, the argument about two apples, 2 times 0 looks like this:
After all, eating an apple 0 times means not eating a single one. It will be clear even to a small child. Like it or not, 0 will come out, two or three can be replaced with absolutely any number and absolutely the same thing will come out. And to put it simply, zero is nothing and when you have there is nothing, then no matter how much you multiply - it's all the same 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 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 follows another important rule:
You can't divide by zero!
This rule, too, has been stubbornly hammered into our heads since childhood. We just know that it’s impossible and that’s all without worrying our heads extra information. If you are suddenly asked the question, for what reason 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 school curriculum, because there is not so much controversy and controversy around this rule.
Everyone just memorized the rule and does 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 full of the above, and all other manipulations with numbers are built from them. That is, the entry 10: 2 is an abbreviation of the equation 2 * x = 10. Therefore, the entry 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
To not divide by 0!
Cut 1 as you like, along,
Just don't divide by 0!
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Division by zero. Fascinating math
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 impossibility of dividing by zero bright to that example. And allowed arithmetic operations with zero can be performed using generally accepted definitions.
History of Zero
Zero is the reference point in all standard systems calculus. 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 several 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 bench. 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 primary school are actually not 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 standard example for subtraction: 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 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 given task 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 an infinite number pick one number. 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 school mathematics. Studied in technical universities mathematical analysis slightly expands the concept of problems that have no solution. For example, to already famous expression 0:0 new ones are added that have no solution in school courses mathematics:
It is impossible to solve such expressions by elementary methods. But higher mathematics thanks to additional features for a number 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 the conditional infinitesimal variable. And the expressions in which, when substituting desired value division by zero is obtained, 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 trigonometric functions their expressions tend to be reduced to the first wonderful 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 - French mathematician, founder french school mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. AT 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 ∞:∞.
Math: long division and multiplication
Multiplication and division of single-digit numbers will not be difficult for any student who has learned the multiplication table. It is included in the 2nd grade math curriculum. Another thing is when it is necessary to perform mathematical operations with multi-digit numbers. They begin such actions in mathematics lessons in grade 3. Parsing new theme"Division and multiplication in a column"
Multiplication of multi-digit numbers
Divide and multiply complex numbers the easiest way is a column. To do this, you need the digits of the number: hundreds, tens, units:
235 = 200 (hundreds) + 30 (tens) + 5 (ones).
We will need this for correct notation numbers when multiplied.
When writing two numbers that need to be multiplied, they are written one under the other, placing the numbers in digits (units under units, tens under tens). When multiplying a multi-digit number by a single-digit number, there will be no difficulties:
The recording is done like this: 
The calculation is carried out from the end - from the category of units. When multiplying by the first digit - from the category of units - the record is also carried out from the end:
- 3 x 5 = 15, write down 5 (ones), tens (1) remember;
- 2 x 5 \u003d 10 and 1 ten that we remembered, only 11, we write down 1 (tens), we remember hundreds (1);
- since we don’t have further digits in the example, we write down hundreds (1 - which was remembered).
The next step is to multiply by the second digit (tens place):
Since we multiplied by a number from the tens place, we will start writing in the same way, from the end, starting from the second place on the right (where the tens place is).
1. you need to write down the multiplication in a column by digits;
2. make calculations starting from units;
3. write down the total by digits - if we multiply by a figure from the rank of units - we start the recording from the last column, from the rank - tens - from this column and keep the record.
The rule that applies to multiplication in a column by a two-digit number also applies to numbers with large quantity discharges.
To make it easier to remember the rules for writing multiplication examples multi-digit numbers in a column, you can make cards by highlighting different colors different ranks.
If numbers are multiplied in a column with zeros at the end, they are not taken into account in the calculation, and the record is kept in such a way that significant figure was under the signifier, and the zeros remain to the right. After the calculations, their number is added to the right:
Mathematician Yakov Trakhtenberg developed a system of rapid counting. The Trachtenberg method facilitates multiplication if a certain system of calculations is applied. For example, multiplying by 11. To get the result, you need to add a number to the next one:
2.253 x 11 = (0 + 2) (2 + 2) (2 + 5) (5 + 3) (3 + 0) = 2 + 4 + 7 + 8 + 3 = 24.783.
Proving true is simple: 11 = 10 + 1
2.253 x 10 + 2.253 = 22.530 + 2.253 = 24.783.
