Addition of negative numbers, rules, examples. Posts tagged "addition of negative numbers"

Positive and negative numbers
Coordinate line
Let's go straight. Let's mark point 0 (zero) on it and take this point as the starting point.

We indicate with an arrow the direction of movement in a straight line to the right from the origin of coordinates. In this direction from point 0 we will plot positive numbers.

That is, numbers that are already known to us, except zero, are called positive.

Sometimes positive numbers are written with a “+” sign. For example, "+8".

For brevity, the “+” sign before a positive number is usually omitted and instead of “+8” they simply write 8.

Therefore, “+3” and “3” are the same number, only designated differently.

Let's choose some segment whose length we take as one and move it several times to the right from point 0. At the end of the first segment the number 1 is written, at the end of the second - the number 2, etc.

Putting the unit segment to the left from the origin we get negative numbers: -1; -2; etc.

Negative numbers used to denote various quantities, such as: temperature (below zero), flow - that is, negative income, depth - negative height, and others.

As can be seen from the figure, negative numbers are numbers already known to us, only with a minus sign: -8; -5.25, etc.

The number 0 is neither positive nor negative.

The number axis is usually positioned horizontally or vertically.

If the coordinate line is located vertically, then the direction up from the origin is usually considered positive, and the direction down from the origin is negative.

The arrow indicates the positive direction.

The straight line marked:
. origin (point 0);
. unit segment;
. the arrow indicates the positive direction;
called coordinate line or number axis.

Opposite numbers on a coordinate line
Let us mark two points A and B on the coordinate line, which are located at the same distance from point 0 on the right and left, respectively.

In this case, the lengths of the segments OA and OB are the same.

This means that the coordinates of points A and B differ only in sign.

Points A and B are also said to be symmetrical about the origin.
The coordinate of point A is positive “+2”, the coordinate of point B has a minus sign “-2”.
A (+2), B (-2).

Numbers that differ only in sign are called opposite numbers. The corresponding points of the numerical (coordinate) axis are symmetrical relative to the origin.

Every number has only one opposite number. Only the number 0 does not have an opposite, but we can say that it is the opposite of itself.

The notation "-a" means the opposite number of "a". Remember that a letter can hide either a positive number or a negative number.

Example:
-3 is the opposite number of 3.

We write it as an expression:
-3 = -(+3)

Example:
-(-6) is the opposite number to the negative number -6. So -(-6) is a positive number 6.

We write it as an expression:
-(-6) = 6

Addition negative numbers
The addition of positive and negative numbers can be analyzed using the number line.

It is convenient to perform the addition of small modulo numbers on a coordinate line, mentally imagining how the point denoting the number moves along the number axis.

Let's take some number, for example, 3. Let's denote it on the number axis by point A.

Let's add the positive number 2 to the number. This will mean that point A must be moved two unit segments in the positive direction, that is, to the right. As a result, we get point B with coordinate 5.
3 + (+ 2) = 5

In order to add a negative number (- 5) to a positive number, for example, 3, point A must be moved 5 units of length in the negative direction, that is, to the left.

In this case, the coordinate of point B is - 2.

So, the order of addition rational numbers using a number axis would be:
. mark a point A on the coordinate line with a coordinate equal to the first term;
. move it a distance equal to modulus the second term in the direction that corresponds to the sign in front of the second number (plus - move to the right, minus - to the left);
. the point B obtained on the axis will have a coordinate that will be equal to the sum of these numbers.

Example.
- 2 + (- 6) =

Moving from point - 2 to the left (since there is a minus sign in front of 6), we get - 8.
- 2 + (- 6) = - 8

Adding numbers with the same signs
Adding rational numbers can be easier if you use the concept of modulus.

Let's say we need to add numbers that have the same signs.
To do this, we discard the signs of the numbers and take the modules of these numbers. Let's add the modules and put the sign in front of the sum that was common to these numbers.

Example.

An example of adding negative numbers.
(- 3,2) + (- 4,3) = - (3,2 + 4,3) = - 7,5

To add numbers of the same sign, you need to add their modules and put in front of the sum the sign that was before the terms.

Adding numbers with different signs
If the numbers have different signs, then we act somewhat differently than when adding numbers with the same signs.
. We discard the signs in front of the numbers, that is, we take their modules.
. From the larger module we subtract the smaller one.
. Before the difference we put the sign that was in the number with a larger module.

An example of adding a negative and a positive number.
0,3 + (- 0,8) = - (0,8 - 0,3) = - 0,5

An example of adding mixed numbers.

To add numbers of different signs you need:
. subtract the smaller module from the larger module;
. Before the resulting difference, put the sign of the number with the larger modulus.

Subtracting Negative Numbers
As you know, subtraction is the opposite of addition.
If a and b are positive numbers, then subtracting the number b from the number a means finding a number c that, when added to the number b, gives the number a.
a - b = c or c + b = a

The definition of subtraction holds true for all rational numbers. That is subtracting positive and negative numbers can be replaced by addition.

To subtract another from one number, you need to add the opposite number to the one being subtracted.

