Multiplication. Multiplication of numbers with different signs, rule, examples

) And the denominator on the denominator (we get a denominator of the work).

Formula multiplication fractions:

For example:

Before proceeding with multiplication of numerals and denominators, it is necessary to check the possibility of cutting the fraction. If it turns out to shorten the fraction, then you will be easier to carry out calculations.

Division of ordinary fraction on the fraction.

Division fractions with the participation of a natural number.

It's not as scary as it seems. As in the case of adding, we translate an integer in the fraction with a unit in the denominator. For example:

Multiplying mixed fractions.

Rules of multiplication of fractions (mixed):

we transform mixed fractions into the wrong;
reduce the numerals and denominators of fractions;
reducing the fraction;
if you got the wrong fraction, we transform the wrong fraction into a mixed one.

Note! To multiply the mixed fraction on another mixed fraction, you need to begin, lead them to the mind of the wrong fractions, and then multiply by the rule of multiplication of ordinary fractions.

The second method of multiplication of the fraction on a natural number.

It is more convenient to use the second way of multiplying an ordinary fraction for a number.

Note! For multiplication of fractions on natural number A denominator is needed to divide the number, and the numerator is left unchanged.

From the above, the example is clear that this option is more convenient for use when the denoter of the fraction is divided without a residue on a natural number.

Multi-storey fractions.

In high school classes, three-story (or more) fractions are found. Example:

To bring such a fraction to the usual mind, use division after 2 points:

Note!In dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, eg:

When dividing units on any fraction, the result will the same fraction, only inverted:

Practical tips when multiplying and dividing fractions:

1. The most important in working with fractional expressions is accuracy and attentiveness. All calculations do carefully and gently, concentrately and clearly. Better write down a few unnecessary lines in the drafts, than getting confused in the calculations in the mind.

2. In tasks with different species fractions - go to the form of ordinary fractions.

3. All fractions reducing until it is impossible to cut.

4. Multi-storey fractional expressions We are looking into the form of ordinary, using the division after 2 points.

5. Unit of fraction divide in mind, just turning the fraction.

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

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Multiplication rule numbers with different signs

Multiplying a positive number on a negative, as well as a negative on a positive, is carried out at the following multiply multiplication rules with different signs: To multiply the numbers with different signs, you need to multiply, and to put a minus sign before the work received.

We write this rule In letterproof. For any positive actual number A and a valid negative number -B equality. a · (-b) \u003d - (| a | · | b |) , as well as for a negative number, and a positive number B rightly equality (-A) · b \u003d - (| a | · | b |) .

The rule of multiplication of numbers with different signs is fully consistent with properties of action with valid numbers. Indeed, on their basis it is easy to show that for real and positive numbers a and b is the chain of the equalities of the form a · (-b) + a · b \u003d a · ((- b) + b) \u003d a · 0 \u003d 0, which proves that A · (-b) and A · b are opposite numbers, whence the equality A · (-b) \u003d - (A · b). And from it follows the fairness of the multiplication rule.

It should be noted that the voiced rule of multiplication of numbers with different signs is fair both for real numbers and for rational numbers and for integers. This follows from the fact that actions with rational and integers possess the same properties that were used in the proof above.

It is clear that multiplication of numbers with different signs according to the obtained rule is reduced to multiplying positive numbers.

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

Examples of multiplication of numbers with different signs

We will analyze the solutions of several examples of multiplication of numbers with different signs. Let's start with a simple case to focus on the steps of the rule, and not on the computational difficulties.

Example.

Perform a multiplication of a negative number -4 to a positive number 5.

Decision.

According to the rule of multiplication of numbers with different signs, we first need to multiply the modules of the initial multipliers. Module -4 is 4, and the module 5 is 5, and the multiplication of natural numbers 4 and 5 gives 20. Finally, it remains to put a minus sign before the number it obtained, we have -20. The multiplication is completed.

A brief solution can be written as follows: (-4) · 5 \u003d - (4 · 5) \u003d - 20.

Answer:

(-4) · 5 \u003d -20.

When multiplying fractional numbers with different signs you need to be able to perform multiplication of ordinary fractions, multiplying decimal fractions and their combinations with natural and mixed numbers.

Example.

Spend multiplication of numbers with different signs 0, (2) and.

Decision.

After transferring the periodic decimal fraction in an ordinary fraction, as well as by performing a transition from a mixed number to incorrect fraction, from the original work We will come to the product of ordinary fractions with different signs of the species. This product according to the rule of multiplication of numbers with different signs is equal. It remains only to multiply ordinary fractions in brackets, we have .

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

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Multiplication rule numbers with different signs

We write this rule in the alphabet. For any positive actual number A and a valid negative number -B equality. a · (-b) \u003d - (| a | · | b |) , as well as for a negative number, and a positive number B rightly equality (-A) · b \u003d - (| a | · | b |) .

