How to add 2 numbers with different signs. Posts tagged "addition of numbers with different signs"
>>Math: Adding numbers with different signs
33. Addition of numbers with different signs
If the air temperature was equal to 9 °C, and then it changed to - 6 °C (i.e., decreased by 6 °C), then it became equal to 9 + (- 6) degrees (Fig. 83).
To add the numbers 9 and - 6 using , you need to move point A (9) to the left by 6 unit segments (Fig. 84). We get point B (3).
This means 9+(- 6) = 3. The number 3 has the same sign as the term 9, and its module equal to the difference between the moduli of terms 9 and -6.
Indeed, |3| =3 and |9| - |- 6| = = 9 - 6 = 3.
If the same air temperature of 9 °C changed by -12 °C (i.e., decreased by 12 °C), then it became equal to 9 + (-12) degrees (Fig. 85). Adding the numbers 9 and -12 using the coordinate line (Fig. 86), we get 9 + (-12) = -3. The number -3 has the same sign as the term -12, and its module is equal to the difference between the modules of the terms -12 and 9.
Indeed, | - 3| = 3 and | -12| - | -9| =12 - 9 = 3.
To add two numbers with different signs, you need to:
1) subtract the smaller one from the larger module of the terms;
2) put in front of the resulting number the sign of the term whose modulus is greater.
Usually, the sign of the sum is first determined and written, and then the difference in modules is found.
For example:
1) 6,1+(- 4,2)= +(6,1 - 4,2)= 1,9,
or shorter 6.1+(- 4.2) = 6.1 - 4.2 = 1.9;
When adding positive and negative numbers you can use micro calculator. To enter a negative number into a microcalculator, you need to enter the modulus of this number, then press the “change sign” key |/-/|. For example, to enter the number -56.81, you need to press the keys sequentially: | 5 |, | 6 |, | ¦ |, | 8 |, | 1 |, |/-/|. Operations on numbers of any sign are performed on a microcalculator in the same way as on positive numbers.
For example, the sum -6.1 + 3.8 is calculated using program
? The numbers a and b have different signs. What sign will the sum of these numbers have if the larger module is negative?
if the smaller modulus is negative?
if the larger modulus is a positive number?
if the smaller modulus is a positive number?
Formulate a rule for adding numbers with different signs. How to enter a negative number into a microcalculator?
TO 1045. The number 6 was changed to -10. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is it equal to sum 6 and -10?
1046. The number 10 was changed to -6. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of 10 and -6?
1047. The number -10 was changed to 3. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 3?
1048. The number -10 was changed to 15. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 15?
1049. In the first half of the day the temperature changed by - 4 °C, and in the second half - by + 12 °C. By how many degrees did the temperature change during the day?
1050. Perform addition:
1051. Add:
a) to the sum of -6 and -12 the number 20;
b) to the number 2.6 the sum is -1.8 and 5.2;
c) to the sum -10 and -1.3 the sum of 5 and 8.7;
d) to the sum of 11 and -6.5 the sum of -3.2 and -6.
1052. Which number is 8; 7.1; -7.1; -7; -0.5 is the root equations- 6 + x = -13.1?
1053. Guess the root of the equation and check:
a) x + (-3) = -11; c) m + (-12) = 2;
b) - 5 + y=15; d) 3 + n = -10.
1054. Find the meaning of the expression:
1055. Follow the steps using a microcalculator:
a) - 3.2579 + (-12.308); d) -3.8564+ (-0.8397) +7.84;
b) 7.8547+ (- 9.239); e) -0.083 + (-6.378) + 3.9834;
c) -0.00154 + 0.0837; e) -0.0085+ 0.00354+ (- 0.00921).
P 1056. Find the value of the sum:
1057. Find the meaning of the expression:
1058. How many integers are located between the numbers:
a) 0 and 24; b) -12 and -3; c) -20 and 7?
1059. Imagine the number -10 as the sum of two negative terms so that:
a) both terms were integers;
b) both terms were decimal fractions;
c) one of the terms was a regular ordinary fraction.
1060. What is the distance (in unit segments) between the points of the coordinate line with coordinates:
a) 0 and a; b) -a and a; c) -a and 0; d) a and -Za?
