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 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;
- 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 identical 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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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 this rule down in letter form. For any positive real number a and any negative real number −b, the equality 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 for both real numbers, both 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 simple case, to focus on the rule steps rather than the computational complexities.
Example.
Multiply the negative number −4 by the 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.
By converting a periodic decimal fraction into a common fraction, and also by converting 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 of multiplying numbers with different signs, is equal to . All that remains is to multiply the ordinary fractions in brackets, we have 
.
Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional Expressions long time considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
Modern look simple fractional remainders, the parts of which are separated precisely horizontal line, first contributed to Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions are multiplied with different denominators.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next, you need to remember how fractional numbers are multiplied with same denominators. The very rule of this process is easy to formulate independently: the result of multiplication simple fractions with the same denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in essence, new denominator there is a square of one of the originally existing ones.
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 formed number under the fractional line will be the product of different numbers and, naturally, the square of one numerical expression it is impossible to name it.
It is worth considering the multiplication of fractions with different denominators using 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 .
The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers next to each other worth multipliers You cannot abbreviate above or below the fractional line.
Along with simple fractional numbers, there is a concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided 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.
The example uses multiplication of a number by ordinary fractional part , the rule for this action can be written as:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Convert mixed numbers to improper fractions and obtain the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way of representing a mixed fraction as an improper fraction, it can also be represented as general formula:
a bc = a*b+ c/c, where is the denominator new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in reverse side. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplication improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex problems. math problems in various program variations. A sufficient number of such services offer their assistance in counting multiplication of fractions with different numbers in denominators - so-called online calculators for calculating fractions. They are capable of not only multiplying, but also producing all the other simplest arithmetic operations with ordinary fractions and mixed numbers. It’s easy to work with; you fill in the appropriate fields on the site page and select the sign mathematical operation and click “calculate”. The program calculates automatically.
Subject arithmetic operations with fractional numbers is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but whole fractional expressions , but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well learned basic knowledge give complete confidence in successful decision most complex tasks.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of a person to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
) and denominator by denominator (we get the denominator of the product).
Formula for multiplying fractions:
For example:
Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.
Dividing a common fraction by a fraction.
Dividing fractions involving natural numbers.
It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:
Multiplying mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper fractions;
- multiplying the numerators and denominators of fractions;
- reduce the fraction;
- If you get an improper fraction, then we convert the improper fraction into a mixed fraction.
Note! To multiply mixed fraction to another mixed fraction, you must first convert them to the form of improper fractions, and then multiply them according to the rule for multiplying ordinary fractions.
The second way to multiply a fraction by a natural number.
It may be more convenient to use the second method of multiplication common fraction per number.
Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.
Multistory fractions.
In high school, three-story (or more) fractions are often encountered. Example:
To bring such a fraction to its usual form, use division through 2 points:
Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, For example:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.
2. In tasks with different types fractions - go to the form of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.
5. Divide a unit by a fraction in your head, simply turning the fraction over.