In this article we will talk about addition negative numbers . First, we give a rule for adding negative numbers and prove it. After that, we will analyze characteristic examples addition of negative numbers.
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Negative addition rule
Before giving the formulation of the rule for adding negative numbers, let's turn to the material of the article positive and negative numbers. There we mentioned that negative numbers can be perceived as debt, and in this case determines the amount of this debt. Therefore, the addition of two negative numbers is the addition of two debts.
This conclusion makes it possible to understand negative addition rule. To add two negative numbers, you need:
- stack their modules;
- put a minus sign in front of the received amount.
Let's write down the rule for adding negative numbers −a and −b in literal form: (−a)+(−b)=−(a+b).
It is clear that the voiced rule reduces the addition of negative numbers to the addition of positive numbers (the modulus of a negative number is a positive number). It is also clear that the result of adding two negative numbers is a negative number, as evidenced by the minus sign that is placed in front of the sum of the moduli.
The rule for adding negative numbers can be proved based on properties of actions with real numbers(or the same properties of operations with rational or integer numbers). To do this, it suffices to show that the difference between the left and right parts of the equality (−a)+(−b)=−(a+b) is equal to zero.
Since subtracting a number is the same as adding the opposite number (see the rule for subtracting integers), then (−a)+(−b)−(−(a+b))=(−a)+(−b)+(a+b). By virtue of the commutative and associative properties of addition, we have (−a)+(−b)+(a+b)=(−a+a)+(−b+b). Since the sum of opposite numbers is equal to zero, then (−a+a)+(−b+b)=0+0 , and 0+0=0 due to the property of adding a number to zero. This proves the equality (−a)+(−b)=−(a+b) , and hence the rule for adding negative numbers.
It remains only to learn how to apply the rule of adding negative numbers in practice, which we will do in the next paragraph.
Examples of Adding Negative Numbers
Let's analyze examples of adding negative numbers. Let's start with the simplest case - the addition of negative integers, the addition will be carried out according to the rule considered in the previous paragraph.
Example.
Add negative numbers -304 and -18007 .
Solution.
Let's follow all the steps of the rule of adding negative numbers.
First, we find the modules of the added numbers: and 
. Now you need to add the resulting numbers, here it is convenient to perform column addition: 
Now we put a minus sign in front of the resulting number, as a result we have −18 311 .
We write the whole solution in short form: (−304)+(−18 007)= −(304+18 007)=−18 311 .
Answer:
−18 311 .
Addition of negative rational numbers depending on the numbers themselves, can be reduced either to the addition of natural numbers, or to the addition of ordinary fractions, or to the addition of decimal fractions.
Example.
Add a negative number and a negative number −4,(12) .
Solution.
According to the rule of adding negative numbers, you first need to calculate the sum of modules. The modules of the added negative numbers are 2/5 and 4,(12) respectively. The addition of the obtained numbers can be reduced to the addition ordinary fractions. To do this, we translate the periodic decimal fraction into an ordinary fraction:. So 2/5+4,(12)=2/5+136/33 . Now let's execute
Negative numbers are numbers with a minus sign (-), for example -1, -2, -3. Reads like: minus one, minus two, minus three.
Application example negative numbers is a thermometer showing the temperature of the body, air, soil or water. AT winter time when it is very cold outside, the temperature is negative (or, as the people say, "minus").
For example, -10 degrees cold:
The usual numbers that we considered earlier, such as 1, 2, 3, are called positive. Positive numbers are numbers with a plus sign (+).
When writing positive numbers, the + sign is not written down, which is why we see the numbers 1, 2, 3 that are familiar to us. But it should be borne in mind that these positive numbers look like this: +1, +2, +3.
Lesson contentThis is a straight line on which all numbers are located: both negative and positive. As follows:
Shown here are numbers from -5 to 5. In fact, the coordinate line is infinite. The figure shows only a small fragment of it.
The numbers on the coordinate line are marked as dots. Oily in the picture black dot is the starting point. The countdown starts from zero. To the left of the reference point, negative numbers are marked, and to the right, positive ones.
The coordinate line continues indefinitely on both sides. Infinity in mathematics is denoted by the symbol ∞. The negative direction will be denoted by the symbol −∞, and positive symbol+∞. Then we can say that all numbers from minus infinity to plus infinity are located on the coordinate line:
Each point on the coordinate line has its own name and coordinate. Name is any Latin letter. Coordinate is a number that indicates the position of a point on this line. Simply put, the coordinate is the same number that we want to mark on the coordinate line.
For example, point A(2) reads as "point A with coordinate 2" and will be denoted on the coordinate line as follows:
Here A is the name of the point, 2 is the coordinate of the point A.