M 1061. Radii of geographical parallels earth's surface, on which the cities of Athens and Moscow are located, are respectively 5040 km and 3580 km (Fig. 87). How much shorter is the Moscow parallel than the Athens parallel?
1062. Write an equation to solve the problem: “A field with an area of 2.4 hectares was divided into two sections. Find square each site, if it is known that one of the sites:
a) 0.8 hectares more than another;
b) 0.2 hectares less than another;
c) 3 times more than another;
d) 1.5 times less than another;
e) constitutes another;
e) is 0.2 of the other;
g) constitutes 60% of the other;
h) is 140% of the other.”
1063. Solve the problem:
1) On the first day, the travelers traveled 240 km, on the second day 140 km, on the third day they traveled 3 times more than on the second, and on the fourth day they rested. How many kilometers did they travel on the fifth day, if over 5 days they drove an average of 230 km per day?
2) Father’s monthly income is 280 rubles. My daughter's scholarship is 4 times less. How much does a mother earn per month if there are 4 people in the family? younger son- a schoolboy and each person receives an average of 135 rubles?
1064. Follow these steps:
1) (2,35 + 4,65) 5,3:(40-2,9);
2) (7,63-5,13) 0,4:(3,17 + 6,83).
1066. Present each of the numbers as a sum of two equal terms:
1067. Find the value of a + b if:
a) a= -1.6, b = 3.2; b) a=- 2.6, b = 1.9; V) 
1068. There were 8 apartments on one floor of a residential building. 2 apartments had a living area of 22.8 m2, 3 apartments - 16.2 m2, 2 apartments - 34 m2. What living area did the eighth apartment have if on this floor on average each apartment had 24.7 m2 of living space?
1069. The freight train consisted of 42 cars. There were 1.2 times more covered cars than platforms, and the number of tanks was equal to the number of platforms. How many cars of each type were on the train?
1070. Find the meaning of the expression 
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
Mathematics planning, textbooks and books online, courses and tasks in mathematics for grade 6 download
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year guidelines discussion programs Integrated LessonsFractions are ordinary numbers and can also be added and subtracted. But due to the fact that they contain a denominator, more complex rules than for integers.
Let's consider the simplest case, when there are two fractions with same denominators. Then:
To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.
Within each expression, the denominators of the fractions are equal. By definition of adding and subtracting fractions we get:
As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.
But even in such simple actions, people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Get rid of bad habit Adding the denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Many people also make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus and where to put a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Let's look at all this with specific examples:
Task. Find the meaning of the expression:
In the first case, everything is simple, but in the second, let’s add minuses to the numerators of the fractions:
What to do if the denominators are different
Directly adding fractions with different denominators it is forbidden. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are discussed in the lesson “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:
Task. Find the meaning of the expression:
In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.
What to do if a fraction has an integer part
I can please you: having different denominators in fractions is not the most great evil. Much more errors occur when the whole part is highlighted in the addend fractions.
Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use the simple diagram below:
- Convert all fractions containing an integer part to improper ones. We obtain normal terms (even with different denominators), which are calculated according to the rules discussed above;
- Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
- If this is all that was required in the problem, we perform the inverse transformation, i.e. We get rid of an improper fraction by highlighting the whole part.
Rules for transition to improper fractions and highlighting an entire part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:
Task. Find the meaning of the expression:
Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:
To simplify the calculations, I have skipped some obvious steps in the last examples.
A small note about the last two examples, where fractions with the integer part highlighted are subtracted. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.
Re-read this sentence again, look at the examples - and think about it. This is where beginners make a huge number of mistakes. They love to give such tasks to tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.
Summary: general calculation scheme
In conclusion I will give general algorithm, which will help you find the sum or difference of two or more fractions:
- If one or more fractions have an integer part, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
- If possible, shorten the result. If the fraction is incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, immediately before writing down the answer.
ADDING AND SUBTRACTING
numbers with different signs
To ensure that the student, in less time than before, masters a large amount of knowledge, thorough and effective - this is one of the main tasks of modern pedagogy. In this regard, there is a need to begin studying new things by repeating old, already studied, known material on a given topic. In order for the repetition to proceed quickly and in order to have the most obvious connection between the new and the old, it is necessary to organize the recording of the studied material in a special way when explaining.