It should be noted that the voiced rule of multiplication of numbers with different signs is fair both for valid numbers and for rational numbers And for integers. This follows from the fact that actions with rational and integers possess the same properties that were used in the proof above.

It is clear that multiplication of numbers with different signs according to the obtained rule is reduced to multiplying positive numbers.

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

Examples of multiplication of numbers with different signs

Perform a multiplication of a negative number -4 to a positive number 5.

A brief solution can be written as follows: (-4) · 5 \u003d - (4 · 5) \u003d - 20.

(-4) · 5 \u003d -20.

When multiplying fractional numbers with different signs, you need to be able to perform multiplication of ordinary fractions, multiplying decimal fractions and combinations with natural and mixed numbers.

Spend multiplication of numbers with different signs 0, (2) and.

After transferring the periodic decimal fractions In an ordinary fraction, as well as by performing a transition from a mixed number to incorrect fraction, we will come from the original work to the product of ordinary fractions with different signs of the species. This product according to the rule of multiplication of numbers with different signs is equal. It remains only to multiply 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 figure it out with multiplication and division.

Suppose we need to multiply +3 to -4. How to do it?

Let's consider such a case. Three people climbed into debt, and every 4 dollar debt. What is the total debt? In order to find it, it is necessary to fold all three debt: 4 dollars + 4 dollars + 4 dollars \u003d 12 dollars. We decided that the addition of three numbers 4 is indicated as 3 × 4. Since in this case we are talking about the debt, before 4 there is a sign "-". We know that the total debt is equal to 12 dollars, so now our task has the form 3x (-4) \u003d - 12.

We will get the same result if, by the condition of the task, each of the four people has a debt of 3 dollars. In other words, (+4) x (-3) \u003d - 12. And since the order of the factors does not matter, we obtain (-4) x (+3) \u003d - 12 and (+4) x (-3) \u003d - 12.

Let's summarize the results. With multiplying one positive and one negative number, the result will always be a negative number. The numerical amount of the answer will be the same as in the case of positive numbers. Work (+4) x (+3) \u003d + 12. The presence of a sign "-" affects only a sign, but does not affect the numerical value.

And how to multiply two negative numbers?

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

Perhaps we will go different ways. In multiplication, when a sign of one of the multipliers changes the sign of the work. If we change signs from both multipliers, we must change twice sign of workFirst, with a positive on a negative, and then on the contrary, with a negative on a positive, that is, the work will have an initial sign.

Consequently, it is quite logical, although it is slightly strange that (-3) x (-4) \u003d + 12.

Sign position When multiply changed in this way:

positive number x positive number \u003d positive number;
negative number x positive number \u003d negative number;
positive number x Negative number \u003d negative number;
negative number x negative number \u003d positive number.

In other words, multipliering 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 valid for the opposite multiplication - for.

You can easily make sure by spending reverse operations multiplication. If in each of the examples given above, you multiply the private per divider, then get divisible, and make sure it has the same sign, for example (-3) x (-4) \u003d (+ 12).

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This article gives detailed review division numbers with different signs. At first, the number of dividing numbers with different signs is given. The examples of dividing positive numbers on negative and negative numbers are disassembled below.

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The division rule of numbers with different signs

In the article, the division of integers was obtained by the rule of division of integers with different signs. It can be extended to rational numbers, and for valid numbers, repeating all the arguments from this article.

So, the division rule of numbers with different signs It has the following wording: To split a positive number to a negative or negative number on a positive, it is necessary to divide into the divider module, and to put a minus sign before the obtained number.

We write this division rule with the help of letters. If the numbers A and B have different signs, then the formula is valid a: B \u003d - | A |: | B | .

Of the voiced rules, it is clear that the result of dividing numbers with different signs is a negative number. Indeed, since the divisory module and the divider module is positive, then their private has a positive number, and the minus sign makes it a negative number.

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

You can bring another formulation of the rules for dividing numbers with different signs: To divide the number A to the number B, you need to multiply by the number B -1, the reverse number B. I.e, a: B \u003d A · B -1 .

This rule can be used when it is possible to go beyond the set of integers (as far as not every integer has the opposite). In other words, it applies to the set of rational, as well as on a variety of valid numbers.

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

The same rule is used in dividing negative numbers.

It remains to consider how this division rule of numbers with different signs is used when solving examples.

Examples of dividing numbers with different signs

Consider solutions of several characteristic examples of dividing numbers with different signsTo assimilate the principle of application of rules from the previous paragraph.

Divide the negative number -35 per positive number 7.

The division rule of numbers with different signs prescribes first to find the dividera and divider modules. The number -35 module is 35, and the number 7 module is 7. Now we need to divide the divisory module on the divider module, that is, it is necessary to divide 35 to 7. Remembering how the division of natural numbers is performed, we obtain 35: 7 \u003d 5. Remained last step The rules for dividing numbers with different signs - to put minus before the number obtained, we have -5.

That's all the solution :.