Calculation algorithms for different numbers are different, but they allow you to perform calculations quickly.
Video "Column multiplication"
Division of multi-digit numbers
Dividing by a column may seem difficult for children, but remembering the algorithm is not difficult. Consider the division of multi-digit numbers by single digit:
215: 5 = ?
The calculation is written as follows: 
Under the divisor we will write the result. The division is performed as follows: we compare the leftmost digit of the dividend with the divisor: 2 is less than 5, we cannot divide 2 by 5, so we take one more digit: 21 is greater than 5, when dividing it turns out: 20: 5 = 4 (remainder 1)
We demolish the following figure to the resulting remainder: we get 15. 15 is more than 5, we divide: 15: 5 = 3
The solution will look like this:
This is how division is made without a remainder. According to the same algorithm, division into a column with a remainder is performed with the only difference that in last entry it will not be zero, but the remainder.
If it is necessary to divide three-digit numbers in a column by two-digit, the procedure will be the same as when dividing by a single-digit number.
Here are some examples for division:
Similarly, the calculation is carried out when dividing a multi-digit number by a two-digit number with a remainder: 853: 15 = 50 and (3) the remainder 
Pay attention to this entry: if intermediate calculations the result is 0, but the example is not completely solved, zero is not written down, but the next digit is immediately demolished, and the calculation continues.
It will help to learn the rules for dividing multi-digit numbers in a video tutorial column. Having memorized the algorithm and following the sequence of recording calculations, examples of multiplication and division in a column in grade 4 will no longer seem so complicated.
Important! Follow the record: the digits should be written under the digits, in a column.
Video "Division in a column"
If in grade 2 a child learned the multiplication table, examples of multiplication and division of a two-digit or three digit number in mathematics lessons for grade 4 will not cause him any difficulties.
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Multiplication and division rules
After the multiplication table is learned, the students are explained the rules of multiplication and division, taught to use them when calculating mathematical expressions.
What is multiplication? It's smart addition
When adding and subtracting, multiplying and dividing numbers in simple expressions children do not have difficulties:
In such calculations, you only need to know the rules of addition and subtraction and the multiplication table.
When more start difficult exercises, examples consist of two or more actions, and even with brackets, children have errors when solving. And the main one is wrong order actions.
What's the difference?
Indeed, is it so important - which action in the example to perform first, which second?
If we perform the steps in order, we get:
We got two different answers. But it should not be so, therefore, the order in which actions are performed matters. Especially if the expression contains parentheses:
We are trying to solve it in two ways:
The answers are different, and in order to determine the order of actions, there are brackets in the expression - they show which action must be performed first. So the correct solution would be:
There should be no other solution for the answer in the example.
Which is more important, multiplication or addition?
When solving examples
Arrange the course of action.
Multiply or divide - in first place.
For expressions in which there is not addition or subtraction, but multiplication or division, the same rule applies: all operations with numbers are performed in order, starting from the left:
A more difficult case is when multiplication or division with addition or subtraction occur in one problem. What is the order of calculations then?
If you perform all the steps in order, first division, then addition. As a result, we get:
So the example is correct. What if it contains parentheses?
Anything in parentheses always takes precedence. That's why they stand in the expression. Therefore, the order of calculations in similar expressions will be as follows:
Example:
81: 9 + (6 – 2) + 3 = ?
81: 9 + (6 – 2) + 3 = 16.
And what will be the priority: multiplication - or division, subtraction - or addition, if both actions occur in the task? Nothing, they are equal, in this case the first rule applies - actions are performed one after the other, starting from the left.
Algorithm for solving the expression:
28: (11 – 4) + 18 – (25 – 8) = ?
- 11 – 4 = 7;
- 25 – 8 = 17;
- 28: 7 = 4;
- 4 + 18 = 22;
- 22 – 17 = 5.
Answer: 28: (11 - 4) + 18 - (25 - 8) = 5.
Important! If the expression contains letters, the procedure remains the same.
Round zero is so pretty
But it doesn't mean anything.
In the examples, zero does not occur as a number, but it can be the result of some intermediate action, for example:
When multiplying by 0, the rule says that the result will always be 0. Why? It can be explained simply: what is multiplication? This is the same number, added to its own kind several times. Otherwise:
0 × 5 = 0 + 0 + 0 + 0 + 0 = 0;
Dividing by 0 is meaningless, and dividing zero by any number will always result in 0:
0: 5 = 0.