As an example, I will tell you how I teach students to add and subtract numbers with different signs using a coordinate line. Before studying the topic directly and during lessons in the 5th and 6th grades, I pay a lot of attention to the structure of the coordinate line. Before starting to study the topic “Addition and subtraction of numbers with different signs,” it is necessary that each student firmly knows and is able to answer next questions:
1) How is the coordinate line constructed?
2) How are the numbers located on it?
3) What is the distance from the number 0 to any number?
Students should understand that moving along a straight line to the right leads to an increase in the number, i.e. the addition action is performed, and to the left - to its decrease, i.e. the action of subtracting numbers is performed. So that working with the coordinate line does not cause boredom, there are many games non-standard tasks. For example, this one.
A straight line has been drawn along the highway. Length of one unit segment equals 2 m. everyone moves only along a straight line. On number 3 are Gena and Cheburashka. They went to the same place different sides and stopped at the same time. Gena walked twice as far as Cheburashka and ended up on number 11. What number did Cheburashka end up on? How many meters did Cheburashka walk? Which of them walked slower and by how much?(Non-standard mathematics at school. - M., Laida, 1993, No. 62).
When I am firmly convinced that all students can cope with movements along a straight line, and this is very important, I move directly to teaching adding and subtracting numbers at the same time.
Each student is given a reference note. By analyzing the provisions of the notes and relying on existing geometric visual pictures of the coordinate line, students gain new knowledge. (The outline is shown in the figure). Studying a topic begins by writing down in a notebook the questions that will be discussed.
1 . How to perform addition using a coordinate line? How to find an unknown term? Let's look at the relevant part of the outline??. Let us remember that a add b- it means to increase a on b and movement along the coordinate line occurs to the right. We recall how the components of addition and the laws of addition are named and calculated, as well as the properties of zero during addition. Are these parts?? And?? notes. Therefore, the following questions written in the notebook are:
1). Addition is movement to the right.
SL. + SL. = C; SL. = C - SL.
2). Addition laws:
1) displacement law: a+ b= b+ a;
2) combination law: (a+ b) + c= a+ (b+ c) = (a+ c) + b
3). Properties of zero during addition: a+ 0= a; 0+ a= a; a+ (- a) = 0.
4). Subtraction is a movement to the left.
U. - V. = R.; U. = V. + R.; V. = U. - R.
5). Addition can be replaced by subtraction, and subtraction can be replaced by addition.
4 + 3 = - 1 3 - 4 = -1
4 + 3 = 3 + (- 4) = 3 - 4 = - 1
according to the commutative law of addition
6). This is how the parentheses open:
+ (a+ b+ c) = + a+ b+ c
"gentleman"
- (a + b + c) = - a - b - c
"robber"
2 . Laws of addition.
3 . List the properties of zero during addition.
4 . How to subtract numbers using a coordinate line? Rules for finding unknown subtrahends and minuends.
5 . How do you go from addition to subtraction and from subtraction to addition?
6 . How to open parentheses preceded by: a) plus sign; b) minus sign?
The theoretical material is quite voluminous, but since each part of it is connected and, as it were, “flows” from one another, memorization occurs successfully. Working with notes doesn't end there. Each part of the outline is associated with the text of the textbook, which is read in class. If after this the student believes that the part being analyzed is completely clear to him, then he lightly paints over the text of the summary in the appropriate frame, as if saying: “I understand this.” If there is something unclear, then the frame is not painted over until everything becomes clear. The white part of the notes is the signal “Figure it out!”
The teacher's goal, which should be achieved by the end of the lesson, is this: students, leaving the lesson, should remember that addition is movement along a coordinate line to the right, and subtraction is to the left. All students learned to open brackets. The remaining time of the lesson is devoted to opening the brackets. We open brackets orally and in writing in tasks like:
|
); - 20 + (- 7 + (- 5)). |
||||
Homework assignment. Answer the questions written in the notebook by reading the textbook paragraphs indicated in the notes.
In the next lesson we will practice the algorithm for adding and subtracting numbers. Each student has a card on their desk with instructions:
1) Write down an example.
2) Open the brackets, if any.
3) Draw a coordinate line.
4) Mark the first number on it without scale.
