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 consider a few typical 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?
Divide the mixed number - 3 3 22 by the decimal fraction 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 ordinary 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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Class: 6
“Knowledge is a collection of facts. Wisdom is the ability to use them
The purpose of the lesson: 1) derivation of the rule for multiplying positive and negative numbers; ways of applying these rules in the simplest cases;
2) development of skills to compare, identify patterns, generalize;
3) search for various ways and methods for solving practical problems;
4) make a mini-project. News bulletin.
Equipment: thermometer model, cards for mutual simulator, projector.
During the classes
Greetings. To find out what new topic we will consider today, mental counting will help us. Calculate the examples, replace the answers with letters using "number - letter".
Slide #1 Think a little
Slide 2 Who is this?
The Indian mathematician Brahmagupta, who lived in the 7th century, represented positive numbers as "property", negative numbers as "debts".
He expressed the rules for adding positive and negative numbers as follows:
"The sum of two properties is property":
"The sum of two debts is debt":
And we will learn the rule after we consider the topic "Multiplication of negative and positive numbers"
Your task is to learn how to multiply positive and negative numbers, as well as how to multiply negative numbers.
We will make a mini-project.
Mini project.
News bulletin
"Multiplication of Positive and Negative Numbers"
Group work (4 groups).(The action is placed in a mathematical simulator)
Task 1 (1 group)
The air temperature drops every hour by two degrees. Now the thermometer shows zero degrees. What temperature will it show in three hours? Draw this on a coordinate line. Give similar examples. Make a conclusion and generalize.
Solution:
Since now the temperature is zero degrees and for every hour it drops by 2 degrees, then in 3 hours it will be equal to -6,
(-2) 3=-(2 3)=-6
Task 1 (Group 2)
The air temperature drops every hour by two degrees. Now the thermometer shows zero degrees. What air temperature did the thermometer show 3 hours ago? Draw this on a coordinate line. Make a conclusion.
Solution:
Since the temperature drops by two degrees every hour, and now it is zero degrees, 3 hours ago it was +6.
(-2) (-3)=2 3=6
Task 1 (group 3)
The factory produces 200 men's suits a day. When they began to produce suits of a new style, the fabric consumption per suit was changed by -0.4 m2. How much did the cost of fabric for suits change per day?
Solution:
This means that the cost of fabric for suits per day has changed by - 80.
(-0.4) 200=-(0.4 200)=-80.
Task 1 (Group 4)
The air temperature drops every hour by two degrees. Now the thermometer shows zero degrees. What air temperature did the thermometer show 4 hours ago?
Solution:
Since the temperature drops by two degrees every hour, and now it is zero degrees, then 4 hours ago it was equal to +8, that is
(-2) (-4)=2 4=8
Conclusions (students enter information into the layout of the newsletter).
Slide #4 Think about it.
Primary comprehension and application of the studied.
Work with the table at the board and in the field (using the newsletter layout).
We repeat the rule (questions are asked by students).
Working with the textbook:
- 1 student: No. 1105 (f, h, i) 2 student: No. 1105 (k, l, m)
- No. 1107 (we work in groups) 1 group: a), d);
2nd group: b), e);
Group 3: c), d).
Physical education (2 min.)
We repeat the rule for the equation of positive and negative numbers.
Slide number 5 Task 2
Task 2 (the same for all groups).
Apply the commutative and associative properties, multiply several numbers and conclude:
If the number of negative factors is even, then the product is the number _?_
If the number of negative factors is odd, then the product is the number _?_
Add more information to the newsletter layout.
Slide number 6 Rule of signs.
Determine the sign of the product:
1) "+" "-" "-" "+" "-" "-"
2) "-" "-" "-" "+" "+"
·«+»·«-»·«-»
3) "-" "+" "-" "-" "+" "+"
·«-»·«+»·«-»·«-»·«+»
So, let's go through the entire bulletin and repeat the rules for applying them to solving tasks on the cards.
Trainer (4 options).
Check yourself.
