Posts tagged "addition of negative numbers". Multiplying and dividing negative numbers
In this article we will talk about adding negative numbers. First we give the rule for adding negative numbers and prove it. After this, we will look at typical examples of adding negative numbers.
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Rule for adding negative numbers
Before formulating the rule for adding negative numbers, let us turn to the material in 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 allows us to realize rule for adding negative numbers. To add two negative numbers, need to:
- fold 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 letter form: (−a)+(−b)=−(a+b).
It is clear that the stated 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 modules.
The rule for adding negative numbers can be proven based on properties of operations with real numbers(or the same properties of operations with rational or integer numbers). To do this, it is enough to show that the difference between the left and right sides 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). Due to 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 with zero. This proves the equality (−a)+(−b)=−(a+b) , and hence the rule for adding negative numbers.
All that remains is 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 sort it out examples of adding negative numbers. Let's start with the simplest case - the addition of negative integers; we will carry out the addition according to the rule discussed in the previous paragraph.
Example.
Add the negative numbers −304 and −18007.
Solution.
Let's follow all the steps of the rule for adding negative numbers.
First we find the modules of the numbers being added: 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.
Let's write the entire solution in a short form: (−304)+(−18,007)= −(304+18,007)=−18,311.
Answer:
−18 311 .
Addition of negative rational numbers depending on the numbers themselves, it 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 for adding negative numbers, you first need to calculate the sum of the modules. The modules of the negative numbers being added are equal to 2/5 and 4, (12) respectively. The addition of the resulting numbers can be reduced to addition ordinary fractions. To do this, we convert the periodic decimal fraction into an ordinary fraction: . Thus, 2/5+4,(12)=2/5+136/33. Now let's do it
Now let's deal with multiplication and division.
Let's say we need to multiply +3 by -4. How to do this?
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 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 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 the same 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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Rule for adding negative numbers
If you remember the math lesson and the topic “Adding and subtracting numbers with different signs”, then to add two negative numbers you need:
- perform the addition of their modules;
- add a “–” sign to the received amount.
According to the addition rule, we can write:
$(−a)+(−b)=−(a+b)$.
The rule for adding negative numbers applies to negative integers, rational numbers, and real numbers.
Example 1
Add the negative numbers $−185$ and $−23\789.$
Solution.
Let's use the rule for adding negative numbers.
Let's find the modules of these numbers:
$|-23 \ 789|=23 \ 789$.
Let's add the resulting numbers:
$185+23 \ 789=23 \ 974$.
Let's 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 decimals.
Example 2
Add the negative numbers $-\frac(1)(4)$ and $−7.15$.
Solution.
According to the rule for 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 resulting value and get $–7.4$.
Brief summary of the solution:
$(-\frac(1)(4))+(−7.15)=−(\frac(1)(4)+7.15)=–(0.25+7.15)=−7, $4.
To add a positive and negative number you need:
- calculate the modules of numbers;
- if they are equal, then the original numbers are opposite and their sum is 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 module;
- Before the resulting value, put the sign of the number whose modulus is greater.
compare the resulting numbers:
Adding numbers with opposite signs comes down to subtracting from more positive number smaller negative number.
The rule for adding numbers with opposite signs is true for integers, rationals and real numbers.
Example 3
Add the numbers $4$ and $−8$.
Solution.
You need to add numbers with opposite signs. Let's use the corresponding 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.
Let’s put the sign $“–”$, which we remembered, in front of the resulting number, and we get $−4.$
Brief summary of the solution:
$4+(–8) = –(8–4) = –4$.
Answer: $4+(−8)=−4$.
To add rational numbers with opposite signs, it is convenient to represent them in the form of ordinary or decimal fractions.
Subtracting numbers with different and negative signs
Rule for subtracting negative numbers:
To subtract a negative number $b$ from a number $a$, it is necessary to add the number $−b$ to the minuend $a$, which is the opposite of the subtrahend $b$.
According to the subtraction rule, we can write:
$a−b=a+(−b)$.
This rule is valid for integers, rationals and real numbers. The rule can be used to subtract a negative number from a positive number, from a negative number, and from zero.
Example 4
Subtract the negative number $−5$ from the negative number $−28$.
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 numbers to the form of ordinary fractions, mixed numbers or decimals.
Adding and subtracting 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 of $7$ is $–7$.
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.
Within the framework of this material we will touch upon such important topic, like adding negative numbers. In the first paragraph we will tell you the basic rule for this action, and in the second we will analyze specific examples solving similar problems.
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Basic rule for adding natural numbers
Before we derive the rule, let us remember what we generally know about positive and negative numbers. Previously, we agreed that negative numbers should be perceived as debt, loss. The modulus of a negative number expresses the exact size of this loss. Then the addition of negative numbers can be represented as the addition of two losses.
Using this reasoning, we formulate the basic rule for adding negative numbers.
Definition 1
In order to complete adding negative numbers, you need to add up the values of their modules and put a minus in front of the result. In literal form, the formula looks like (− a) + (− b) = − (a + b) .