5) If the number is followed by a “+” sign, then move to the right, and if there is a “-” sign, then move to the left by as many unit segments as the second term contains. Draw it diagrammatically and put a sign next to the number you are looking for?
6) Ask the question “Where is zero?”
7) Determine the sign of the number that has a question mark, which is a solution, like this: if? is to the right of 0, then the answer has a + sign, but what if? is to the left of 0, then the answer has a sign - . Write the found sign in the answer after the = sign.
8) Mark three segments on the drawing.
9) Find the length of the segment from zero to sign?
Example 1.- 35 + (- 9) = - 35 - 9 = - 44.
1. I copy the example and open the parentheses.
2. I draw a picture and reason like this:
a) I mark - 35 and move to the left by 9 unit segments; I put a sign next to the desired number?;
b) I ask myself: “Where is zero?” I answer: “Zero is to the right - 35 by 35 unit segments, which means the sign of the answer is -, so? to the left of zero";
c) looking for the distance from 0 to the sign?. To do this, I calculate 35 + 9 = 44 and assign the resulting number in response to the - sign.
Example 2.- 35 + 9.
Example 3. 9 - 35.
We solve these examples using similar reasoning to example 1. There cannot be other cases of arrangement of numbers, and each picture corresponds to one of the rules given in the textbook and requiring memorization. It has been verified (and repeatedly) that this method of addition is more rational. In addition, it allows you to add numbers even when the student thinks that he does not remember a single rule. This method It also works when working with fractions, you just need to bring them to a common denominator, and then draw a picture. For example,
Everyone uses the “instruction” card as long as there is a need for it.
Such work replaces the tedious and monotonous action of counting according to the rules of a living and actively working thought. There are many advantages: no need to cram and feverishly figure out which rule to apply; The structure of the coordinate line is easy to remember, and this is both in algebra and in geometry when calculating the value of a segment when a point on a line lies between two other points. This technique is effective both in classes with in-depth study of mathematics and in classes age norm and even in correction classes.
This article is devoted to numbers with different signs. We will break down the material and try to subtract between these numbers. In this paragraph we will get acquainted with the basic concepts and rules that will be useful when solving exercises and problems. The article also presents detailed examples that will help you better understand the material.
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How to do subtraction correctly
To better understand the process of subtraction, we need to start with some basic definitions.
Definition 1
If you subtract the number b from the number a, then this can be transformed as the addition of the number a and - b, where b and − b are numbers with opposite signs.
If we express this rule letters, then it looks like this: a − b = a + (− b), where a and b are any real numbers.
This rule for subtracting numbers with different signs works for real, rational and integer numbers. It can be proven based on the properties of operations with real numbers. Thanks to them, we can represent numbers as several equalities (a + (− b)) + b = a + ((− b) + b) = a + 0 = a. Since addition and subtraction are closely related, the expression a − b = a + (− b) will also be equal. This means that the subtraction rule in question is also true.
This rule, which is used to subtract numbers with different signs, allows you to work with both positive and negative numbers. You can also perform the process of subtracting from a negative number from a positive one, which turns into addition.
In order to consolidate the information received, we will consider typical examples and in practice consider the subtraction rule for numbers with different signs.
Examples of subtraction exercises
Let's reinforce the material by looking at typical examples.
Example 1
You need to subtract 4 from − 16.
In order to perform a subtraction, you should take the number opposite to the one you are subtracting 4, which is − 4. According to the subtraction rule discussed above (− 16) − 4 = (− 16) + (− 4) . Next we must add the resulting negative numbers. We get: (− 16) + (− 4) = − (16 + 4) = − 20. (− 16) − 4 = − 20 .
In order to subtract fractions, you need to represent numbers in ordinary or decimals. It depends on what type of numbers it will be more convenient to carry out calculations with.
Example 2
It is necessary to subtract − 0, 7 from 3 7.
We resort to the rule of subtracting numbers. Replace subtraction with addition: 3 7 - (- 0, 7) = 3 7 + 0, 7.
We add the fractions and get the answer in the form fractional number. 3 7 - (- 0 , 7) = 1 9 70 .
When a number is represented as square root, logarithm, basic and trigonometric functions, then often the result of subtraction can be written in the form numerical expression. To clarify this rule, consider the following example.
