Multiplication of numbers with different signs (6th grade). Fractions

In this article we will deal with multiplying numbers with different signs . Here we will first formulate the rule for multiplying positive and negative numbers, justify it, and then consider the application of this rule when solving examples.

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Rule for multiplying numbers with different signs

Multiplying a positive number by a negative number, as well as a negative number by a positive number, is carried out as follows: the rule for multiplying numbers with different signs: to multiply numbers with different signs, you need to multiply and put a minus sign in front of the resulting product.

Let's write it down this rule in letter form. For any positive real number a and any negative real number −b, the following equality holds: a·(−b)=−(|a|·|b|) , and also for a negative number −a and a positive number b the equality (−a)·b=−(|a|·|b|) .

The rule for multiplying numbers with different signs is fully consistent with properties of operations with real numbers. Indeed, on their basis it is easy to show that for real and positive numbers a and b a chain of equalities of the form a·(−b)+a·b=a·((−b)+b)=a·0=0, which proves that a·(−b) and a·b are opposite numbers, which implies the equality a·(−b)=−(a·b) . And from it follows the validity of the multiplication rule in question.

It should be noted that the stated rule for multiplying numbers with different signs is valid both for real numbers and for rational numbers and for integers. This follows from the fact that operations with rational and integer numbers have the same properties that were used in the proof above.

It is clear that multiplying numbers with different signs according to the resulting rule comes down to multiplying positive numbers.

It remains only to consider examples of the application of the disassembled multiplication rule when multiplying numbers with different signs.

Examples of multiplying numbers with different signs

Let's look at several solutions examples of multiplying numbers with different signs. Let's start with a simple case to focus on the steps of the rule rather than the computational complexity.

Example.

Multiply a negative number −4 by positive number 5 .

Solution.

According to the rule for multiplying numbers with different signs, we first need to multiply the absolute values of the original factors. The modulus of −4 is 4, and the modulus of 5 is 5, and multiplying the natural numbers 4 and 5 gives 20. Finally, it remains to put a minus sign in front of the resulting number, we have −20. This completes the multiplication.

Briefly, the solution can be written as follows: (−4)·5=−(4·5)=−20.

Answer:

(−4)·5=−20.

When multiplying fractions with different signs, you need to be able to multiply ordinary fractions, multiply decimals and their combinations with natural and mixed numbers.

Example.

Multiply numbers with different signs 0, (2) and .

Solution.

Having carried out the conversion of a periodic decimal fraction into an ordinary fraction, as well as having carried out the transition from a mixed number to an improper fraction, from the original product we will come to the product of ordinary fractions with different signs of the form . This product, according to the rule for multiplying numbers with different signs, is equal to . All that remains is to multiply the ordinary fractions in brackets, we have .

In this article we will deal with multiplying numbers with different signs. Here we will first formulate the rule for multiplying positive and negative numbers, justify it, and then consider the application of this rule when solving examples.

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Rule for multiplying numbers with different signs

Let's write this rule down in letter form. For any positive real number a and any negative real number −b, the following equality holds: a·(−b)=−(|a|·|b|) , and also for a negative number −a and a positive number b the equality (−a)·b=−(|a|·|b|) .

It should be noted that the stated rule for multiplying numbers with different signs is valid for both real numbers and rational numbers and for integers. This follows from the fact that operations with rational and integer numbers have the same properties that were used in the proof above.

It is clear that multiplying numbers with different signs according to the resulting rule comes down to multiplying positive numbers.

It remains only to consider examples of the application of the disassembled multiplication rule when multiplying numbers with different signs.

Examples of multiplying numbers with different signs

Let's look at several solutions examples of multiplying numbers with different signs. Let's start with a simple case to focus on the steps of the rule rather than the computational complexity.

Multiply the negative number −4 by the positive number 5.