Answers to cards.
| 1 option | Option 2 | 3 option | 4 option | |
| 1) | 18 | 20 | 24 | 18 |
| 2) | -20 | -18 | -18 | -24 |
| 3) | -24 | 16 | 24 | 18 |
| 4) | 15 | -15 | 1 | -2 |
| 5) | -4 | 0 | -5 | 0 |
| 6) | 0 | 2 | 2 | -5 |
| 7) | -1 | -3 | -1,5 | -3 |
| 8) | -0,8 | -3,5 | -4,8 | 3,6 |
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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The focus of this article is division of negative numbers. First, the rule for dividing a negative number by a negative one is given, its justifications are given, and then examples of dividing negative numbers are given with a detailed description of the solutions.
Page navigation.
Rule for dividing negative numbers
Before giving the rule for dividing negative numbers, let us recall the meaning of the division action. Division in its essence represents finding an unknown factor by a known product and a known other factor. That is, the number c is the quotient of a divided by b when c b=a , and vice versa, if c b=a , then a:b=c .
Rule for dividing negative numbers the following: the quotient of dividing one negative number by another is equal to the quotient of dividing the numerator by the modulus of the denominator.
Let's write down the voiced rule using letters. If a and b are negative numbers, then the equality a:b=|a|:|b| .
The equality a:b=a b −1 is easy to prove, starting from properties of multiplication of real numbers and definitions of reciprocal numbers. Indeed, on this basis, one can write a chain of equalities of the form (a b −1) b=a (b −1 b)=a 1=a, which, by virtue of the sense of division mentioned at the beginning of the article, proves that a · b − 1 is the quotient of dividing a by b .
And this rule allows you to go from dividing negative numbers to multiplication.
It remains to consider the application of the considered rules for dividing negative numbers when solving examples.
Examples of dividing negative numbers
Let's analyze examples of division of negative numbers. Let's start with simple cases, on which we will work out the application of the division rule.
Example.
Divide the negative number −18 by the negative number −3 , then compute the quotient (−5):(−2) .
Solution.
By the rule of division of negative numbers, the quotient of dividing −18 by −3 is equal to the quotient of dividing the moduli of these numbers. Since |−18|=18 and |−3|=3 , then (−18):(−3)=|−18|:|−3|=18:3 , it remains only to perform the division of natural numbers, we have 18:3=6.
We solve the second part of the problem in the same way. Since |−5|=5 and |−2|=2 , then (−5):(−2)=|−5|:|−2|=5:2 . This quotient corresponds to an ordinary fraction 5/2, which can be written as a mixed number.
The same results are obtained using a different rule for dividing negative numbers. Indeed, the number −3 is inversely the number , then 
, now we perform the multiplication of negative numbers: 
. Likewise, .
Answer:
(−18):(−3)=6 and 
.
When dividing fractional rational numbers, it is most convenient to work with ordinary fractions. But, if convenient, then you can divide and final decimal fractions.
Example.
Divide the number -0.004 by -0.25 .
Solution.
The modules of the dividend and divisor are 0.004 and 0.25, respectively, then, according to the rule for dividing negative numbers, we have (−0,004):(−0,25)=0,004:0,25 .
- or perform division of decimal fractions by a column,
- or go from decimals to ordinary fractions, and then divide the corresponding ordinary fractions.
Let's take a look at both approaches.
To divide 0.004 by 0.25 in a column, first move the comma 2 digits to the right, while dividing 0.4 by 25. Now we perform division by a column: 
So 0.004:0.25=0.016 .
And now let's show what the solution would look like if we decided to convert decimal fractions to ordinary ones. Because 
and , then 
, and execute
In this lesson, we will review the rules for adding positive and negative numbers. We will also learn how to multiply numbers with different signs and learn the rules of signs for multiplication. Consider examples of multiplication of positive and negative numbers.
The property of multiplying by zero remains true in the case of negative numbers. Zero multiplied by any number is zero.
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. - M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. - M .: Education, Mathematics Teacher Library, 1989.
Homework
- Internet portal Mnemonica.ru ().
- Internet portal Youtube.com ().
- Internet portal School-assistant.ru ().
- Internet portal Bymath.net ().