Recall other arithmetic operations with zero:
Multiplication and division by one
Mathematical operations with one are different from operations with zero. When a number is multiplied or divided by 1, the original number itself is obtained:
7 x 1 = 7;
7: 1 = 7.
Of course, if you have 7 friends, and each gave you a candy, you will have 7 candies, and if you ate them alone, that is, shared only with yourself, then all of them ended up in your stomach.
Calculations with fractions, powers and complex functions
This is difficult cases computing, which are not covered in elementary school.
Multiplication simple fractions each other is not difficult, it is enough just to multiply the numerator by the numerator, and the denominator by the denominator.
Example:
After reduction we get: \(\) = \(\).
Dividing simple fractions is not as difficult as it seems at first glance. It is enough to transform the problem - turn it into an example with multiplication. To do this is simple - you need to flip the fraction so that the denominator becomes the numerator, and the numerator becomes the denominator.
Example:
If a number is encountered in the problem, represented as a power, its value is calculated before all the others (you can imagine that it is enclosed in brackets - and the actions in brackets are performed first).
Example:
By converting the number represented as a power into a regular expression with the action of multiplication, solving the example turned out to be simple: first multiplication, then subtraction (because it is in brackets) and division.
Since such functions are studied only within the framework of high school, we will not consider them, it is enough just to say that, as in the case of powers, they have priority in the calculation: first, the value of this expression is found, then the calculation order is normal - brackets, multiplication with division, then in order from left to right.
Main rules on the topic
Talking about major and minor mathematical operations, it must be said that the four basic operations can be reduced to two: addition and multiplication. If subtraction and division seem difficult for schoolchildren, they remember the rules of addition and multiplication faster. Indeed, the expression 5 - 2 can be written differently:
In cases with multiplication, rules similar to the properties of addition apply: the product will not change from a rearrangement of factors:
When deciding challenging tasks the first action is the one highlighted in brackets, then the division or multiplication, then all the other actions in order.
When you need to solve examples without brackets, first multiplication or division is performed, then subtraction or addition.
Multiplication and division of integers
When multiplying and dividing integers, several rules apply. AT this lesson we will look at each of them.
When multiplying and dividing integers, pay attention to the signs of the numbers. It will depend on them which rule to apply. You also need to learn a few laws of multiplication and division. Learning these rules will help you avoid some embarrassing mistakes in the future.
Laws of multiplication
Some of the laws of mathematics we considered in the lesson the laws of mathematics. But we have not considered all the laws. There are many laws in mathematics, and it would be wiser to study them sequentially as needed.
First, let's remember what multiplication consists of. Multiplication consists of three parameters: multiplying, multiplier and works. For example, in the expression 3 × 2 = 6, the number 3 is the multiplicand, the number 2 is the multiplier, and the number 6 is the product.
Multiplicand shows what exactly we are increasing. In our example, we increase the number 3.
Factor Shows how many times you need to increase the multiplicand. In our example, the multiplier is the number 2. This multiplier shows how many times you need to increase the multiplier 3. That is, during the multiplication operation, the number 3 will be doubled.
Work this is actually the result of the multiplication operation. In our example, the product is the number 6. This product is the result of multiplying 3 by 2.
The expression 3 × 2 can also be understood as the sum of two triples. Multiplier 2 in this case will show how many times you need to take the number 3:
Thus, if you take the number 3 twice in a row, you get the number 6.
Commutative law of multiplication
The multiplier and the multiplier are called one common word – factors. The commutative law of multiplication looks like this:
From the permutation of the places of the factors, the product does not change.
Let's check if this is the case. Multiply for example 3 by 5. Here 3 and 5 are factors.
Now let's swap the factors:
In both cases, we get the answer 15, which means we can put an equal sign between the expressions 3 × 5 and 5 × 3, since they are equal to the same value:
And with the help of variables displacement law multiplication can be written like this:
where a and b- factors
Associative law of multiplication
This law says that if an expression consists of several factors, then the product will not depend on the order of operations.