According to the rule for multiplying numbers with different signs, we first need to multiply the absolute values of the original factors. Modulus −4 is equal to 4, and modulus 5 is equal to 5, and multiplication natural numbers 4 and 5 gives 20. Finally, it remains to put a minus sign in front of the resulting number, we have −20. This completes the multiplication.

Briefly, the solution can be written as follows: (−4)·5=−(4·5)=−20.

(−4)·5=−20.

When multiplying fractions with different signs, you need to be able to multiply ordinary fractions, multiply decimals and their combinations with natural and mixed numbers.

Multiply numbers with different signs 0, (2) and.

Having carried out the conversion of a periodic decimal fraction into an ordinary fraction, and also having carried out the transition from a mixed number to an improper fraction, from the original product we will come to the product of ordinary fractions with different signs of the form. This product is equal to the rule for multiplying numbers with different signs. All that remains is to multiply the ordinary fractions in brackets, we have .

Separately, it is worth mentioning the multiplication of numbers with different signs, when one or both factors are

Now let's deal with multiplication and division.

Let's say we need to multiply +3 by -4. How to do it?

Let's consider such a case. Three people are in debt and each has $4 in debt. What is the total debt? In order to find it, you need to add up all three debts: 4 dollars + 4 dollars + 4 dollars = 12 dollars. We decided that the addition of three numbers 4 is denoted as 3x4. Since in this case we are talking about debt, there is a “-” sign before the 4. We know that the total debt is $12, so our problem now becomes 3x(-4)=-12.

We will get the same result if, according to the problem, each of the four people has a debt of $3. In other words, (+4)x(-3)=-12. And since the order of the factors does not matter, we get (-4)x(+3)=-12 and (+4)x(-3)=-12.

Let's summarize the results. When you multiply one positive number and one negative number, the result will always be a negative number. The numerical value of the answer will be the same as in the case of positive numbers. Product (+4)x(+3)=+12. The presence of the “-” sign only affects the sign, but does not affect the numerical value.

How to multiply two negative numbers?

Unfortunately, it is very difficult to come up with a suitable real-life example on this topic. It is easy to imagine a debt of 3 or 4 dollars, but it is absolutely impossible to imagine -4 or -3 people who got into debt.

Perhaps we will go a different way. In multiplication, when the sign of one of the factors changes, the sign of the product changes. If we change the signs of both factors, we must change twice work mark, first from positive to negative, and then vice versa, from negative to positive, that is, the product will have an initial sign.

Therefore, it is quite logical, although a little strange, that (-3) x (-4) = +12.

Sign position when multiplied it changes like this:

positive number x positive number = positive number;
a negative number x positive number = negative number;
positive number x negative number = negative number;
negative number x negative number = positive number.

In other words, multiplying two numbers with the same signs, we get a positive number. Multiplying two numbers with different signs, we get a negative number.

The same rule is true for the action opposite to multiplication - for.

You can easily verify this by running inverse multiplication operations. In each of the examples above, if you multiply the quotient by the divisor, you will get the dividend and make sure it has the same sign, for example (-3)x(-4)=(+12).

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This article provides a detailed overview dividing numbers with different signs. First, the rule for dividing numbers with different signs is given. Below are examples of dividing positive numbers by negative and negative numbers by positive.

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Rule for dividing numbers with different signs

In the article division of integers, a rule for dividing integers with different signs was obtained. It can be extended to both rational numbers and real numbers by repeating all the reasoning from the above article.

So, rule for dividing numbers with different signs has the following formulation: to divide a positive number by a negative or a negative number by a positive, you must divide the dividend by the modulus of the divisor, and put a minus sign in front of the resulting number.

Let's write this division rule using letters. If the numbers a and b have different signs, then the formula is valid a:b=−|a|:|b| .

From the stated rule it is clear that the result of dividing numbers with different signs is a negative number. Indeed, since the modulus of the dividend and the modulus of the divisor are positive numbers, their quotient is a positive number, and the minus sign makes this number negative.