Based on this rule, we can conclude that adding negative numbers is similar to adding positive ones, only in the end we must get a negative number, because we must put a minus sign in front of the sum of the modules.
What evidence can be given for this rule? To do this, we need to remember the basic properties of operations with real numbers (or with integers, or with rational numbers - they are the same for all these types of numbers). To prove it, we just need to demonstrate that the difference between the left and right sides of the equality (− a) + (− b) = − (a + b) will be equal to 0.
Subtracting one number from another is the same as adding the same opposite number to it. Therefore, (− a) + (− b) − (− (a + b)) = (− a) + (− b) + (a + b) . Recall that numerical expressions with addition have two main properties - associative and commutative. Then we can conclude that (− a) + (− b) + (a + b) = (− a + a) + (− b + b) . Because, adding opposite numbers, we always get 0, then (− a + a) + (− b + b) = 0 + 0, and 0 + 0 = 0. Our equality can be considered proven, which means we have also proven the rule for adding negative numbers.
In the second paragraph, we will take specific problems where we need to add negative numbers, and we will try to apply the learned rule to them.
Example 1
Find the sum of two negative numbers - 304 and - 18,007.
Solution
Let's perform the steps step by step. First we need to find the modules of the numbers being added: - 304 = 304, - 180007 = 180007. Next we need to perform the addition action, for which we use the column counting method:
All we have left is to put a minus in front of the result and get - 18,311.
Answer: - - 18 311 .
What numbers we have depends on what we can reduce the action of addition to: finding the sum of natural numbers, adding ordinary or decimal fractions. Let's analyze the problem with these numbers.
Example N
Find the sum of two negative numbers - 2 5 and − 4, (12).
Solution
We find the modules of the required numbers and get 2 5 and 4, (12). We got two different fractions. Let us reduce the problem to the addition of two ordinary fractions, for which we represent the periodic fraction in the form of an ordinary one:
4 , (12) = 4 + (0 , 12 + 0 , 0012 + . . .) = 4 + 0 , 12 1 - 0 , 01 = 4 + 0 , 12 0 , 99 = 4 + 12 99 = 4 + 4 33 = 136 33
As a result, we received a fraction that will be easy to add with the first original term (if you have forgotten how to correctly add fractions with different denominators, repeat the relevant material).
2 5 + 136 33 = 2 33 5 33 + 136 5 33 5 = 66 165 + 680 165 = 764 165 = 4 86 105
In the end we got mixed number, in front of which we only have to put a minus. This completes the calculations.
Answer: - 4 86 105 .
Real negative numbers add up in a similar way. The result of such an action is usually written down as a numerical expression. Its value may not be calculated or limited to approximate calculations. So, for example, if we need to find the sum - 3 + (− 5), then we write the answer as - 3 − 5. We have devoted a separate material to the addition of real numbers, in which you can find other examples.
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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 that shows the temperature of the body, air, soil or water. IN winter time, when it is very cold outside, the temperature can be negative (or, as people say, “minus”).
For example, −10 degrees cold:
The ordinary numbers that we looked at 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 we should keep 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. Looks like this:
The numbers shown here are from −5 to 5. In fact, the coordinate line is infinite. The figure shows only a small fragment of it.
Numbers on the coordinate line are marked as dots. Bold in the picture black dot is the starting point. The countdown starts from zero. Negative numbers are marked to the left of the origin, and positive numbers to the right.
The coordinate line continues indefinitely on both sides. Infinity in mathematics is symbolized by the symbol ∞. The negative direction will be indicated by the symbol −∞, and the positive direction by the 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 shows the position of a point on this line. Simply put, a coordinate is the very 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 with coordinate 4"
Here B is the name of the point, 4 is the coordinate of the point B.
Example 3. Point M(−3) reads 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 point M .
Points can be designated by any letters. But it is generally accepted to denote them in capital Latin letters. Moreover, the beginning of the report, which is otherwise called origin usually means big Latin letter O
It is easy to notice that negative numbers lie to the left relative to the origin, and positive numbers lie to the right.
There are phrases such as “the further to the left, the less” And "the further to the right, the more". You probably already guessed what we are talking about. With each step to the left, the number will decrease downward. And with each step to the right the number will increase. An arrow pointing to the right indicates a positive reference direction.
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 five strikes the eye first of all as a number greater than three.
This is due to the fact that −5 is a negative number, 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 further to the left, the less” . And the rule says that any negative number is less than any positive number. It follows that
−5 < 3
"Minus five is less than three"
Rule 2. Of two negative numbers, the one that is located to the left on the coordinate line is smaller.
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 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 further 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 smaller. 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 more 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 further to the right, the more" . And the rule says that zero is greater than any negative number. It follows that
Zero is greater than minus three
Rule 4. Zero is less than any positive number.
For example, let's compare 0 and 4. Zero less, than 4. This is in principle clear and true. But we will try to see this 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 further to the left, the less” . And the rule says that zero is less than any positive number. It follows that
Zero is less than four
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