Positive and negative numbers are studied at the very beginning of the mathematics course, in the sixth grade. Although further learning requires constantly working with these numbers, it is not surprising that as time passes, some little things are forgotten - and people begin to make blunders.
Multiplication and division are some of the most common operations with numbers that have different signs. Let's figure it out and remember how to multiply and divide such numbers among themselves, putting the correct sign in the answer.
Multiplication of numbers with different signs
This rule is one of the simplest in arithmetic.
- If we have a certain positive number “a” in front of us, and it needs to be multiplied by a negative number “z”, then we simply multiply the numbers - and then put a minus sign in front of the result.
- You can also say this - in order to multiply numbers with different signs on each other, you need to multiply the modules of factors among themselves, and then return the minus sign in response.
The following numerical notation is valid for the statement: -а*z = - (|а|*|z|). We also recall that special rules apply for zero - if any number, positive or negative, is multiplied by it, the answer will in any case be equal to zero.
Let's take a couple of simple examples.
- If the expression looks like – 5*6, then you need to solve it as follows: -5*6 = - (|5|*|6|) = - 30.
- If an expression of the following type is - 7*0, then 0 is immediately written in the answer.
Division of numbers with different signs
For such cases, a very simple rule also applies. It is similar to the previous one - if the task requires dividing “-a” by “b”, or “a” by “-b”, then first we take the modules of numbers, their absolute values, and perform the division process without any permutation of the dividend and divisor .
Thus, the quotient is found - and then a minus sign is added to it. It does not matter whether a negative number acts as a dividend, or vice versa, we divide a number with a plus sign by a negative one - the answer will always be with a minus sign. In other words, using the numerical method, we write it like this: -a: b = - (|a| : |b|).
For example, - 10: 2 = - (10:2) = - 5, or 21: (-3) = - (21:3) = - 7. In the end, the division is not at all complicated and comes down to our usual actions on modules numbers.
And just like in the previous case, zero is in a special position. Its presence in the expression automatically gives zero in the answer. And it doesn't matter if it's 0:a or a:0 - both an attempt to divide by zero and a division by zero give the same result.
Now let's deal with multiplication and division.
Suppose we need to multiply +3 by -4. How to do it?
Let's consider such a case. Three people got into 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 + $4 + $4 = $12. We have decided that the addition of three numbers 4 is denoted as 3 × 4. Since in this case we are talking about debt, there is a “-” sign in front of 4. We know the total debt is $12, so now our problem is 3x(-4)=-12.
We will get the same result if, according to the condition of the problem, each of the four people has a debt of 3 dollars. 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 multiplying one positive 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 do you multiply two negative numbers?
Unfortunately, it is very difficult to come up with a suitable example from life on this topic. It's easy to imagine $3 or $4 in debt, but it's completely impossible to imagine -4 or -3 people getting into debt.
Perhaps we will go the other way. In multiplication, changing the sign of one of the factors changes the sign of the product. If we change the signs of both factors, we must change the signs twice product sign, first from positive to negative, and then vice versa, from negative to positive, that is, the product will have its original sign.
Therefore, it is quite logical, although a bit 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 the same sign, 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. If in each of the examples above you multiply the quotient by the divisor, you get the dividend, and make sure it has the same sign, like (-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 to positive.
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Rule for dividing numbers with different signs
In the article division of integers, the rule for dividing integers with different signs was obtained. It can be extended to both rational numbers and real numbers by repeating all the arguments from the specified article.
So, rule for dividing numbers with different signs has the following formulation: in order to divide a positive number by a negative or a negative number by a positive one, it is necessary to divide the dividend by the modulus of the divisor, and put a minus sign in front of the resulting number.
We 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 voiced 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 more positive than the number, then their quotient is a positive number, and the minus sign makes this number negative.
Note that the considered rule 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 reciprocal 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 is applicable on the set of rationals, as well as on the set real numbers.
It is clear that this rule for dividing numbers with different signs allows you to go 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 in solving examples.