Note that the rule considered reduces the division of numbers with different signs to the division of positive numbers.

You can give another formulation of the rule for dividing numbers with different signs: to divide the number a by the number b, you need to multiply the number a by the number b −1, the inverse of the number b. That is, a:b=a b −1 .

This rule can be used when it is possible to go beyond the set of integers (since not every integer has an inverse). In other words, it applies to the set of rational numbers as well as the set of real numbers.

It is clear that this rule for dividing numbers with different signs allows you to move from division to multiplication.

The same rule is used when dividing negative numbers.

It remains to consider how this rule for dividing numbers with different signs is applied when solving examples.

Examples of dividing numbers with different signs

Let us consider solutions to several characteristic examples of dividing numbers with different signs to understand the principle of applying the rules from the previous paragraph.

Divide the negative number −35 by the positive number 7.

The rule for dividing numbers with different signs prescribes first finding the modules of the dividend and divisor. The modulus of −35 is 35, and the modulus of 7 is 7. Now we need to divide the module of the dividend by the module of the divisor, that is, we need to divide 35 by 7. Remembering how division of natural numbers is performed, we get 35:7=5. The last step left in the rule for dividing numbers with different signs is to put a minus in front of the resulting number, we have −5.

Here's the whole solution: .

It was possible to proceed from a different formulation of the rule for dividing numbers with different signs. In this case, we first find the inverse of the divisor 7. This number is the common fraction 1/7. Thus, . It remains to multiply numbers with different signs: . Obviously, we came to the same result.

(−35):7=−5 .

Calculate the quotient 8:(−60) .

According to the rule for dividing numbers with different signs, we have 8:(−60)=−(|8|:|−60|)=−(8:60) . The resulting expression corresponds to a negative ordinary fraction (see the division sign as a fraction bar), you can reduce the fraction by 4, we get .

Let's write down the whole solution briefly: .

When dividing fractional rational numbers with different signs, their dividend and divisor are usually represented as ordinary fractions. This is due to the fact that it is not always convenient to perform division with numbers in other notation (for example, in decimal).

The modulus of the dividend is equal, and the modulus of the divisor is 0,(23) . To divide the modulus of the dividend by the modulus of the divisor, let's move on to ordinary fractions.

IN this lesson Multiplication and division of rational numbers are considered.

Lesson content

Multiplying rational numbers

The rules for multiplying integers also apply to rational numbers. In other words, to multiply rational numbers, you need to be able to

Also, you need to know the basic laws of multiplication, such as: the commutative law of multiplication, the associative law of multiplication, the distributive law of multiplication and multiplication by zero.

Example 1. Find the value of an expression

This is the multiplication of rational numbers with different signs. To multiply rational numbers with different signs, you need to multiply their modules and put a minus in front of the resulting answer.

To clearly see that we are dealing with numbers that have different signs, we enclose each rational number in brackets along with its signs

The modulus of the number is equal to , and the modulus of the number is equal to . Having multiplied the resulting modules as positive fractions, we received the answer, but before the answer we put a minus, as the rule required of us. To ensure this minus before the answer, the multiplication of modules was performed in parentheses, preceded by a minus.

The short solution looks like this:

Example 2. Find the value of an expression

Example 3. Find the value of an expression

This is the multiplication of negative rational numbers. To multiply negative rational numbers, you need to multiply their modules and put a plus in front of the resulting answer

Solution for this example can be written briefly:

Example 4. Find the value of an expression

The solution for this example can be written briefly:

Example 5. Find the value of an expression

This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer

The short solution will look much simpler:

Example 6. Find the value of an expression

Let's convert the mixed number to an improper fraction. Let's rewrite the rest as is

We obtained the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer. The entry with modules can be skipped so as not to clutter the expression

The solution for this example can be written briefly

Example 7. Find the value of an expression

This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer

At first the answer turned out to be an improper fraction, but we highlighted the whole part in it. note that whole part was separated from the fraction module. The resulting mixed number was enclosed in parentheses preceded by a minus sign. This is done to ensure that the requirement of the rule is fulfilled. And the rule required that the answer received be preceded by a minus.