It was possible to proceed from another formulation of the rules for dividing numbers with different signs. In this case, you first find the number, the reverse divider 7. This number is an ordinary fraction of 1/7. In this way, . It remains to perform multiplication of numbers with different signs :. Obviously, we came to the same result.

(−35):7=−5 .

Calculate the private 8: (- 60).

According to the rules of dividing numbers with different signs we have 8:(−60)=−(|8|:|−60|)=−(8:60) . The resulting expression corresponds to a negative ordinary fraction (see the fission sign as a fractional line), you can reduce the fraction on 4, we get .

We write all the decision briefly :.

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

The dividend module is equal, and the divider module is 0, (23). To divide the module divide on the divider module, we turn to ordinary fractions.

Ordinary fractional numbers first meet schoolchildren in grade 5 and accompany them throughout their lives, since in everyday life it is often necessary to consider or use some object not entirely, but separate pieces. The beginning of the study of this topic is a share. Shares are equal partswhich is divided by a particular subject. After all, it is not always possible to express, let's say, the length or price of the goods an integer, should take into account the parts or the share of any measure. Educated from the verb "Dog" - divide into parts, and having the Arab roots, in the VIII century the word "fraction" in Russian originated.

Fractional expressions for a long time considered the most complex section of mathematics. In the XVII century, with the appearance of first-legislers in mathematics, they were called "broken numbers", which was very difficult to appear in the understanding of people.

Modern appearance Simple fractional residues, parts of which are separated by the horizontal feature, first contributed to Fibonacci - Leonardo Pisa. His works dated in 1202. But the purpose of this article is simply and understandably explain to the reader, as a multiplication of mixed fractions with different denominators.

Multiplication of fractions with different denominators

Initially, it is worth determining varieties of fractions:

correct;
incorrect;
mixed.

Next, it is necessary to remember how multiplication of fractional numbers with the same denominants occurs. The rule of this process itself is easy to formulate independently: the result of multiplication of simple fractions with the same denominants is a fractional expression, the numerator of which has a product of numerals, and the denominator is a product of data denominators. That is, in fact, the new denominator is the square of one of the existing initially.

When multiplying simple fractions with different denominators For two or more factors, the rule does not change:

a /b. * C /d. = A * C / b * d.

The only difference is that educated number Under a fractional feature will be the product of different numbers and, naturally, the square of one numerical expression It is impossible to call it.

It is worth considering the multiplication of fractions with different denominators on the examples:

8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .

Examples use methods for reducing fractional expressions. You can reduce only the numbers of the number with the numbers of the denominator, nearby factories above the fractional feature or under it cannot be cut.

Along with simple fractional numbers, there is a concept of mixed fractions. The mixed number consists of an integer and fractional part, that is, it is the sum of these numbers:

1 4/ 11 =1 + 4/ 11.

How to multiply

A few examples are offered for consideration.

2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.

In the example, the multiplication of the number on ordinary fractional part, Count the rule for this action by the formula:

a * b /c. = A * b /c.

In fact, such a product is the sum of the same fractional residues, and the number of terms indicates this natural number. Private case:

4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.

There is another option to solve the multiplication of the number on the fractional residue. It is easy to just divide the denominator to this number:

d * E /f. = E /f: D.

It is useful to use this technique when the denominator is divided into a natural number without a residue or, as they say, a focus.

Translate mixed numbers into incorrect fractions and get a product of the previously described:

1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.

This example involves a way of representation mixed fraci In the wrong, it can also be represented as general formula:

a. B.c. = A * B + C / C, where the denominator of the new fraction is formed by multiplying the integer part with the denominator and when it is additioned with the numerator of the original fractional residue, and the denominator remains the same.

This process works in reverse side. To highlight the whole part and fractional residue, it is necessary to divide the numerator of the incorrect fraction on its denominator "Corner".

Multiplying irregular fractions Made a generally accepted way. When the record goes under a single fractional feature, as needed to make a reduction in fractions to reduce such a number and easier to calculate the result.

On the Internet there are many assistants to solve even complex math problems in different variations programs. Sufficient number Such services offer their help with a multiplication score of fractions with different numbers in denominators - the so-called online calculators for calculating fractions. They are capable not only multiplying, but also to produce all the other simple arithmetic operations with ordinary fractions and mixed numbers. It is easy to work with it, the corresponding fields are filled on the site page, the sign of the mathematical action is selected and the "calculate" is pressed. The program considers automatically.

The theme of arithmetic action with fractional numbers is relevant throughout the training of middle and senior schoolchildren. In high school, there are no longer the simplest species, but whole fractional expressions, but knowledge of the rules for transformation and calculations obtained earlier are applied in primeval form. Well-learned basic knowledge give full confidence in successful decision The most complex tasks.

In conclusion, it makes sense to bring the word Lev Nikolayevich Tolstoy, who wrote: "A person eating a fraction. Increase its number - their advantages - not in human power, but everyone can reduce its denominator - his opinion about himself, and this decrease is to get closer to its perfection. "