Examples of dividing numbers with different signs
Let us consider solutions of several characteristic examples of dividing numbers with different signs to grasp the principle of applying the rules from the previous paragraph.
Example.
Divide the negative number −35 by the positive number 7 .
Solution.
The rule for dividing numbers with different signs prescribes first to find 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 modulus of the dividend by the modulus of the divisor, that is, we need to divide 35 by 7. Remembering how the division of natural numbers is performed, we get 35:7=5. The last step of the rule for dividing numbers with different signs remains - put a minus in front of the resulting number, we have -5.
Here is the whole solution: .
One could proceed from a different formulation of the rule for dividing numbers with different signs. In this case, we first find the number that is the reciprocal of the divisor 7. This number is the common fraction 1/7. Thus, . It remains to perform the multiplication of numbers with different signs: . Obviously, we came to the same result.
Answer:
(−35):7=−5 .
Example.
Calculate the quotient 8:(−60) .
Solution.
By the rule 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 division sign as a fraction bar), you can reduce the fraction by 4, we get 
.
We write down the whole solution briefly: .
Answer:
.
When dividing fractions rational numbers with different signs, their usual dividend and divisor are represented as ordinary fractions. This is due to the fact that it is not always convenient to perform division with numbers in a different notation (for example, in decimal).
Example.
Solution.
The modulus of the dividend is , 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.
Let's translate a mixed number into an ordinary fraction: 
, and
In this article, we will look at dividing positive numbers by negative numbers and vice versa. We will give a detailed analysis of the rule for dividing numbers with different signs, and also give examples.
Rule for dividing numbers with different signs
The rule for integers with different signs, obtained in the article on the division of integers, is also valid for rational and real numbers. Let us give a more general formulation of this rule.
Rule for dividing numbers with different signs
When dividing a positive number by a negative one and vice versa, you need to divide the dividend modulus by the divisor modulus, and write the result with a minus sign.
In literal form, it looks like this:
a ÷ - b = - a ÷ b
A ÷ b = - a ÷ b .
Dividing numbers with different signs always results in a negative number. The considered rule, in fact, reduces the division of numbers with different signs to the division of positive numbers, since the modules of the dividend and divisor are positive.
Another equivalent mathematical formulation of this rule is:
a ÷ b = a b - 1
To divide the numbers a and bhaving different signs, you need to multiply the number a by the reciprocal of the number b, that is, b - 1. This formulation is applicable on the set of rational and real numbers, it allows you to go from division to multiplication.
Let us now consider how to apply the theory described above in practice.
How to divide numbers with different signs? Examples
Below we will look at a few characteristic examples.
Example 1. How to divide numbers with different signs?
Divide - 35 by 7.
First, let's write the modules of the dividend and divisor:
35 = 35 , 7 = 7 .
Now let's separate the modules:
35 7 = 35 7 = 5 .
We add a minus sign in front of the result and get the answer:
Now let's use a different formulation of the rule and calculate the reciprocal of 7 .
Now let's do the multiplication:
35 1 7 = - - 35 1 7 = - 35 7 = - 5 .
Example 2. How to divide numbers with different signs?
If we divide fractional numbers with rational signs, the dividend and divisor must be represented as ordinary fractions.
Example 3. How to divide numbers with different signs?
Let's divide mixed number- 3 3 22 on decimal 0 , (23) .
The modules of the dividend and the divisor are respectively 3 3 22 and 0 , (23) . Converting 3 3 22 to a common fraction, we get:
3 3 22 = 3 22 + 3 22 = 69 22 .
We can also represent the divisor as a common fraction:
0 , (23) = 0 , 23 + 0 , 0023 + 0 , 000023 = 0 , 23 1 - 0 , 01 = 0 , 23 0 , 99 = 23 99 .
Now we divide common fractions, perform reductions and get the result:
69 22 ÷ 23 99 = - 69 22 99 23 = - 3 2 9 1 = - 27 2 = - 13 1 2 .
In conclusion, consider the case when the dividend and divisor are irrational numbers and are written as roots, logarithms, powers, etc.