The solution for this example can be written briefly:

Example 8. Find the value of an expression

First, let's multiply and and multiply the resulting number with the remaining number 5. We'll skip the entry with modules so as not to clutter the expression.

Answer: expression value equals −2.

Example 9. Find the meaning of the expression:

Let's translate mixed numbers to improper fractions:

We got the multiplication of negative rational numbers. Let's multiply the modules of these numbers and put a plus in front of the resulting answer. The entry with modules can be skipped so as not to clutter the expression

Example 10. Find the value of an expression

The expression consists of several factors. According to the associative law of multiplication, if an expression consists of several factors, then the product will not depend on the order of actions. This allows us to evaluate a given expression in any order.

Let's not reinvent the wheel, but calculate this expression from left to right in the order of the factors. Let's skip the entry with modules so as not to clutter the expression

Third action:

Fourth action:

Answer: the value of the expression is

Example 11. Find the value of an expression

Let's remember the law of multiplication by zero. This law states that a product is equal to zero if at least one of the factors is equal to zero.

In our example, one of the factors is equal to zero, so without wasting time we answer that the value of the expression is equal to zero:

Example 12. Find the value of an expression

The product is equal to zero if at least one of the factors is equal to zero.

In our example, one of the factors is equal to zero, so without wasting time we answer that the value of the expression equals zero:

Example 13. Find the value of an expression

You can use the order of actions and first calculate the expression in brackets and multiply the resulting answer with a fraction.

You can also use the distributive law of multiplication - multiply each term of the sum by a fraction and add the resulting results. We will use this method.

According to the order of operations, if an expression contains addition and multiplication, then the multiplication must be performed first. Therefore, in the resulting new expression, let’s put in brackets those parameters that must be multiplied. This way we can clearly see which actions to perform earlier and which later:

Third action:

Answer: expression value equals

The solution for this example can be written much shorter. It will look like this:

It is clear that this example could be solved even in one’s mind. Therefore, you should develop the skill of analyzing an expression before solving it. It is likely that it can be solved mentally and save a lot of time and nerves. And in tests and exams, as you know, time is very valuable.

Example 14. Find the value of the expression −4.2 × 3.2

This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer

Notice how the modules of rational numbers were multiplied. In this case, to multiply the moduli of rational numbers, it took .

Example 15. Find the value of the expression −0.15 × 4

This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer

Notice how the modules of rational numbers were multiplied. In this case, in order to multiply the moduli of rational numbers, it was necessary to be able to.

Example 16. Find the value of the expression −4.2 × (−7.5)

This is the multiplication of negative rational numbers. Let's multiply the modules of these numbers and put a plus in front of the resulting answer

Division of rational numbers

The rules for dividing integers also apply to rational numbers. In other words, to be able to divide rational numbers, you need to be able to

Otherwise, the same methods for dividing ordinary and decimal fractions are used. To divide a common fraction by another fraction, you need to multiply the first fraction by the reciprocal of the second fraction.

And to divide decimal to another decimal fraction, you need to move the decimal point in the dividend and in the divisor to the right by as many digits as there are after the decimal point in the divisor, then perform the division as with a regular number.

Example 1. Find the meaning of the expression:

This is the division of rational numbers with different signs. To calculate such an expression, you need to multiply the first fraction by the reciprocal of the second.

So, let's multiply the first fraction by the reciprocal of the second.

We obtained the multiplication of rational numbers with different signs. And we already know how to calculate such expressions. To do this, you need to multiply the moduli of these rational numbers and put a minus in front of the resulting answer.