For example, the expression 3 × 2 × 4 consists of several factors. To calculate it, you can multiply 3 and 2, then multiply the resulting product by the remaining number 4. It will look like this:
3 x 2 x 4 = (3 x 2) x 4 = 6 x 4 = 24
This was the first solution. The second option is to multiply 2 and 4, then multiply the resulting product by the remaining number 3. It will look like this:
3 x 2 x 4 = 3 x (2 x 4) = 3 x 8 = 24
In both cases, we get the answer 24. Therefore, between the expressions (3 × 2) × 4 and 3 × (2 × 4) we can put an equal sign, since they are equal to the same value:
(3 x 2) x 4 = 3 x (2 x 4)
and with the help of variables, the associative law of multiplication can be written as follows:
a × b × c = (a × b) × c = a × (b × c)
where instead of a, b, c can be any number.
Distributive law of multiplication
The distributive law of multiplication allows you to multiply a sum by a number. To do this, each term of this sum is multiplied by this number, then the results are added.
For example, let's find the value of the expression (2 + 3) × 5
The expression in brackets is the sum. This amount must be multiplied by the number 5. To do this, each term of this sum, that is, the numbers 2 and 3, must be multiplied by the number 5, then add the results:
(2 + 3) × 5 = 2 × 5 + 3 × 5 = 10 + 15 = 25
So the value of the expression (2 + 3) × 5 is 25 .
With the help of variables, the distributive law of multiplication is written as follows:
(a + b) × c = a × c + b × c
where instead of a, b, c can be any number.
The law of multiplication by zero
This law says that if in any multiplication there is at least one zero, then the answer will be zero.
The product is zero if at least one of the factors zero.
For example, the expression 0 × 2 is zero
In this case, the number 2 is a multiplier and shows how many times you need to increase the multiplicand. That is, how many times to increase zero. Literally, this expression is read as "increase zero twice." But how can you double zero if it's zero?
In other words, if “nothing” is doubled, or even a million times, it will still be “nothing”.
And if in the expression 0 × 2 we swap the factors, again we get zero. We know this from the previous displacement law:
Examples of applying the law of multiplication by zero:
2 x 5 x 0 x 9 x 1 = 0
In the last two examples, there are several factors. Seeing zero in them, we immediately put zero in the answer, applying the law of multiplication by zero.
We have considered the basic laws of multiplication. Next, consider the multiplication of integers.
Integer multiplication
Example 1 Find the value of the expression −5 × 2
This is the multiplication of numbers with different signs. −5 is negative and 2 is positive. For such cases, the following rule should be applied:
To multiply numbers with different signs, you need to multiply their modules, and put a minus before the received answer.
−5 × 2 = − (|−5| × |2|) = − (5 × 2) = − (10) = −10
Usually written shorter: −5 × 2 = −10
Any multiplication can be represented as a sum of numbers. For example, consider the expression 2 × 3. It equals 6.
multiplier in given expression is the number 3. This multiplier shows how many times you need to increase the two. But the expression 2 × 3 can also be understood as the sum of three deuces:
The same thing happens with the expression −5 × 2. This expression can be represented as a sum
And the expression (-5) + (-5) is equal to -10, and we know this from the last lesson. This is the addition of negative numbers. Recall that the result of adding negative numbers is a negative number.
Example 2 Find the value of the expression 12 × (−5)
This is the multiplication of numbers with different signs. 12 - positive number, (−5) is negative. Again, we apply the previous rule. We multiply the modules of numbers and put a minus before the received answer:
12 × (−5) = − (|12| × |−5|) = − (12 × 5) = − (60) = −60
Usually written shorter: 12 × (−5) = −60
Example 3 Find the value of the expression 10 × (−4) × 2
This expression consists of several factors. First, multiply 10 and (−4), then multiply the resulting number by 2. Along the way, apply the previously studied rules:
10 × (−4) = −(|10| × |−4|) = −(10 × 4) = (−40) = −40
Second action:
−40 × 2 = −(|−40 | × | 2|) = −(40 × 2) = −(80) = −80
So the value of the expression 10 × (−4) × 2 is −80
Usually written shorter: 10 × (-4) × 2 = -40 × 2 = -80
Example 4 Find the value of the expression (−4) × (−2)
This is the multiplication of negative numbers. In such cases, the following rule should apply:
To multiply negative numbers, you need to multiply their modules and put a plus in front of the received answer.
(−4) × (−2) = |−4| × |−2| = 4 × 2 = 8
Plus, by tradition, we do not write down, so we just write down the answer 8.
