Any number multiplied by 0 is equal. Fun math

Even at school, teachers tried to hammer into our heads the simplest rule: “Any number multiplied by zero equals zero!”, - but still a lot of controversy constantly arises around him. Some people just remember the rule and don’t bother themselves with the question “why?” “You can’t and that’s it, because they said so at school, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.

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Who's right in the end?

During these disputes, both people with opposing points of view look at each other like a ram and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams, resting their horns on each other. The only difference between them is that one is slightly less educated than the other.

Most often, those who consider this rule to be incorrect try to appeal to logic in this way:

I have two apples on my table, if I put zero apples on them, that is, I don’t put a single one, then my two apples will not disappear! The rule is illogical!

Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 = 2. So let’s discard this conclusion right away - it is illogical, although it has the opposite purpose - to call to 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:

1. 25×3 = 75
2. 25 + 25 + 25 = 75
3. 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 a deep meaning in this number. After all, zero, which has the meaning of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy about multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimal fractions, this is done both before and after the decimal point.

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. This will be clear to even the smallest child. Whatever one may say, the result will be 0, two or three can be replaced with absolutely any number and the result will be absolutely the same. And to put it simply, then zero is nothing, and when do you have there is nothing, then no matter how much you multiply, it’s still the same will be zero. There is no such thing as magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.

Division

From all of the above, another important rule follows:

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 most will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions surrounding this rule.

Everyone simply memorized the rule and did not divide by zero, not suspecting that the answer was hidden on the surface. Addition, multiplication, division and subtraction are unequal; of the above, only multiplication and addition are valid, and all other manipulations with numbers are built from them. That is, the notation 10: 2 is an abbreviation of the equation 2 * x = 10. This means that the notation 10: 0 is the same abbreviation for 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be a priori incorrect.

Let me tell you,

So as not to divide by 0!

Cut 1 as you want, lengthwise,

Just don't divide by 0!

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

Zero widely used in decimal fractions to determine the values ​​of the “empty” places, both before and after the decimal point. In addition, one of the fundamental rules of arithmetic is associated with it, which states that zero cannot be divided. Its logic, strictly speaking, stems from the very essence of this number: indeed, it is impossible to imagine that some value different from it (and it itself too) would be divided into “nothing”.

Calculation examples

WITH zero all arithmetic operations are carried out, and as its “partners” they can use integers, ordinary and decimal fractions, and all of them can have both positive and negative values. Let us give examples of their implementation and some explanations for them.

ADDITION

When adding zero to a certain number (both integer and fractional, both positive and negative), its value remains absolutely unchanged.

Example 1

twenty four plus zero equals twenty-four.

Example 2

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

MULTIPLICATION

When multiplying any number (integer, fraction, positive or negative) by zero it turns out zero.

Example 1

Five hundred eighty six times zero equals zero.

Example 2

Zero multiplied by one hundred thirty-five point six seven equals zero.

Example 3

Zero multiply by zero equals zero.

DIVISION

The rules for dividing numbers by each other in cases where one of them is a zero differ depending on what role the zero itself plays: a dividend or a divisor?

In cases where zero represents the dividend, the result is always equal to it, regardless of the value of the divisor.

Example 1

Zero divided by two hundred sixty five equals zero.

Example 2

Zero divided by seventeen five hundred ninety-six equals zero.

0:

= 0

Divide zero to zero According to the rules of mathematics, it is impossible. This means that when performing such a procedure, the quotient is uncertain. Thus, in theory, it can represent absolutely any number.

0: 0 = 8 because 8 × 0 = 0

In mathematics there is a problem like division of zero by zero, does not make any sense, since its result is an infinite set. This statement, however, is true if no additional data is provided that could affect the final result.

These, if present, should consist of indicating the degree of change in the magnitude of both the dividend and the divisor, and even before the moment when they turned into zero. If this is defined, then an expression such as zero divide by zero, in the vast majority of cases some meaning can be attached.

This lesson will look at how to perform multiplication and division by numbers of the form 10, 100, 0.1, 0.001. Various examples on this topic will also be solved.

Exercise. How to multiply the number 25.78 by 10?

The decimal notation of a given number is a shorthand notation for the amount. It is necessary to describe it in more detail:

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

It turns out that...

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

Exercise. Multiply 25.486 by 100.

