Evgeniy Shiryaev, teacher and head of the Mathematics Laboratory of the Polytechnic Museum, told AiF.ru about division by zero:
1. Jurisdiction of the issue
Agree, what makes the rule especially provocative is the ban. How can this not be done? Who banned? What about our civil rights?
Neither the Constitution of the Russian Federation, nor the Criminal Code, nor even the charter of your school objects to the intellectual action that interests us. This means there is no ban legal force, and nothing prevents you from trying to divide something by zero right here, on the pages of AiF.ru. For example, a thousand.
2. Let's divide as taught
Remember, when you first learned how to divide, the first examples were solved by checking multiplication: the result multiplied by the divisor had to be the same as the divisible. If it didn’t match, they didn’t decide.
Example 1. 1000: 0 =...
Let's forget about the forbidden rule for a moment and make several attempts to guess the answer.
Incorrect ones will be cut off by the check. Try the following options: 100, 1, −23, 17, 0, 10,000. For each of them, the check will give the same result:
100 0 = 1 0 = − 23 0 = 17 0 = 0 0 = 10,000 0 = 0
By multiplying zero, everything turns into itself and never into a thousand. The conclusion is easy to formulate: no number will be tested. That is, no number can be the result of dividing a non-zero number by zero. Such division is not prohibited, but simply has no result.
3. Nuance
We almost missed one opportunity to refute the ban. Yes, we admit that a non-zero number cannot be divided by 0. But maybe 0 itself can?
Example 2. 0: 0 = ...
What are your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor 0 is equal to the dividend 0.
More options! 1? Fits too. And −23, and 17, and that’s it. In this example, the test will be positive for any number. And to be honest, the solution in this example should be called not a number, but a set of numbers. Everyone. And it doesn’t take long to agree that Alice is not Alice, but Mary Ann, and both of them are a rabbit’s dream.
4. What about higher mathematics?
The problem has been resolved, the nuances have been taken into account, the dots have been placed, everything has become clear - the answer to the example with division by zero cannot be a single number. Solving such problems is hopeless and impossible. Which means... interesting! Take two.
Example 3. Figure out how to divide 1000 by 0.
But no way. But 1000 can be easily divided by other numbers. Well, let's at least do what we can, even if we change the task at hand. And then, you see, we get carried away, and the answer will appear by itself. Let’s forget about zero for a minute and divide by one hundred:
A hundred is far from zero. Let's take a step towards it by decreasing the divisor:
1000: 25 = 40,
1000: 20 = 50,
1000: 10 = 100,
1000: 8 = 125,
1000: 5 = 200,
1000: 4 = 250,
1000: 2 = 500,
1000: 1 = 1000.
The dynamics are obvious: the closer the divisor is to zero, the larger the quotient. The trend can be observed further by moving to fractions and continuing to reduce the numerator:
It remains to note that we can get as close to zero as we like, making the quotient as large as we like.
In this process there is no zero and there is no last quotient. We indicated the movement towards them by replacing the number with a sequence converging to the number we are interested in:
This implies a similar replacement for the dividend:
1000 ↔ { 1000, 1000, 1000,... }
It’s not for nothing that the arrows are double-sided: some sequences can converge to numbers. Then we can associate the sequence with its numerical limit.
Let's look at the sequence of quotients:
It grows unlimitedly, not striving for any number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:
Comparison with the numbers of sequences that have a limit allows us to propose a solution to the third example:
When elementwise dividing a sequence converging to 1000 into a sequence of positive numbers, converging to 0, we obtain a sequence converging to ∞.
5. And here is the nuance with two zeros
What is the result of dividing two sequences of positive numbers that converge to zero? If they are the same, then the unit is identical. If the dividend sequence converges to zero faster, then in the quotient the sequence has a zero limit. And when the elements of the divisor decrease much faster than those of the dividend, the sequence of the quotient will grow greatly:
Uncertain situation. And that’s what it’s called: uncertainty of type 0/0 . When mathematicians see sequences that fit such uncertainty, they do not rush to divide the two identical numbers at each other, but figure out which of the sequences runs faster to zero and how exactly. And each example will have its own specific answer!
6. In life
Ohm's law relates current, voltage and resistance in a circuit. It is often written in this form:
Let us allow ourselves to neglect the neat physical understanding and formally look at the right-hand side as the quotient of two numbers. Let's imagine that we decide school task on electricity. The condition gives the voltage in volts and resistance in ohms. The question is obvious, the solution is in one action.