In such a situation, the quotient is written as a numerical expression, which is simplified as much as possible. If necessary, its approximate value is calculated with the required accuracy.
Example 4. How to divide numbers with different signs?
Divide the numbers 5 7 and - 2 3 .
According to the rule for dividing numbers with different signs, we write the equality:
5 7 ÷ - 2 3 = - 5 7 ÷ - 2 3 = - 5 7 ÷ 2 3 = - 5 7 2 3 .
Let's get rid of the irrationality in the denominator and get the final answer:
5 7 2 3 = - 5 4 3 14 .
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§ 1 Multiplication of positive and negative numbers
In this lesson, we will get acquainted with the rules for multiplying and dividing positive and negative numbers.
It is known that any product can be represented as a sum of identical terms.
The term -1 must be added 6 times:
(-1)+(-1)+(-1) +(-1) +(-1) + (-1) =-6
So the product of -1 and 6 is -6.
The numbers 6 and -6 are opposite numbers.
Thus, we can conclude:
When multiplying -1 by natural number get its opposite number.
For negative numbers, as well as for positive ones, the commutative law of multiplication is fulfilled:
If a natural number is multiplied by -1, then the opposite number will also be obtained.
Multiplying any non-negative number by 1 results in the same number.
For example:
For negative numbers, this statement is also true: -5 ∙1 = -5; -2 ∙ 1 = -2.
Multiplying any number by 1 results in the same number.
We have already seen that when minus 1 is multiplied by a natural number, the opposite number will be obtained. When multiplying a negative number, this statement is also true.
For example: (-1) ∙ (-4) = 4.
Also -1 ∙ 0 = 0, the number 0 is the opposite of itself.
When you multiply any number by minus 1, you get its opposite number.
Let's move on to other cases of multiplication. Let's find the product of the numbers -3 and 7.
The negative factor -3 can be replaced by the product of -1 and 3. Then the associative multiplication law can be applied:
1 ∙ 21 = -21, i.e. the product of minus 3 and 7 is minus 21.
When multiplying two numbers with different signs, a negative number is obtained, the modulus of which is equal to the product of the moduli of the factors.
What is the product of numbers with the same sign?
We know that when you multiply two positive numbers, you get a positive number. Find the product of two negative numbers.
Let's replace one of the factors with a product with a factor minus 1.
We apply the rule we have derived, when multiplying two numbers with different signs, a negative number is obtained, the modulus of which is equal to the product of the moduli of the factors,
get -80.
Let's formulate the rule:
When multiplying two numbers with the same signs, a positive number is obtained, the modulus of which is equal to the product of the moduli of the factors.
§ 2 Division of positive and negative numbers
Let's move on to division.
By selection we find the roots of the following equations:
y ∙ (-2) = 10. 5 ∙ 2 = 10, so x = 5; 5 ∙ (-2) = -10, so a = 5; -5 ∙ (-2) = 10, so y = -5.
Let us write down the solutions of the equations. In each equation, the factor is unknown. We find the unknown factor by dividing the product by the known factor, we have already selected the values of the unknown factors.
Let's analyze.
When dividing numbers with the same signs (and these are the first and second equations), a positive number is obtained, the modulus of which is equal to the quotient of the moduli of the dividend and divisor.
When dividing numbers with different signs (this is the third equation), a negative number is obtained, the modulus of which is equal to the quotient of the moduli of the dividend and divisor. Those. when dividing positive and negative numbers, the sign of the quotient is determined by the same rules as the sign of the product. And the modulus of the quotient is equal to the quotient of the modulus of the dividend and divisor.
Thus, we have formulated the rules for multiplication and division of positive and negative numbers.
List of used literature:
- Mathematics. Grade 6: lesson plans for the textbook by I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilin. – Mnemosyne, 2009.
- Mathematics. Grade 6: student textbook educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
- Mathematics. Grade 6: textbook for students of educational institutions./N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M.: Mnemosyne, 2013.
- Mathematics Handbook - http://lyudmilanik.com.ua
- Handbook for students in high school http://shkolo.ru