Multiplying by 100 is the same as multiplying by 10 twice. In other words, you need to move the decimal point to the right twice:

Exercise. Divide 25.78 by 10.

As in the previous case, you need to present the number 25.78 as a sum:

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

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

Exercise. Divide 124.478 by 100.

Dividing by 100 is the same as dividing by 10 twice, so the decimal point moves left 2 places:

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

Conversely, if a decimal fraction needs to be divided by 10, 100, 1000, and so on, you need to move the decimal point to the left by as many positions as there are zeros in the multiplier.

Example 1

Multiplying by 100 means moving the decimal place two places to the right.

After the shift, you can find that there are no more digits after the decimal point, which means that the fractional part is missing. Then there is no need for a comma, the number is an integer.

Example 2

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

Now multiplying by 10,000 is easy:

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

Equivalent decimal notations

Entry 52 means the following:

If we put 0 in front, we get entry 052. These entries are equivalent.

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

Now let's look at the decimal fraction:

If you assign zero, you get:

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

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

Example 3

Since division by 100 occurs, it is necessary to move the decimal point 2 positions to the left. There are no numbers left to the left of the decimal point. A whole part is missing. This notation is often used by programmers. In mathematics, if there is no whole part, then they put a zero in its place.

Example 4

You need to move it to the left by three positions, but there are only two positions. If you write several zeros in front of a number, it will be an equivalent notation.

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

Example 5

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

Multiplying and dividing by numbers 10, 100, 1000 is a very simple procedure. The situation is exactly the same with the numbers 0.1, 0.01, 0.001.

Example. Multiply 25.34 by 0.1.

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

Similarly, multiplying by 0.01 is dividing by 100:

Example. 5.235 divided by 0.1.

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

That is, to divide by 0.1, you need to move the decimal point to the right one position, which is equivalent to multiplying by 10.

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

Dividing by 10 and multiplying by 0.1 are the same thing. The comma needs to be moved to the right by 1 position:

Division by zero in mathematics, division in which the divisor is zero. Such a division can be formally written ⁄ 0, where is the dividend.

In ordinary arithmetic (with real numbers), this expression does not make sense, since:

• for ≠ 0 there is no number that when multiplied by 0 gives, therefore no number can be taken as the quotient ⁄ 0;
• at = 0, division by zero is also undefined, since any number when multiplied by 0 gives 0 and can be taken as the quotient 0 ⁄ 0.

Historically, one of the first references to the mathematical impossibility of assigning the value ⁄ 0 is contained in George Berkeley's critique of infinitesimal calculus.

Logical errors

Since when we multiply any number by zero, we always get zero as a result, when we divide both parts of the expression × 0 = × 0, which is true regardless of the value of and, by 0 we get the expression =, which is incorrect in the case of arbitrarily specified variables. Since zero can be specified not explicitly, but in the form of a rather complex mathematical expression, for example in the form of the difference of two values ​​reduced to each other through algebraic transformations, such a division can be a rather unobvious error. The imperceptible introduction of such a division into the process of proof in order to show the identity of obviously different quantities, thereby proving any absurd statement, is one of the varieties of mathematical sophism.

In computer science

In programming, depending on the programming language, the data type, and the value of the dividend, attempting to divide by zero can have different consequences. The consequences of division by zero in integer and real arithmetic are fundamentally different:

• Attempt integer division by zero is always a critical error that makes further execution of the program impossible. It either throws an exception (which the program can handle itself, thereby avoiding a crash), or causes the program to stop immediately, displaying an uncorrectable error message and possibly the contents of the call stack. In some programming languages, such as Go, integer division by a zero constant is considered a syntax error and causes the program to compile abnormally.
• IN real arithmetic consequences can be different in different languages:

• throwing an exception or stopping the program, as with integer division;
• obtaining a special non-numeric value as a result of an operation. In this case, the calculations are not interrupted, and their result can subsequently be interpreted by the program itself or the user as a meaningful value or as evidence of incorrect calculations. A widely used principle is that when dividing like ⁄ 0, where ≠ 0 is a floating point number, the result is equal to positive or negative (depending on the sign of the dividend) infinity - or, and when = 0 the result is a special value NaN (abbr. . from the English “not a number” - “not a number”). This approach is adopted in the IEEE 754 standard, which is supported by many modern programming languages.