Now let's look at the definition of superconductivity: this is the property of some metals to have zero electrical resistance.
Well, let's solve the problem for a superconducting circuit? Just set it up R= 0 it won’t work, physics throws up interesting task, which obviously stands behind scientific discovery. And the people who managed to divide by zero in this situation received Nobel Prize. It’s useful to be able to bypass any prohibitions!
Very often, many people wonder why division by zero cannot be used? In this article we will talk in great detail about where this rule came from, as well as what actions can be performed with a zero.
In contact with
Zero can be called one of the most interesting numbers. This number has no meaning, it means emptiness in literally words. However, if a zero is placed next to any number, then the value of this number will become several times greater.
The number itself is very mysterious. I used it again ancient people Mayan. For the Mayans, zero meant “beginning,” and counting calendar days also started from scratch.
Very interesting fact is that the zero sign and the uncertainty sign were similar. By this, the Mayans wanted to show that zero is the same identical sign as uncertainty. In Europe, the designation zero appeared relatively recently.
Many people also know the prohibition associated with zero. Anyone will say that you can't divide by zero. Teachers at school say this, and children usually take their word for it. Usually, children are either simply not interested in knowing this, or they know what will happen if, having heard an important prohibition, they immediately ask, “Why can’t you divide by zero?” But when you get older, your interest awakens, and you want to know more about the reasons for this ban. However, there is reasonable evidence.
Actions with zero
First you need to determine what actions can be performed with zero. Exists several types of actions:
- Addition;
- Multiplication;
- Subtraction;
- Division (zero by number);
- Exponentiation.
Important! If you add zero to any number during addition, then this number will remain the same and will not change its numerical value. The same thing happens if you subtract zero from any number.
When multiplying and dividing things are a little different. If multiply any number by zero, then the product will also become zero.
Let's look at an example:
Let's write this as an addition:
There are five zeros in total, so it turns out that
Let's try to multiply one by zero. The result will also be zero.
Zero can also be divided by any other number that is not equal to it. In this case, the result will be , the value of which will also be zero. The same rule applies to negative numbers. If zero is divided by a negative number, the result is zero.
You can also construct any number to the zero degree. In this case, the result will be 1. It is important to remember that the expression “zero in zero degree"is absolutely pointless. If you try to raise zero to any power, you get zero. Example:
We use the multiplication rule and get 0.
So is it possible to divide by zero?
So, here we come to the main question. Is it possible to divide by zero? at all? And why can’t we divide a number by zero, given that all other actions with zero exist and are applied? To answer this question it is necessary to turn to higher mathematics.
Let's start with the definition of the concept, what is zero? School teachers They say that zero is nothing. Emptiness. That is, when you say that you have 0 handles, it means that you have no handles at all.
In higher mathematics, the concept of “zero” is broader. It does not mean emptiness at all. Here zero is called uncertainty because if we do a little research, it turns out that when we divide zero by zero, we can end up with any other number, which may not necessarily be zero.
Did you know that those are simple arithmetic operations that you studied at school are not so equal to each other? The most basic actions are addition and multiplication.
For mathematicians, the concepts of “” and “subtraction” do not exist. Let's say: if you subtract three from five, you will be left with two. This is what subtraction looks like. However, mathematicians would write it this way:
Thus, it turns out that the unknown difference is a certain number that needs to be added to 3 to get 5. That is, you don’t need to subtract anything, you just need to find the appropriate number. This rule applies to addition.
Things are a little different with rules of multiplication and division. It is known that multiplication by zero leads to a zero result. For example, if 3:0=x, then if you reverse the entry, you get 3*x=0. And a number that was multiplied by 0 will give zero in the product. It turns out that there is no number that would give any value other than zero in the product with zero. This means that division by zero is meaningless, that is, it fits our rule.
But what happens if you try to divide zero itself by itself? Let's take x as something indefinite number. The resulting equation is 0*x=0. It can be solved.
If we try to take zero instead of x, we will get 0:0=0. It would seem logical? But if we try to take any other number, for example, 1, instead of x, we will end up with 0:0=1. The same situation will happen if we take any other number and plug it into the equation.
In this case, it turns out that we can take any other number as a factor. The result will be infinite set different numbers. Sometimes division by 0 in higher mathematics still makes sense, but then usually a certain condition appears, thanks to which we can still choose one suitable number. This action is called "uncertainty disclosure." In ordinary arithmetic, division by zero will again lose its meaning, since we will not be able to choose one number from the set.