Example 3
It is necessary to subtract the number 5 from the number - 2.
Let's use the subtraction rule described above. Let's take the opposite number to subtract 5 - this is − 5. According to working with numbers with different signs - 2 - 5 = - 2 + (- 5) .
Now let's do the addition: we get - 2 + (- 5) = 2 + 5.
The resulting expression is the result of subtracting the original numbers with different signs: - 2 + 5.
The value of the resulting expression can be calculated as accurately as possible only if necessary. For detailed information You can explore other sections related to this topic.
If you notice an error in the text, please highlight it and press Ctrl+Enter
In this lesson we will learn adding and subtracting integers, as well as rules for their addition and subtraction.
Recall that integers are all positive and negative numbers, as well as the number 0. For example, the following numbers are integers:
−3, −2, −1, 0, 1, 2, 3
Positive numbers are easy, and. Unfortunately, the same cannot be said about negative numbers, which confuse many beginners with their minuses in front of each number. As practice shows, mistakes made due to negative numbers frustrate students the most.
Lesson contentExamples of adding and subtracting integers
The first thing you should learn is to add and subtract integers using a coordinate line. It is not at all necessary to draw a coordinate line. It is enough to imagine it in your thoughts and see where the negative numbers are located and where the positive ones are.
Let's consider the simplest expression: 1 + 3. The value of this expression is 4:
This example can be understood using a coordinate line. To do this, from the point where the number 1 is located, you need to move three steps to the right. As a result, we will find ourselves at the point where the number 4 is located. In the figure you can see how this happens:
The plus sign in the expression 1 + 3 tells us that we should move to the right in the direction of increasing numbers.
Example 2. Let's find the value of the expression 1 − 3.
The value of this expression is −2
This example can again be understood using a coordinate line. To do this, from the point where the number 1 is located, you need to move to the left three steps. As a result, we will find ourselves at the point where the negative number −2 is located. In the picture you can see how this happens:
The minus sign in the expression 1 − 3 tells us that we should move to the left in the direction of decreasing numbers.
In general, you need to remember that if addition is carried out, then you need to move to the right in the direction of increase. If subtraction is carried out, then you need to move to the left in the direction of decrease.
Example 3. Find the value of the expression −2 + 4
The value of this expression is 2
This example can again be understood using a coordinate line. To do this, from the point where the negative number −2 is located, you need to move four steps to the right. As a result, we will find ourselves at the point where the positive number 2 is located.
It can be seen that we have moved from the point where the negative number −2 is located to right side four steps, and ended up at the point where the positive number 2 is located.
The plus sign in the expression −2 + 4 tells us that we should move to the right in the direction of increasing numbers.
Example 4. Find the value of the expression −1 − 3
The value of this expression is −4
This example can again be solved using a coordinate line. To do this, from the point where the negative number −1 is located, you need to move to the left three steps. As a result, we will find ourselves at the point where the negative number −4 is located
It can be seen that we have moved from the point where the negative number −1 is located to left side three steps, and ended up at the point where the negative number −4 is located.
The minus sign in the expression −1 − 3 tells us that we should move to the left in the direction of decreasing numbers.
Example 5. Find the value of the expression −2 + 2
The value of this expression is 0
This example can be solved using a coordinate line. To do this, from the point where the negative number −2 is located, you need to move two steps to the right. As a result, we will find ourselves at the point where the number 0 is located
It can be seen that we have moved from the point where the negative number −2 is located to the right side by two steps and ended up at the point where the number 0 is located.
The plus sign in the expression −2 + 2 tells us that we should move to the right in the direction of increasing numbers.
Rules for adding and subtracting integers
To add or subtract integers, it is not at all necessary to imagine a coordinate line every time, much less draw it. It is more convenient to use ready-made rules.
When applying the rules, you need to pay attention to the sign of the operation and the signs of the numbers that need to be added or subtracted. This will determine which rule to apply.
Example 1. Find the value of the expression −2 + 5
Here a positive number is added to a negative number. In other words, numbers with different signs are added. −2 is a negative number, and 5 is a positive number. For such cases, the following rule applies:
To add numbers with different signs, you need to subtract the smaller module from the larger module, and before the resulting answer put the sign of the number whose module is larger.