Accidental division by zero in a computer program can sometimes cause expensive or dangerous malfunctions in the hardware controlled by the program. For example, on September 21, 1997, as a result of a division by zero in the computerized control system of the US Navy cruiser USS Yorktown (CG-48), all electronic equipment in the system turned off, causing the ship's propulsion system to stop operating.

see also

Notes

Function = 1 ⁄ . When it tends to zero from the right, it tends to infinity; when tends to zero from the left, tends to minus infinity

If you divide any number by zero on a regular calculator, it will give you the letter E or the word Error, that is, “error.”

In a similar case, the computer calculator writes (in Windows XP): “Division by zero is prohibited.”

Everything is consistent with the rule known from school that you cannot divide by zero.

Let's figure out why.

Division is the mathematical operation inverse to multiplication. Division is determined through multiplication.

Divide a number a(divisible, for example 8) by number b(divisor, for example the number 2) - means finding such a number x(quotient), when multiplied by a divisor b it turns out the dividend a(4 2 = 8), that is a divide by b means solving the equation x · b = a.

The equation a: b = x is equivalent to the equation x · b = a.

We replace division with multiplication: instead of 8: 2 = x we ​​write x · 2 = 8.

8: 2 = 4 is equivalent to 4 2 = 8

18: 3 = 6 is equivalent to 6 3 = 18

20: 2 = 10 is equivalent to 10 2 = 20

The result of division can always be checked by multiplication. The result of multiplying a divisor by a quotient must be the dividend.

Let's try to divide by zero in the same way.

For example, 6: 0 = ... We need to find a number that, when multiplied by 0, will give 6. But we know that when multiplied by zero, we always get zero. There is no number that, when multiplied by zero, gives something other than zero.

When they say that dividing by zero is impossible or prohibited, they mean that there is no number corresponding to the result of such division (dividing by zero is possible, but dividing is not :)).

Why do they say in school that you can’t divide by zero?

Therefore in definition operation of dividing a by b immediately emphasizes that b ≠ 0.

If everything written above seemed too complicated to you, then just give it a try: Dividing 8 by 2 means finding out how many twos you need to take to get 8 (answer: 4). Dividing 18 by 3 means finding out how many threes you need to take to get 18 (answer: 6).

Dividing 6 by zero means finding out how many zeros you need to take to get 6. No matter how many zeros you take, you will still get a zero, but you will never get 6, i.e., division by zero is undefined.

An interesting result is obtained if you try to divide a number by zero on an Android calculator. The screen will display ∞ (infinity) (or - ∞ if dividing by a negative number). This result is incorrect because the number ∞ does not exist. Apparently, programmers confused completely different operations - dividing numbers and finding the limit of a number sequence n/x, where x → 0. When dividing zero by zero, NaN (Not a Number) will be written.

“You can’t divide by zero!” - Most schoolchildren learn this rule by heart, without asking questions. All children know what “you can’t” 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 not possible.

The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as valid: 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 objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem completely differently. There is no subtraction, there is only addition. Therefore the entry 5 - 3 means a number that, when added to a number 3 will give a number 5 . That is 5 - 3 is simply a shorthand version of the equation: x + 3 = 5. There is no subtraction in this equation.

Division by zero

There is only a task - to find a suitable number.

The same is true with multiplication and division. Record 8: 4 can be understood as the result of dividing eight objects into four equal piles. But in reality this is 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. Record 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 it always works out 0 . This is an inherent property of zero, strictly speaking, part of its definition.

Such a number that, when multiplied by 0 will give something other than zero, it simply does not exist. That is, our problem has no solution. (Yes, this happens; not every problem has a solution.) Which means the records 5: 0 does not correspond to any specific number, and it simply does not mean anything and therefore has no meaning. The meaninglessness of this entry is briefly expressed by saying that you cannot divide by zero.

The most attentive readers in this place will certainly ask: is it possible to divide zero by zero?

Indeed, 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. We get 0 1 = 0. Right? 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 choose any one of them. That is, we cannot say which number the entry corresponds to 0: 0 . And if so, then we are forced to admit that this entry also makes no 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 possible solutions to the equation 0 x = 0; In such cases, mathematicians talk about “unfolding uncertainty,” but such cases do not occur in arithmetic.)

This is the peculiarity of the division operation. More precisely, the operation of multiplication and the number associated with it have zero.