Usually written shorter (−4) × (−2) = 8
The question arises why, when multiplying negative numbers, a positive number suddenly turns out. Let's try to prove that (−4) × (−2) equals 8 and nothing else.
First, we write the following expression:
Let's enclose it in brackets:
Let's add our expression (−4) × (−2) to this expression. Let's put it in parentheses too:
We equate all this to zero:
(4 × (−2)) + ((−4) × (−2)) = 0
Now the fun begins. The bottom line is that we must calculate the left side of this expression, and as a result get 0.
So the first product (4 × (−2)) is −8. Let's write the number −8 in our expression instead of the product (4 × (−2))
Now, instead of the second product, we temporarily put an ellipsis
Now let's look carefully at the expression −8 + […] = 0. What number should be used instead of the ellipsis in order for equality to be observed? The answer suggests itself. Instead of an ellipsis, there should be a positive number 8 and no other. Only in this way will equality be maintained. Because −8 + 8 equals 0.
We return to the expression −8 + ((−4) × (−2)) = 0 and instead of the product ((−4) × (−2)) we write the number 8
Example 5 Find the value of the expression −2 × (6 + 4)
We apply the distributive law of multiplication, that is, we multiply the number −2 by each term of the sum (6 + 4)
−2 × (6 + 4) = (−2 × 6) + (−2 × 4)
Now let's evaluate the expressions in brackets. Then we add up the results. Along the way, apply the previously learned rules. The entry with modules can be omitted so as not to clutter up the expression
−2 × 6 = −(2 × 6) = −(12) = −12
−2 × 4 = −(2 × 4) = −(8) = −8
Third action:
So the value of the expression −2 × (6 + 4) is −20
Usually written shorter: −2 × (6 + 4) = (−12) + (−8) = −20
Example 6 Find the value of the expression (−2) × (−3) × (−4)
The expression consists of several factors. First, we multiply the numbers -2 and -3, and the resulting product is multiplied by the remaining number -4. We skip the entry with modules so as not to clutter up the expression
So the value of the expression (−2) × (−3) × (−4) is −24
Usually written shorter: (−2) × (−3) × (−4) = 6 × (−4) = −24
Division laws
Before dividing integers, it is necessary to study two laws of division.
First of all, let's remember what division consists of. The division consists of three parameters: divisible, divider and private. For example, in expression 8: 2 = 4, 8 is the dividend, 2 is the divisor, 4 is the quotient.
Dividend shows exactly what we share. In our example, we are dividing the number 8.
Divider Shows how many parts to divide the dividend. In our example, the divisor is the number 2. This divisor shows how many parts to divide the dividend 8. That is, during the division operation, the number 8 will be divided into two parts.
Private is the actual result of the division operation. In our example, the quotient is 4. This quotient is the result of dividing 8 by 2.
Can't divide by zero
Any number cannot be divided by zero. This is because division is the inverse of multiplication. For example, if 2 × 6 = 12, then 12:6 = 2
It can be seen that the second expression is written in reverse order.
Now we will do the same for the expression 5 × 0. We know from the laws of multiplication that the product is equal to zero if at least one of the factors is equal to zero. So the expression 5 × 0 is also zero
If we write this expression in reverse order, we get:
The answer immediately catches the eye is 5, which is the result of dividing zero by zero. It's impossible and stupid.
Another similar expression can be written in reverse order, for example 2 × 0 = 0
In the first case, dividing zero by zero, we got 5, and in the second case, 2. That is, each time dividing zero by zero, we can get different meanings, which is unacceptable.
The second explanation is that dividing the dividend by the divisor means finding a number that, when multiplied by the divisor, will give the dividend.
For example, the expression 8: 2 means to find a number that, when multiplied by 2, will give 8
Here, instead of the ellipsis, there should be a number that, when multiplied by 2, gives the answer 8. To find this number, it is enough to write this expression in reverse order:
Now imagine that you need to find the value of the expression 5: 0. In this case, 5 is the dividend, 0 is the divisor. To divide 5 by 0 means to find a number that, when multiplied by 0, will give 5
Here, instead of the ellipsis, there should be a number that, when multiplied by 0, gives the answer 5. But there is no number that, when multiplied by zero, gives 5.
The expression […] × 0 = 5 contradicts the law of multiplication by zero, which states that the product is equal to zero when at least one of the factors is equal to zero.
