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 owed 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 slices 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 puzzle 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.

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. 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.

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 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 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.

www.bolshoyvopros.ru

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.

Solution.

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 divide as we divide natural numbers. 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.

Solution.

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.

Because 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:

www.mathematics-repetition.com

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

This video tutorial is available by subscription

Do you already have a subscription? To come in

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

Multiplying numbers by 10, 100

The 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.

The 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

The 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:

The 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 to this example is constructed in a similar way: 0.1 is expressed as an ordinary fraction, and dividing by is the same as multiplying by 10:

That is, to divide by 0.1, you need to shift the comma to the right 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

Output

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 works. 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”.

FROM zero all arithmetic operations are carried out, and integers, ordinary and decimal fractions can be used as its “partners”, and all of them can have both positive and negative values. 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.

• Forms of tax declarations We bring to your attention declaration forms for all types of taxes and fees: 1. Income tax. Attention from 10.02.2014 the income tax report is submitted according to new sample declarations approved by order of the Ministry of Revenue No. 872 of 30.12.2013.1. 1. Income tax return […]
• Squaring the sum and square the difference rules Purpose: To derive formulas for squaring the sum and difference of expressions. Expected results: to learn how to use the formulas of the square of the sum and the square of the difference. Type of lesson: problem presentation lesson. I. Presentation of the topic and purpose of the lesson II. Work on the topic of the lesson When multiplying […]
• What is the difference between privatization of an apartment with minor children and privatization without children? Features of their participation, documents Any real estate transactions require close attention of the participants. Especially if you plan to privatize an apartment with minor children. In order for it to be recognized as valid, and […]
• The amount of the state duty for an old-style passport for a child under 14 years old and where to pay it Appeal to government agencies for any service is always accompanied by payment of a state fee. To apply for a foreign passport, you also need to pay a federal fee. How much is the size […]
• How to fill out an application form for replacing a passport at 45 Russian passports must be replaced when the age mark is reached - 20 or 45 years. To receive a public service, you must submit an application in the prescribed form, attach the necessary documents and pay the […]
• How and where to issue a donation for a share in an apartment Many citizens are faced with such a legal procedure as donating real estate that is in shared ownership. There is quite a lot of information on how to issue a donation for a share in an apartment correctly, and it is not always reliable. Before starting, […]

If you break the generally accepted rules in the world of science, you can get the most unexpected results.

Ever since school, teachers have been telling us that there is one rule in mathematics that cannot be broken. It sounds like this: "You can't divide by zero!"

Why is it that such a familiar number for us, which we so often encounter in everyday life, causes so many difficulties when performing a simple arithmetic operation like division?

Let's look into this issue.

If we divide one number into all smaller numbers, then as a result we will get more and more large values. For example

Thus, it turns out that if we divide by a number tending to zero, then we will get the largest result tending to infinity.

Does this mean that if we divide our number by zero, we get infinity?

This sounds logical, but all we know is that if we divide by a number close in value to zero, then the result will only tend to infinity and this does not mean that when divided by zero, we will end up with infinity . Why is it so?

First, we need to understand what the arithmetic operation of division is. So, if we divide 20 by 10, then this will mean how many times we will need to add the number 10 to get 20 as a result, or what number we need to take twice to get 20.

In general, division is the inverse of multiplication. For example, multiplying any number by X, we can ask the question: "Is there a number that we need to multiply by the result obtained in order to find out the original value of X?" And if there is such a number, then it will be the inverse value for X. For example, if we multiply 2 by 5, we get 10. If after that we multiply 10 by one fifth, then again we get 2:

So 1/5 is the reciprocal of 5, the reciprocal of 10 is 1/10.

As you have already noticed, as a result of multiplying a certain number by its reciprocal number, the answer will always be one. And in the event that you want to divide some number by zero, then you will need to find its reciprocal number, which should be equal to one divided by zero.

This will mean that when multiplied by zero, one should turn out, and since it is known that if you multiply any number by 0, you get 0, then this is impossible and zero does not have a reciprocal number.

Is it possible to come up with something to get around this contradiction?

Previously, mathematicians have already found ways to circumvent mathematical rules, because in the past, according to mathematical rules, it was impossible to obtain the value of the square root of a negative number, then it was proposed to denote such square roots with imaginary numbers. As a result, a new branch of mathematics about complex numbers appeared.

So why don't we also try to introduce a new rule, according to which one divided by zero would be denoted by an infinity sign and see what happens?

Suppose we don't know anything about infinity. In this case, if we start from the reciprocal of zero, then multiplying zero by infinity, we should get one. And if we add to this one more value of zero divided by infinity, then the result should be the number two:

In accordance with the distributive law of mathematics, the left side of the equation can be represented as:

and since 0+0=0, then our equation will take the form 0*∞=2, due to the fact that we have already determined 0*∞=1, it turns out that 1=2.

This sounds ridiculous. However, such an answer also cannot be considered completely wrong, since such calculations simply do not work for ordinary numbers. For example, division by zero is used in the Riemann sphere, but in a completely different way, and this is a completely different story ...

In short, division by zero in the usual way does not end well, but nevertheless this should not become a hindrance to our experiments in the field of mathematics, perhaps we will be able to open new areas for research.

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 owed 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 slices 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 puzzle 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.

Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still there is a lot of controversy around him. Someone just 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.

In contact with

Who is right in the end

During these disputes, both people, having opposite points of view, look at each other like a ram, and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams resting 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.

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:

1. 25x3=75
2. 25 + 25 + 25 = 75
3. 25x3 = 25 + 25 + 25

From this equation follows the conclusion, that multiplication is a simplified addition.

What is zero

Any person from childhood knows: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the 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 determine empty digits in decimal fractions, this is done both before and after the decimal point.

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 secrets, as ancient scientists 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.

Going back to the very beginning, the argument about two apples, 2 times 0 looks like this:

• If you eat two apples five times, then eaten 2×5 = 2+2+2+2+2 = 10 apples
• If you eat two of them three times, then eaten 2 × 3 = 2 + 2 + 2 = 6 apples
• If you eat two apples zero times, then nothing will be eaten - 2x0 = 0x2 = 0+0 = 0

After all, eating an apple 0 times means not eating a single one. This will be clear even to the smallest 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.

Division

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 is impossible and that's it, without stuffing our heads with unnecessary 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 the school curriculum, because there are not so many disputes and contradictions 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 for the equation 2 * x = 10. So, 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!