The rule for multiplying any number by zero. School mathematics course: why you can’t divide by zero at school

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 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.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 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 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 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 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 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 careful physical understanding and formally look at right 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 it won’t work, physics throws up interesting task, behind which there is obviously 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!

“You cannot 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’s very interesting and important to know why you can’t.

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.

We'll look at subtraction, for example. What does 5 - 3 mean? 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 notation 5 - 3 means a number that, when added to the number 3, will give the number 5. That is, 5 - 3 is simply a shorthand notation of the equation: x 3 = 5. There is no subtraction in this equation. There is only a task - to find a suitable number.

The same is true with multiplication and division. Entry 8:4 can be understood as the result of dividing eight items into four equal piles. But in reality, it's just a shorthand form of the equation 4 * x = 8.

This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Recording 5: 0 is an abbreviation for 0 * x = 5. That is, this task is to find a number that, when multiplied by 0, will give 5. But we know that when multiplied by 0, we always get 0. This is an inherent property of zero, strictly speaking , part of its definition.

There is no such number that when multiplied by 0 will give something other than zero. That is, our problem has no solution. (Yes, this happens; not every problem has a solution.) This means that the entry 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? In fact, the equation 0 * x = 0 can be solved safely. For example, we can take x = 0, and then we get 0 * 0 = 0. So, 0: 0=0? But let's not rush. Let's try to take x = 1. We get 0 * 1 = 0. right? So 0:0 = 1? But this way 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 what number the entry 0:0 corresponds to. 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 possible options solutions to the equation 0 * x = 0; in such cases, mathematicians talk about “Unlocking 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 at lectures on mathematics at the university, first of all, they will teach you exactly this.

Why can’t you divide by zero? “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 does 5 – 3 mean? 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 notation 5 – 3 means a number that, when added to the number 3, will give the number 5. That is, 5 – 3 is simply an abbreviated notation of the equation: x + 3 = 5. There is no subtraction in this equation. There is only a task - to find a suitable number.The same is true with multiplication and division. Entry 8:4 can be understood as the result of dividing eight items into four equal piles. But it's really just a shortened form of the equation 4 x = 8.This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Recording 5: 0 is an abbreviation for 0 x = 5. That is, this task is to find a number that, when multiplied by 0, will give 5. But we know that when multiplied by 0, the result is always 0. This is an inherent property of zero, strictly speaking , part of its definition.There is no such number that when multiplied by 0 will give something other than zero. That is, our problem has no solution. (Yes, this happens; not every problem has a solution.) This means that the entry 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 can be solved safely. For example, we can take x = 0, and then we get 0 · 0 = 0. So, 0: 0=0? But let's not rush. Let's try to take x = 1. We get 0 · 1 = 0. Correct? So 0:0 = 1? But this way 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 to which number the entry 0:0 corresponds. 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 “revealing 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.

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 wrong one in the case of arbitrarily given variable expression= . 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 by algebraic transformations, such a division can be a rather unobvious mistake. 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 languages programming.

Random division by zero in computer program sometimes causes expensive or dangerous malfunctions in program-controlled equipment. 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 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 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.

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 the truest sense of the word. 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 to the power of zero” is absolutely meaningless. 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 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 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 some indefinite number as x. 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 an infinite number 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 a large complex of 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.