Let's complete this example to the end. The entry with modules can be skipped so as not to clutter the expression

So the value of the expression is

The detailed solution is as follows:

A short solution would look like this:

Example 2. Find the value of an expression

This is the division of rational numbers with different signs. To calculate this expression, you need to multiply the first fraction by the reciprocal of the second.

The reciprocal of the second fraction is the fraction . Let's multiply the first fraction by it:

A short solution would look like this:

Example 3. Find the value of an expression

This is the division of negative rational numbers. To calculate this expression, you again need to multiply the first fraction by the reciprocal of the second.

The reciprocal of the second fraction is the fraction . Let's multiply the first fraction by it:

We got the multiplication of negative rational numbers. We already know how such an expression is calculated. You need to multiply the moduli of rational numbers and put a plus in front of the resulting answer.

Let's finish this example to the end. You can skip the entry with modules so as not to clutter the expression:

Example 4. Find the value of an expression

To calculate this expression, you need to multiply the first number −3 by the inverse fraction of .

The inverse of a fraction is the fraction . Multiply the first number −3 by it

Example 6. Find the value of an expression

To calculate this expression, you need to multiply the first fraction by the number reciprocal of number 4.

The reciprocal of the number 4 is a fraction. Multiply the first fraction by it

Example 5. Find the value of an expression

To calculate this expression, you need to multiply the first fraction by the inverse of −3

The inverse of −3 is a fraction. Let's multiply the first fraction by it:

Example 6. Find the value of the expression −14.4: 1.8

This is the division of rational numbers with different signs. To calculate this expression, you need to divide the module of the dividend by the module of the divisor and put a minus before the resulting answer

Notice how the module of the dividend was divided by the module of the divisor. In this case, to do it correctly, it was necessary to be able to.

If you don't want to mess around with decimals (and this happens often), then these , then convert these mixed numbers into improper fractions, and then do the division itself.

Let's calculate the previous expression −14.4: 1.8 this way. Let's convert decimals to mixed numbers:

Now let’s convert the resulting mixed numbers into improper fractions:

Now you can do division directly, namely, divide a fraction by a fraction. To do this, you need to multiply the first fraction by the inverse fraction of the second:

Example 7. Find the value of an expression

Let's convert the decimal fraction −2.06 to an improper fraction, and multiply this fraction by the reciprocal of the second fraction:

Multistory fractions

You can often come across an expression in which the division of fractions is written using a fraction line. For example, the expression could be written as follows:

What is the difference between the expressions and ? There's really no difference. These two expressions carry the same meaning and an equal sign can be placed between them:

In the first case, the division sign is a colon and the expression is written on one line. In the second case, the division of fractions is written using a fraction line. The result is a fraction that people agree to call multi-storey.

When encountering such multi-story expressions, you need to apply the same rules for dividing ordinary fractions. The first fraction must be multiplied by the reciprocal of the second.

It is extremely inconvenient to use such fractions in a solution, so you can write them in an understandable form using a colon rather than a fractional line as a division sign.

For example, let's write a multi-story fraction in an understandable form. To do this, you first need to figure out where the first fraction is and where the second is, because it is not always possible to do this correctly. Multistory fractions have several fraction lines that can be confusing. The main fraction line, which separates the first fraction from the second, is usually longer than the rest.

After determining the main fractional line, you can easily understand where the first fraction is and where the second is:

Example 2.

We find the main fraction line (it is the longest) and see that the integer −3 is divided by a common fraction

And if we mistakenly took the second fractional line as the main one (the one that is shorter), then it would turn out that we are dividing the fraction by the integer 5. In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the dividend in this In this case, the number is −3, and the divisor is the fraction .

Example 3. Let's write the multi-level fraction in an understandable form

We find the main fraction line (it is the longest) and see that the fraction is divided by the integer 2

And if we mistakenly took the first fractional line as the leading one (the one that is shorter), then it would turn out that we are dividing the integer −5 by the fraction. In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the dividend in this case the fraction is , and the divisor is the integer 2.