Example 2 Point B(4) reads as "point B at coordinate 4"
Here B is the name of the point, 4 is the coordinate of the point b.
Example 3 The point M(−3) is read as "point M with coordinate minus three" and will be denoted on the coordinate line as follows:
Here M is the name of the point, −3 is the coordinate of the point M .
Points can be denoted by any letters. But it is generally accepted to designate them with capital Latin letters. Moreover, the beginning of the report, which is otherwise called origin usually denoted by a capital letter O
It is easy to see that negative numbers lie to the left of the origin, and positive numbers to the right.
There are phrases like "the more to the left, the less" and "the more to the right, the more". You probably already guessed what we are talking about. With each step to the left, the number will decrease downwards. And with each step to the right, the number will increase. The arrow pointing to the right indicates the positive direction of counting.
Comparing negative and positive numbers
Rule 1 Any negative number is less than any positive number.
For example, let's compare two numbers: −5 and 3. Minus five less than three, despite the fact that the five catches the eye in the first place, as a number greater than three.
This is because −5 is negative and 3 is positive. On the coordinate line, you can see where the numbers −5 and 3 are located
It can be seen that −5 lies to the left, and 3 to the right. And we said that "the more to the left, the less" . And the rule says that any negative number is less than any positive number. Hence it follows that
−5 < 3
"Minus five is less than three"
Rule 2 Of the two negative numbers, the smaller one is the one located to the left on the coordinate line.
For example, let's compare the numbers -4 and -1. minus four less than minus one.
This is again due to the fact that on the coordinate line −4 is located more to the left than −1
It can be seen that -4 lies to the left, and -1 to the right. And we said that "the more to the left, the less" . And the rule says that of two negative numbers, the one that is located to the left on the coordinate line is less. Hence it follows that
Minus four is less than minus one
Rule 3 Zero is greater than any negative number.
For example, let's compare 0 and −3. Zero more than minus three. This is due to the fact that on the coordinate line 0 is located to the right than −3
It can be seen that 0 lies to the right, and −3 to the left. And we said that "the more to the right, the more" . And the rule says that zero is greater than any negative number. Hence it follows that
Zero is greater than minus three
Rule 4 Zero is less than any positive number.
For example, compare 0 and 4. Zero less than 4. In principle, this is clear and true. But we will try to see it with our own eyes, again on the coordinate line:
It can be seen that on the coordinate line 0 is located to the left, and 4 to the right. And we said that "the more to the left, the less" . And the rule says that zero is less than any positive number. Hence it follows that
Zero is less than four
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Addition of negative numbers.
The sum of negative numbers is a negative number. The module of the sum is equal to the sum of the modules of the terms.
Let's see why the sum of negative numbers will also be a negative number. The coordinate line will help us with this, on which we will perform the addition of the numbers -3 and -5. Let's mark a point on the coordinate line corresponding to the number -3.
To the number -3 we need to add the number -5. Where do we go from the point corresponding to the number -3? That's right, to the left! For 5 single segments. We mark the point and write the number corresponding to it. This number is -8.
So, when adding negative numbers using a coordinate line, we are always to the left of the reference point, therefore, it is clear that the result of adding negative numbers is also a negative number.
Note. We added the numbers -3 and -5, i.e. found the value of the expression -3+(-5). Usually, when adding rational numbers, they simply write down these numbers with their signs, as if listing all the numbers that need to be added. Such a notation is called an algebraic sum. Apply (in our example) record: -3-5=-8.
Example. Find the sum of negative numbers: -23-42-54. (Agree that this entry is shorter and more convenient like this: -23+(-42)+(-54))?
We decide according to the rule of adding negative numbers: we add the modules of the terms: 23+42+54=119. The result will be with a minus sign.
They usually write it down like this: -23-42-54 \u003d -119.
Adding numbers with different signs.
The sum of two numbers with different signs has the sign of the addend with a large modulus. To find the modulus of the sum, you need to subtract the smaller modulus from the larger modulus.
Let's perform the addition of numbers with different signs using the coordinate line.
1) -4+6. It is required to add the number -4 to the number 6. We mark the number -4 with a point on the coordinate line. The number 6 is positive, which means that from the point with coordinate -4 we need to go to the right by 6 unit segments. We ended up to the right of the origin (from zero) by 2 unit segments.
The result of the sum of the numbers -4 and 6 is positive number 2:
— 4+6=2. How could you get the number 2? Subtract 4 from 6, i.e. subtract the smaller one from the larger one. The result has the same sign as the term with a large modulus.
2) Let's calculate: -7+3 using the coordinate line. We mark the point corresponding to the number -7. We go to the right by 3 unit segments and get a point with coordinate -4. We were and remained to the left of the origin: the answer is a negative number.