So, let's see which module is bigger:
The modulus of the number 5 is greater than the modulus of the number −2. The rule requires subtracting the smaller one from the larger module. Therefore, we must subtract 2 from 5, and before the resulting answer put the sign of the number whose modulus is greater.
The number 5 has a larger modulus, so the sign of this number will be in the answer. That is, the answer will be positive:
−2 + 5 = 5 − 2 = 3
Usually written shorter: −2 + 5 = 3
Example 2. Find the value of the expression 3 + (−2)
Here, as in the previous example, numbers with different signs are added. 3 is a positive number, and −2 is a negative number. Note that −2 is enclosed in parentheses to make the expression clearer. This expression is much easier to understand than the expression 3+−2.
So, let's apply the rule for adding numbers with different signs. As in the previous example, we subtract the smaller module from the larger module and before the answer we put the sign of the number whose module is greater:
3 + (−2) = |3| − |−2| = 3 − 2 = 1
The modulus of the number 3 is greater than the modulus of the number −2, so we subtracted 2 from 3, and before the resulting answer we put the sign of the number whose modulus is greater. The number 3 has a larger modulus, which is why the sign of this number is included in the answer. That is, the answer is positive.
Usually written shorter 3 + (−2) = 1
Example 3. Find the value of the expression 3 − 7
In this expression, a larger number is subtracted from a smaller number. In such a case the following rule applies:
To subtract a larger number from a smaller number, you need to more subtract the lesser and put a minus in front of the resulting answer.
3 − 7 = 7 − 3 = −4
There is a slight catch to this expression. Let us remember that the equal sign (=) is placed between quantities and expressions when they are equal to each other.
The value of the expression 3 − 7, as we learned, is −4. This means that any transformations that we will perform in this expression must be equal to −4
But we see that at the second stage there is an expression 7 − 3, which is not equal to −4.
To correct this situation, you need to put the expression 7 − 3 in brackets and put a minus in front of this bracket:
3 − 7 = − (7 − 3) = − (4) = −4
In this case, equality will be observed at each stage:
After the expression has been calculated, the parentheses can be removed, which is what we did.
So to be more precise the solution should look like this:
3 − 7 = − (7 − 3) = − (4) = − 4
This rule can be written using variables. It will look like this:
a − b = − (b − a)
A large number of parentheses and operation signs can complicate the solution of a seemingly simple problem, so it is more advisable to learn how to write such examples briefly, for example 3 − 7 = − 4.
In fact, adding and subtracting integers comes down to nothing more than addition. This means that if you need to subtract numbers, this operation can be replaced by addition.
So, let's get acquainted with the new rule:
Subtracting one number from another means adding to the minuend a number that is opposite to the one being subtracted.
For example, consider the simplest expression 5 − 3. On initial stages studying mathematics, we put an equal sign and wrote down the answer:
But now we are progressing in our study, so we need to adapt to the new rules. The new rule says that subtracting one number from another means adding to the minuend the same number as the subtrahend.
Let's try to understand this rule using the example of expression 5 − 3. The minuend in this expression is 5, and the subtrahend is 3. The rule says that in order to subtract 3 from 5, you need to add to 5 a number that is the opposite of 3. The opposite of the number 3 is −3. Let's write a new expression:
And we already know how to find meanings for such expressions. This is the addition of numbers with different signs, which we looked at earlier. To add numbers with different signs, we subtract the smaller module from the larger module, and before the resulting answer we put the sign of the number whose module is greater:
5 + (−3) = |5| − |−3| = 5 − 3 = 2
The modulus of the number 5 is greater than the modulus of the number −3. Therefore, we subtracted 3 from 5 and got 2. The number 5 has a larger modulus, so we put the sign of this number in the answer. That is, the answer is positive.
At first, not everyone is able to quickly replace subtraction with addition. This is because positive numbers are written without the plus sign.
For example, in the expression 3 − 1, the minus sign indicating subtraction is an operation sign and does not refer to one. One in this case is a positive number, and it has its own plus sign, but we don’t see it, since a plus is not written before positive numbers.
Therefore, for clarity, this expression can be written as follows:
(+3) − (+1)
For convenience, numbers with their own signs are placed in brackets. In this case, replacing subtraction with addition is much easier.