Despite the fact that multi-level fractions are inconvenient to work with, we will encounter them very often, especially when studying higher mathematics.

Naturally, it takes additional time and space to translate a multi-story fraction into an understandable form. Therefore, you can use more quick method. This method is convenient and the output allows you to get a ready-made expression in which the first fraction has already been multiplied by the reciprocal fraction of the second.

This method is implemented as follows:

If the fraction is four-story, for example, then the number located on the first floor is raised to the top floor. And the figure located on the second floor is raised to the third floor. The resulting numbers must be connected with multiplication signs (×)

As a result, bypassing the intermediate notation, we obtain a new expression in which the first fraction has already been multiplied by the reciprocal fraction of the second. Convenience and that's it!

To avoid mistakes when using this method, you can follow the following rule:

From first to fourth. From second to third.

In the rule we're talking about about the floors. The figure from the first floor must be raised to the fourth floor. And the figure from the second floor needs to be raised to the third floor.

Let's try to calculate a multi-story fraction using the above rule.

So, we raise the number located on the first floor to the fourth floor, and raise the number located on the second floor to the third floor

As a result, bypassing the intermediate notation, we obtain a new expression in which the first fraction has already been multiplied by the reciprocal fraction of the second. Next, you can use your existing knowledge:

Let's try to calculate a multi-level fraction using a new scheme.

There are only the first, second and fourth floors. There is no third floor. But we do not deviate from the basic scheme: we raise the figure from the first floor to the fourth floor. And since there is no third floor, we leave the number located on the second floor as is

As a result, bypassing the intermediate notation, we received a new expression in which the first number −3 has already been multiplied by the reciprocal fraction of the second. Next, you can use your existing knowledge:

Let's try to calculate the multi-story fraction using the new scheme.

There are only the second, third and fourth floors. There is no first floor. Since there is no first floor, there is nothing to go up to the fourth floor, but we can raise the figure from the second floor to the third:

As a result, bypassing the intermediate notation, we received a new expression in which the first fraction has already been multiplied by the inverse of the divisor. Next, you can use your existing knowledge:

Using Variables

If the expression is complex and it seems to you that it will confuse you in the process of solving the problem, then part of the expression can be put into a variable and then work with this variable.

Mathematicians often do this. A complex problem is broken down into easier subtasks and solved. Then the solved subtasks are collected into one single whole. This creative process and this is something one learns over the years through hard training.

The use of variables is justified when working with multi-level fractions. For example:

Find the value of an expression

So, there is a fractional expression in the numerator and in the denominator of which fractional expressions. In other words, we are again faced with a multi-story fraction, which we do not like so much.

The expression in the numerator can be entered into a variable with any name, for example:

But in mathematics, in such a case, it is customary to name variables using capital Latin letters. Let's not break this tradition, and denote the first expression with a big Latin letter A

And the expression in the denominator can be denoted by the capital letter B

Now our original expression takes the form . That is, we made a replacement numerical expression to a letter, having previously entered the numerator and denominator into the variables A and B.

Now we can separately calculate the values of variable A and the value of variable B. We will insert the finished values into the expression.

Let's find the value of the variable A

Let's find the value of the variable B

Now let’s substitute their values into the main expression instead of variables A and B:

We have obtained a multi-story fraction in which we can use the scheme “from the first to the fourth, from the second to the third,” that is, raise the number located on the first floor to the fourth floor, and raise the number located on the second floor to the third floor. Further calculations will not be difficult:

Thus, the value of the expression is −1.

Of course we have considered simplest example, but our goal was to learn how we can use variables to make things easier for ourselves, to minimize errors.

Note also that the solution for this example can be written without using variables. It will look like

This solution is faster and shorter, and in this case it makes more sense to write it this way, but if the expression turns out to be complex, consisting of several parameters, brackets, roots and powers, then it is advisable to calculate it in several stages, entering part of its expressions into variables.

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