— 7+3=-4. We could get this result as follows: we subtracted the smaller one from the larger module, i.e. 7-3=4. As a result, the sign of the term with a larger module was set: |-7|>|3|.
Examples. Calculate: a) -4+5-9+2-6-3; b) -10-20+15-25.
Negative addition rule
If you recall the math lesson and the topic “Addition and subtraction of numbers with different signs”, then to add two negative numbers you need:
- perform the addition of their modules;
- add the sign "-" to the received amount.
According to the addition rule, we can write:
$(−a)+(−b)=−(a+b)$.
The negative addition rule applies to negative integers, rational numbers, and real numbers.
Example 1
Add negative numbers $−185$ and $−23 \ 789.$
Solution.
Let's use the rule of adding negative numbers.
Let's find the modules of these numbers:
$|-23 \ 789|=23 \ 789$.
Let's add the resulting numbers:
$185+23 \ 789=23 \ 974$.
We put the sign $"–"$ in front of the found number and get $−23 \ 974$.
Brief solution: $(−185)+(−23 \ 789)=−(185+23 \ 789)=−23 \ 974$.
Answer: $−23 \ 974$.
When adding negative rational numbers, they must be converted to the form natural numbers, ordinary or decimal fractions.
Example 2
Add the negative numbers $-\frac(1)(4)$ and $−7.15$.
Solution.
According to the rule of adding negative numbers, you first need to find the sum of the modules:
$|-\frac(1)(4)|=\frac(1)(4)$;
It is convenient to reduce the obtained values to decimal fractions and perform their addition:
$\frac(1)(4)=0.25$;
$0,25+7,15=7,40$.
Let's put the sign $"-"$ in front of the received value and get $-7.4$.
Solution summary:
$(-\frac(1)(4))+(−7.15)=−(\frac(1)(4)+7.15)=–(0.25+7.15)=−7, 4$.
To add positive and negative numbers:
- calculate modules of numbers;
- if they are equal, then the original numbers are opposite and their sum is equal to zero;
- if they are not equal, then you need to remember the sign of the number whose modulus is greater;
subtract the smaller one from the larger one;
- before the received value, put the sign of the number whose modulus is greater.
compare the received numbers:
Adding numbers with opposite signs is reduced to subtracting from a larger positive number a smaller negative number.
The rule of adding numbers with opposite signs is carried out for integer, rational and real numbers.
Example 3
Add the numbers $4$ and $−8$.
Solution.
You need to add numbers with opposite signs. Let's use the appropriate addition rule.
Let's find the modules of these numbers:
The modulus of the number $−8$ is greater than the modulus of the number $4$, i.e. remember the sign $"-"$.
We put the sign $"–"$, which we memorized, in front of the resulting number, and we get $−4.$
Solution summary:
$4+(–8) = –(8–4) = –4$.
Answer: $4+(−8)=−4$.
To add rational numbers with opposite signs, it is convenient to represent them as ordinary or decimal fractions.
Subtraction of numbers with different and negative signs
Rule for subtracting negative numbers:
To subtract a negative number $b$ from the number $a$, it is necessary to add to the minuend $a$ the number $−b$, which is the opposite of the subtracted $b$.
According to the subtraction rule, we can write:
$a−b=a+(−b)$.
This rule is valid for integer, rational and real numbers. The rule can be used when subtracting a negative number from a positive number, from a negative number, and from zero.
Example 4
Subtract from the negative number $−28$ the negative number $−5$.
Solution.
The opposite number for the number $–5$ is the number $5$.
According to the rule for subtracting negative numbers, we get:
$(−28)−(−5)=(−28)+5$.
Let's add numbers with opposite signs:
$(−28)+5=−(28−5)=−23$.
Answer: $(−28)−(−5)=−23$.
When subtracting negative fractional numbers, it is necessary to convert the numbers to the form of ordinary fractions, mixed numbers or decimals.
Addition and subtraction of numbers with different signs
The rule for subtracting numbers with opposite signs is the same as the rule for subtracting negative numbers.
Example 5
Subtract the positive number $7$ from the negative number $−11$.
Solution.
The opposite number for the number $7$ is the number $–7$.
According to the rule for subtracting numbers with opposite signs, we get:
$(−11)−7=(–11)+(−7)$.
Let's add negative numbers:
$(−11)+(–7)=−(11+7)=−18$.
Brief solution: $(−28)−(−5)=(−28)+5=−(28−5)=−23$.
Answer: $(−11)−7=−18$.
When subtracting fractional numbers with different signs, it is necessary to convert the numbers to the form of ordinary or decimal fractions.
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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