In the expression (+3) − (+1), the number being subtracted is (+1), and the opposite number is (−1).
Let's replace subtraction with addition and instead of the subtrahend (+1) we write the opposite number (−1)
(+3) − (+1) = (+3) + (−1)
Further calculations will not be difficult.
(+3) − (+1) = (+3) + (−1) = |3| − |−1| = 3 − 1 = 2
At first glance, it might seem like what’s the point in these extra movements if you can use the good old method to put an equal sign and immediately write down the answer 2. In fact, this rule will help us out more than once.
Let's solve the previous example 3 − 7 using the subtraction rule. First, let's bring the expression to a clear form, assigning each number its own signs.
Three has a plus sign because it is a positive number. The minus sign indicating subtraction does not apply to seven. Seven has a plus sign because it is a positive number:
Let's replace subtraction with addition:
(+3) − (+7) = (+3) + (−7)
Further calculation is not difficult:
(+3) − (−7) = (+3) + (-7)
= −(|−7| − |+3|) = −(7 − 3) = −(4) = −4
Example 7. Find the value of the expression −4 − 5
Again we have a subtraction operation. This operation must be replaced by addition. To the minuend (−4) we add the number opposite to the subtrahend (+5). Opposite number for the subtrahend (+5) it is the number (−5).
(−4) − (+5) = (−4) + (−5)
We have come to a situation where we need to add negative numbers. For such cases, the following rule applies:
To add negative numbers, you need to add their modules and put a minus in front of the resulting answer.
So, let’s add up the modules of numbers, as the rule requires us to do, and put a minus in front of the resulting answer:
(−4) − (+5) = (−4) + (−5) = |−4| + |−5| = 4 + 5 = −9
The entry with modules must be enclosed in brackets and a minus sign must be placed before these brackets. This way we will provide a minus that should appear before the answer:
(−4) − (+5) = (−4) + (−5) = −(|−4| + |−5|) = −(4 + 5) = −(9) = −9
The solution for this example can be written briefly:
−4 − 5 = −(4 + 5) = −9
or even shorter:
−4 − 5 = −9
Example 8. Find the value of the expression −3 − 5 − 7 − 9
Let's bring the expression to a clear form. Here, all numbers except −3 are positive, so they will have plus signs:
(−3) − (+5) − (+7) − (+9)
Let's replace subtractions with additions. All minuses, except the minus in front of the three, will change to pluses, and all positive numbers will change to the opposite:
(−3) − (+5) − (+7) − (+9) = (−3) + (−5) + (−7) + (−9)
Now let's apply the rule for adding negative numbers. To add negative numbers, you need to add their modules and put a minus in front of the resulting answer:
(−3) − (+5) − (+7) − (+9) = (−3) + (−5) + (−7) + (−9) =
= −(|−3| + |−5| + |−7| + |−9|) = −(3 + 5 + 7 + 9) = −(24) = −24
The solution to this example can be written briefly:
−3 − 5 − 7 − 9 = −(3 + 5 + 7 + 9) = −24
or even shorter:
−3 − 5 − 7 − 9 = −24
Example 9. Find the value of the expression −10 + 6 − 15 + 11 − 7
Let's bring the expression to a clear form:
(−10) + (+6) − (+15) + (+11) − (+7)
There are two operations here: addition and subtraction. We leave addition unchanged, and replace subtraction with addition:
(−10) + (+6) − (+15) + (+11) − (+7) = (−10) + (+6) + (−15) + (+11) + (−7)
Observing, we will perform each action in turn, based on the previously learned rules. Entries with modules can be skipped:
First action:
(−10) + (+6) = − (10 − 6) = − (4) = − 4
Second action:
(−4) + (−15) = − (4 + 15) = − (19) = − 19
Third action:
(−19) + (+11) = − (19 − 11) = − (8) = −8
Fourth action:
(−8) + (−7) = − (8 + 7) = − (15) = − 15
Thus, the value of the expression −10 + 6 − 15 + 11 − 7 is −15
Note. It is not at all necessary to bring the expression into a understandable form by enclosing numbers in parentheses. When does addiction occur? negative numbers, you can skip this step as it is time consuming and can be confusing.
So, to add and subtract integers, you need to remember the following rules:
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