Well, the most meticulous ones, having read this far, may ask: why does it happen that you can’t divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. You can answer it only by becoming familiar with the formal mathematical definitions of numerical sets and operations on them. It's not that difficult, but for some reason it's not taught in school. But in mathematics lectures at the university, this is what you will be taught first of all.

The division function is not defined for a range where the divisor is zero. You can divide, but the result is not certain

You can't divide by zero. Secondary school grade 2 mathematics.

If my memory serves me correctly, then zero can be represented as an infinitesimal value, so there will be infinity. And the school “zero - nothing” is just a simplification; there are so many of them in school mathematics). But it’s impossible without them, everything will happen in due time.

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Division by zero

Quotient from division by zero there is no number other than zero.

The reasoning here is as follows: since in this case no number can satisfy the definition of a quotient.

Let's write, for example,

Whatever number you try (say, 2, 3, 7), it is not suitable because:

\[ 2 0 = 0 \]

\[ 3 0 = 0 \]

\[ 7 0 = 0 \]

What happens if you divide by 0?

etc., but you need to get 2,3,7 in the product.

We can say that the problem of dividing a non-zero number by zero has no solution. However, a number other than zero can be divided by a number as close to zero as desired, and the closer the divisor is to zero, the larger the quotient. So, if we divide 7 by

\[ \frac(1)(10), \frac(1)(100), \frac(1)(1000), \frac(1)(10000) \]

then we get the quotients 70, 700, 7000, 70,000, etc., which increase without limit.

Therefore, they often say that the quotient of 7 divided by 0 is “infinitely large”, or “equal to infinity”, and write

\[ 7: 0 = \infin \]

The meaning of this expression is that if the divisor approaches zero and the dividend remains equal to 7 (or approaches 7), then the quotient increases without limit.

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

History of zero

Zero is the reference point in all standard number systems. Europeans began using this number relatively recently, but the sages of ancient India used zero a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Mayan numerical system. These American people used the duodecimal number system, and the first day of each month began with a zero. It is interesting that among the Mayans the sign denoting “zero” completely coincided with the sign denoting “infinity”. Thus, the ancient Mayans concluded that these quantities are identical and unknowable.

Mathematical operations with zero

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

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

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

Multiplication: Any number multiplied by 0 produces 0 (a*0=0).

Division: Zero can be divided by any number not equal to zero. In this case, the value of such a fraction will be 0. And division by zero is prohibited.

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

Zero to any power is equal to 0 (0 a = 0).

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

Paradoxes of mathematics

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

The thing is that the usual arithmetic operations that schoolchildren learn in primary school are, in fact, not nearly as equal as we think. All simple number operations can be reduced to two: addition and multiplication. These actions constitute the essence of the very concept of number, and other operations are built on the use of these two.

Addition and Multiplication

Let's take a standard subtraction example: 10-2=8. At school they consider it simply: if you subtract two from ten subjects, eight remain. But mathematicians look at this operation completely differently. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x+2=10. To mathematicians, the unknown difference is simply the number that needs to be added to two to make eight. And no subtraction is required here, you just need to find the appropriate numerical value.

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

Examples for division by 0

This is where it becomes a little clear why you can’t divide by zero. Multiplication and division by zero follow their own rules. All examples of dividing this quantity can be formulated as 6:0 = x. But this is an inverted notation of the expression 6 * x=0. But, as you know, any number multiplied by 0 gives only 0 in the product. This property is inherent in the very concept of zero value.

It turns out that there is no such number that, when multiplied by 0, gives any tangible value, that is, this problem has no solution. You should not be afraid of this answer; it is a natural answer for problems of this type. It's just that the 6:0 record doesn't make any sense and it can't explain anything. In short, this expression can be explained by the immortal “division by zero is impossible.”

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

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

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

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, new ones are added to the already known expression 0:0, which do not have solutions 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 using elementary methods. But higher mathematics, thanks to additional possibilities for a number of similar examples, provides final solutions. This is especially evident in the consideration of problems from the theory of limits.

Unlocking Uncertainty

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

As you can see in the example, simply reducing a fraction leads its value to a completely rational answer.

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

L'Hopital method

In some cases, the limits of expressions can be replaced by the limits of their derivatives. Guillaume L'Hopital - French mathematician, founder of the French school of mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule looks like this.