So writing the expression […] × 0 = 5 in reverse order, dividing 5 by 0 makes no sense. That's why they say you can't divide by zero.
Using variables this law is written as follows:
At b ≠ 0
Number a can be divided by a number b, provided that b not equal to zero.
private property
This law says that if the dividend and the divisor are multiplied or divided by the same number, then the quotient will not change.
For example, consider the expression 12: 4. The value of this expression is 3
Let's try to multiply the dividend and the divisor by the same number, for example, by the number 4. If we believe the quotient property, we should again get the number 3 in the answer
(12×4) : (4×4)
(12 × 4) : (4 × 4) = 48: 16 = 3
Now let's try not to multiply, but to divide the dividend and the divisor by the number 4
(12: 4) : (4: 4)
(12: 4) : (4: 4) = 3: 1 = 3
Received a response 3.
We see that if the dividend and the divisor are multiplied or divided by the same number, then the quotient does not change.
Division of integers
Example 1 Find the value of expression 12: (−2)
This is the division of numbers with different signs. 12 is a positive number, (−2) is negative. In such cases, you need
12: (−2) = −(|12| : |−2|) = −(12: 2) = −(6) = −6
Usually written shorter than 12: (−2) = −6
Example 2 Find the value of the expression −24: 6
This is the division of numbers with different signs. −24 is negative, 6 is positive. In such cases, again, divide the dividend modulus by the divisor modulus, and put a minus sign in front of the received answer.
−24: 6 = −(|−24| : |6|) = −(24: 6) = −(4) = −4
Usually written shorter than -24: 6 = -4
Example 3 Find the value of the expression (−45) : (−5)
This is the division of negative numbers. In such cases, you need divide the dividend modulus by the divisor modulus, and put a plus sign in front of the received answer.
(−45) : (−5) = |−45| : |−5| = 45: 5 = 9
Usually written shorter (−45) : (−5) = 9
Example 4 Find the value of the expression (−36) : (−4) : (−3)
According to the order of operations, if the expression contains only multiplication or division, then all actions must be performed from left to right in the order in which they appear.
Divide (−36) by (−4), and divide the resulting number by (−3)
First action:
(−36) : (−4) = |−36| : |−4| = 36: 4 = 9
9: (−3) = −(|−9| : |−3|) = −(9: 3) = −(3) = −3
Usually written shorter (−36) : (−4) : (−3) = 9: (−3) = −3
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Everyone remembers from school that you cannot divide by zero. Younger students are never told why they shouldn't do it. They just offer to take it for granted along with other prohibitions like “you can’t put your fingers in sockets” or “you shouldn’t ask stupid questions to adults.”
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 impossibility of dividing by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.
Algebraic explanation for the impossibility of dividing by zero
Algebraically, you can't divide by zero because it doesn't make any sense. Let's take two arbitrary numbers, a and b, and multiply them by zero. a × 0 is zero and b × 0 is zero. It turns out that a × 0 and b × 0 are equal, because the product in both cases is equal to zero. Thus, we can write the equation: 0 × a = 0 × b. Now suppose we can divide by zero: we divide both sides of the equation by zero and we get that a = b. It turns out that if we allow the operation of division by zero, then all numbers are the same. But 5 is not equal to 6, and 10 is not equal to ½. Uncertainty arises, about which teachers prefer not to tell inquisitive elementary school students.
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 said, 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.
Explanation of the impossibility of dividing by zero in terms of mathematical analysis
In high school, they study the theory of limits, which also speaks of the impossibility of dividing by zero. This number is interpreted there as “indefinitely indefinitely small value". So if we consider the equation 0 × X = 0 within the framework of this theory, we will find that X cannot be found because for this we would have to divide zero by zero. And this also does not make any sense, since both the dividend and the divisor in this case are indefinite quantities, therefore, it is impossible to draw a conclusion about their equality or inequality.
When can you divide by zero?
Unlike schoolchildren, students of technical universities can divide by zero. An operation that is impossible in algebra can be performed in other areas of mathematical knowledge. They have new additional terms tasks that allow this action. Dividing by zero will be possible for those who listen to a course of lectures on non-standard analysis, study the Dirac delta function and become familiar with the extended complex plane.
History of Zero
Zero is the reference point in all standard number systems. The use of the number by Europeans is relatively recent, but the sages of 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.
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 possibilities 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 expanding the limit 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 - 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.