Important! You cannot divide zero by zero.
Zero and infinity
Infinity can be found very often in higher mathematics. Since it is simply not important for schoolchildren to know that there are also mathematical operations with infinity, teachers cannot properly explain to children why it is impossible to divide by zero.
Students begin to learn basic mathematical secrets only in the first year of institute. Higher mathematics provides large complex problems that have no solution. The most famous problems are problems with infinity. They can be solved using mathematical analysis.
Can also be applied to infinity elementary mathematical operations: addition, multiplication by number. Usually they also use subtraction and division, but in the end they still come down to two simple operations.
Even at school, teachers tried to hammer into our heads the simplest rule: “Any number multiplied by zero equals zero!”, – but still a lot of controversy constantly arises around him. Some people just remember the rule and don’t bother themselves with the question “why?” “You can’t and that’s it, because they said so at school, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.
Who's right in the end?
During these disputes, both people having opposite points sight, look at each other like a ram, and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams, resting their horns on each other. The only difference between them is that one is slightly less educated than the other.
Most often, those who consider this rule to be incorrect try to appeal to logic in this way:
I have two apples on my table, if I put zero apples on them, that is, I don’t put a single one, then my two apples will not disappear! The rule is illogical!
Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 = 2. So let’s discard this conclusion right away - it is illogical, although it has reverse target- call for logic.
What is multiplication
Originally the multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies that the number is natural. Thus, any number with multiplication can be reduced to this equation:
- 25×3 = 75
- 25 + 25 + 25 = 75
- 25×3 = 25 + 25 + 25
From this equation it follows that 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 differently - 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 meaning of emptiness, standing next to any natural number, multiplies it tenfold. Hence all the controversy about multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimals, this is done both before and after the decimal point.
Is it possible to multiply by emptiness?
You can multiply by zero, but it is useless, because, whatever one may say, even when multiplying negative numbers, you will still get zero. It’s enough just to remember this simple rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as ancient scientists believed. Below we will give the most logical explanation that this multiplication is useless, because when you multiply a number by it, you will still get the same thing - zero.
Returning to the very beginning, to the argument about two apples, 2 times 0 looks like this:
- If you eat two apples five times, then you eat 2×5 = 2+2+2+2+2 = 10 apples
- If you eat two of them three times, then you eat 2×3 = 2+2+2 = 6 apples
- If you eat two apples zero times, then nothing will be eaten - 2×0 = 0×2 = 0+0 = 0
After all, eating an apple 0 times means not eating a single one. It will be clear even to yourself to a small child. Whatever one may say, the result will be 0, two or three can be replaced with absolutely any number and the result will be absolutely the same. And to put it simply, then zero is nothing, and when do you have there is nothing, then no matter how much you multiply, it’s still the same will be zero. There is no such thing as magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.
Division
From all of the above, another thing follows important rule:
You can't divide by zero!
This rule has also been persistently drilled into our heads since childhood. We just know that it’s impossible to do everything without filling our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the question. simple question from school curriculum, because there is not so much controversy and controversy surrounding this rule.
Everyone simply memorized the rule and did not divide by zero, not suspecting that the answer was hidden on the surface. Addition, multiplication, division and subtraction are unequal; of the above, only multiplication and addition are valid, and all other manipulations with numbers are built from them. That is, the entry 10: 2 is an abbreviation of the equation 2 * x = 10. This means that 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,
So as not to divide by 0!
Cut 1 as you want, lengthwise,
Just don't divide by 0!
Mathematicians have a specific sense of humor and some questions related to calculations are no longer taken seriously. It’s not always clear whether they are trying to explain to you in all seriousness why you can’t divide by zero or whether this is just another joke. But the question itself is not so obvious; if in elementary mathematics one can reach its solution purely logically, then in higher mathematics there may well be other initial conditions.
When did zero appear?
The number zero is fraught with many mysteries:
- IN Ancient Rome They didn’t know this number; the reference system began with I.
- For the right to be called the progenitors of zero for a long time Arabs and Indians argued.
- Studies of the Mayan culture have shown that this ancient civilization could well have been the first in terms of using zero.
- Zero has nothing numerical value, even minimal.
- It literally means nothing, the absence of things to count.
In the primitive system there was no particular need for such a figure; the absence of something could be explained using words. But with the emergence of civilizations, human needs also increased in terms of architecture and engineering.
To implement more complex calculations and it was necessary to introduce new functions a number that would indicate complete absence anything.
Is it possible to divide by zero?
There are two diametrically opposed opinions:
At school, still in junior classes They teach that you should never divide by zero. This is explained extremely simply:
- Let's imagine that you have 20 tangerine slices.
- By dividing them by 5, you will give 4 slices to five friends.
- Dividing by zero will not work, because the process of division between someone will not happen.
Of course, this is a figurative explanation, largely simplified and not entirely consistent with reality. But it explains in an extremely accessible way the meaninglessness of dividing something by zero.
After all, in fact, in this way one can denote the fact of the absence of division. Why complicate it? mathematical calculations and also write down the absence of division?
Can zero be divided by a number?
From the point of view of applied mathematics, any division that involves a zero does not make much sense. But school books are clear in their opinion:
- Zero can be divided.
- Any number can be used for division.
- You can't divide zero by zero.
The third point may cause slight bewilderment, since just a few paragraphs above it was indicated that such a division is quite possible. In fact, it all depends on the discipline in which you are doing the calculations.
In this case, it is really better for schoolchildren to write that expression cannot be determined , and, therefore, it does not make sense. But in some branches of algebraic science it is allowed to write such an expression, dividing zero by zero. Especially when we're talking about O computers and programming languages.
The need to divide zero by a number may arise when solving any equalities and searching for initial values. But in that case, the answer will always be zero. Here, as with multiplication, no matter what number you divide zero by, you won’t end up with more than zero. Therefore, if you notice this treasured number in a huge formula, try to quickly “figure out” whether all the calculations will come down to a very simple solution.
If infinity is divided by zero
It was necessary to mention infinitely large and infinitesimal values a little earlier, because this also opens up some loopholes for division, including using zero. That's true, and there's a little catch here, because infinitesimal value and complete absence of value are different concepts.
But this small difference in our conditions can be neglected; ultimately, calculations are carried out using abstract quantities:
- The numerators must contain an infinity sign.
- The denominators are a symbolic image of a value tending to zero.
- The answer will be infinity, representing an infinitely large function.
It should be noted that we are still talking about symbolic representation indefinitely small function, not about using zero. Nothing has changed with this sign; it still cannot be divided into, only as very, very rare exceptions.
For the most part, zero is used to solve problems that are in purely theoretical plane. Perhaps, after decades or even centuries, all modern computing will find practical use, and they will provide some kind of grandiose breakthrough in science.
In the meantime, most mathematical geniuses only dream of worldwide recognition. The exception to these rules is our compatriot, Perelman. But he is known for solving a truly epoch-making problem with the proof of the Poinqueré conjecture and for his extravagant behavior.
Paradoxes and the meaninglessness of division by zero
Dividing by zero, for the most part, makes no sense:
- Division is represented as inverse function of multiplication.
- We can multiply any number by zero and get zero as an answer.
- By the same logic, one could divide any number by zero.
- Under such conditions, it would be easy to come to the conclusion that any number multiplied or divided by zero is equal to any other number on which this operation was performed.
- Recline mathematical operation and we get the most interesting conclusion - any number is equal to any number.
In addition to creating such incidents, division by zero has no practical significance , from the word in general. Even if it is possible to perform this action, it will not be possible to obtain any new information.
From point of view elementary mathematics, during division by zero, the whole object is divided zero times, that is, not a single time. Simply put - no fission process occurs, therefore, there cannot be a result of this event.
Being in the same company as a mathematician, you can always ask a couple of banal questions, for example, why you can’t divide by zero and get an interesting and understandable answer. Or irritation, because this is probably not the first time a person has been asked this. And not even in the tenth. So take care of your mathematician friends, don’t force them to repeat one explanation a hundred times.
Video: divide by zero
In this video, mathematician Anna Lomakova will tell you what happens if you divide a number by zero and why this cannot be done, from a mathematical point of view:
Evgeniy SHIRYAEV, teacher and head of the Mathematics Laboratory of the Polytechnic Museum, told AiF about division by zero:
1. Jurisdiction of the issue
Agree, what makes the rule especially provocative is the ban. How can this not be done? Who banned? What about our civil rights?
Neither the Constitution, nor the Criminal Code, nor even the charter of your school objects to the intellectual action that interests us. This means that the ban has no legal force, and nothing prevents you from trying to divide something by zero right here, on the pages of AiF. For example, a thousand.
2. Let's divide as taught
Remember, when you first learned how to divide, the first examples were solved with a multiplication check: the result multiplied by the divisor had to coincide with the dividend. It didn’t match - they didn’t decide.
Example 1. 1000: 0 =...
Let's forget about the forbidden rule for a moment and make several attempts to guess the answer.
Incorrect ones will be cut off by the check. Try the following options: 100, 1, −23, 17, 0, 10,000. For each of them, the check will give the same result:
100 0 = 1 0 = − 23 0 = 17 0 = 0 0 = 10,000 0 = 0
By multiplying zero, everything turns into itself and never into a thousand. The conclusion is easy to formulate: no number will pass the test. That is, no number can be the result of dividing a non-zero number by zero. Such division is not prohibited, but simply has no result.
3. Nuance
We almost missed one opportunity to refute the ban. Yes, we admit that a non-zero number cannot be divided by 0. But maybe 0 itself can?
Example 2. 0: 0 = ...
What are your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor 0 is equal to the dividend 0.
More options! 1? Fits too. And −23, and 17, and that’s it. In this example, the test will be positive for any number. And, to be honest, the solution in this example should be called not a number, but a set of numbers. Everyone. And it doesn’t take long to agree that Alice is not Alice, but Mary Ann, and both of them are a rabbit’s dream.
4. What about higher mathematics?
The problem has been resolved, the nuances have been taken into account, the dots have been placed, everything has become clear - the answer to the example with division by zero cannot be a single number. Solving such problems is hopeless and impossible. Which means... interesting! Take two.
Example 3. Figure out how to divide 1000 by 0.
But no way. But 1000 can be easily divided by other numbers. Well, let's at least do what works, even if we change the task. And then, you see, we get carried away, and the answer will appear by itself. Let’s forget about zero for a minute and divide by one hundred:
A hundred is far from zero. Let's take a step towards it by decreasing the divisor:
1000: 25 = 40,
1000: 20 = 50,
1000: 10 = 100,
1000: 8 = 125,
1000: 5 = 200,
1000: 4 = 250,
1000: 2 = 500,
1000: 1 = 1000.
The dynamics are obvious: the closer the divisor is to zero, the larger the quotient. The trend can be observed further by moving to fractions and continuing to reduce the numerator:
It remains to note that we can get as close to zero as we like, making the quotient as large as we like.
In this process there is no zero and there is no last quotient. We indicated the movement towards them by replacing the number with a sequence converging to the number we are interested in:
This implies a similar replacement for the dividend:
1000 ↔ { 1000, 1000, 1000,... }
It’s not for nothing that the arrows are double-sided: some sequences can converge to numbers. Then we can associate the sequence with its numerical limit.
Let's look at the sequence of quotients:
It grows unlimitedly, not striving for any number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:
Comparison with the numbers of sequences that have a limit allows us to propose a solution to the third example:
When elementwise dividing a sequence converging to 1000 by a sequence of positive numbers converging to 0, we obtain a sequence converging to ∞.
5. And here is the nuance with two zeros
What is the result of dividing two sequences of positive numbers that converge to zero? If they are the same, then the unit is identical. If a dividend sequence converges to zero faster, then in particular it is a sequence with a zero limit. And when the elements of the divisor decrease much faster than those of the dividend, the sequence of the quotient will grow greatly:
Uncertain situation. And that’s what it’s called: uncertainty of type 0/0 . When mathematicians see sequences that fit such uncertainty, they do not rush to divide two identical numbers by each other, but figure out which of the sequences runs faster to zero and how exactly. And each example will have its own specific answer!
6. In life
Ohm's law relates current, voltage and resistance in a circuit. It is often written in this form:
Let us allow ourselves to neglect the neat physical understanding and formally look at the right-hand side as the quotient of two numbers. Let's imagine that we are solving a school problem on electricity. The condition gives the voltage in volts and resistance in ohms. The question is obvious, the solution is in one action.
Now let's look at the definition of superconductivity: this is the property of some metals to have zero electrical resistance.
Well, let's solve the problem for a superconducting circuit? Just set it up R= 0 If it doesn’t work out, physics throws up an interesting problem, behind which, obviously, there is a scientific discovery. And the people who managed to divide by zero in this situation received the Nobel Prize. It’s useful to be able to bypass any prohibitions!