Or, in another way, we can say that subtracting the number b is the same as addition, but with the opposite number to b.
a - b = a + (- b)

Example.
6 - 8 = 6 + (- 8) = - 2

Example.
0 - 2 = 0 + (- 2) = - 2

It is worth remembering the expressions below.
0 - a = - a
a - 0 = a
a - a = 0

Rules for subtracting negative numbers
As can be seen from the examples above, subtracting a number b is an addition with the opposite number of b.
This rule holds true not only when subtracting a smaller number from a larger number, but also allows you to subtract from a smaller number larger number, that is, you can always find the difference between two numbers.

The difference can be a positive number, a negative number, or a zero number.

Examples of subtracting negative and positive numbers.
. - 3 - (+ 4) = - 3 + (- 4) = - 7
. - 6 - (- 7) = - 6 + (+ 7) = 1
. 5 - (- 3) = 5 + (+ 3) = 8
It is convenient to remember the sign rule, which allows you to reduce the number of parentheses.
The plus sign does not change the sign of the number, so if there is a plus in front of the parenthesis, the sign in the parentheses does not change.
+ (+ a) = + a

+ (- a) = - a

The minus sign in front of the parentheses reverses the sign of the number in the parentheses.
- (+ a) = - a

- (- a) = + a

From the equalities it is clear that if there are identical signs before and inside the brackets, then we get “+”, and if the signs are different, then we get “-”.
(- 6) + (+ 2) - (- 10) - (- 1) + (- 7) = - 6 + 2 + 10 + 1 - 7 = - 13 + 13 = 0

The rule of signs is preserved even if there is not one number in brackets, but algebraic sum numbers.
a - (- b + c) + (d - k + n) = a + b - c + d - k + n

Please note that if there are several numbers in brackets and there is a minus sign in front of the brackets, then the signs in front of all the numbers in these brackets must change.

To remember the rule of signs, you can create a table for determining the signs of a number.
Sign rule for numbers

Or learn a simple rule.

Two negatives make an affirmative,
Plus times minus equals minus.

Multiplying Negative Numbers
Using the concept of the modulus of a number, we formulate the rules for multiplying positive and negative numbers.

Multiplying numbers with the same signs
The first case that you may encounter is the multiplication of numbers with the same signs.
To multiply two numbers with the same signs:
. multiply the modules of numbers;
. put a “+” sign in front of the resulting product (when writing the answer, the “plus” sign before the first number on the left can be omitted).

Examples of multiplying negative and positive numbers.
. (- 3) . (- 6) = + 18 = 18
. 2 . 3 = 6

Multiplying numbers with different signs
The second possible case is the multiplication of numbers with different signs.
To multiply two numbers with different signs, you need to:
. multiply the modules of numbers;
. Place a “-” sign in front of the resulting work.
Examples of multiplying negative and positive numbers.
. (- 0,3) . 0,5 = - 1,5
. 1,2 . (- 7) = - 8,4

Rules for multiplication signs
Remembering the sign rule for multiplication is very simple. This rule coincides with the rule for opening parentheses.

Two negatives make an affirmative,
Plus times minus equals minus.

In “long” examples, in which there is only a multiplication action, the sign of the product can be determined by the number of negative factors.

At even number of negative factors, the result will be positive, and with odd quantity - negative.
Example.
(- 6) . (- 3) . (- 4) . (- 2) . 12 . (- 1) =

There are five negative factors in the example. This means that the sign of the result will be “minus”.
Now let's calculate the product of the moduli, not paying attention to the signs.
6 . 3 . 4 . 2 . 12 . 1 = 1728

The end result of multiplying the original numbers will be:
(- 6) . (- 3) . (- 4) . (- 2) . 12 . (- 1) = - 1728

Multiplying by zero and one
If among the factors there is a number zero or positive one, then the multiplication is performed according to known rules.
. 0 . a = 0
. a. 0 = 0
. a. 1 = a
Examples:
. 0 . (- 3) = 0
. 0,4 . 1 = 0,4
Negative unity (- 1) plays a special role when multiplying rational numbers.

When multiplied by (- 1), the number is reversed.

IN literal expression this property can be written:
a. (- 1) = (- 1) . a = - a

When adding, subtracting and multiplying rational numbers together, the order of operations established for positive numbers and zero is maintained.

An example of multiplying negative and positive numbers.

Dividing negative numbers
It's easy to understand how to divide negative numbers by remembering that division is the inverse of multiplication.

If a and b are positive numbers, then dividing the number a by the number b means finding a number c that, when multiplied by b, gives the number a.

This definition of division applies to any rational numbers as long as the divisors are non-zero.

Therefore, for example, dividing the number (- 15) by the number 5 means finding a number that, when multiplied by the number 5, gives the number (- 15). This number will be (- 3), since
(- 3) . 5 = - 15

Means

(- 15) : 5 = - 3

Examples of dividing rational numbers.
1. 10: 5 = 2, since 2 . 5 = 10
2. (- 4) : (- 2) = 2, since 2 . (- 2) = - 4
3. (- 18) : 3 = - 6, since (- 6) . 3 = - 18
4. 12: (- 4) = - 3, since (- 3) . (- 4) = 12

From the examples it is clear that the quotient of two numbers with the same signs is a positive number (examples 1, 2), and the quotient of two numbers with different signs is a negative number (examples 3,4).

Rules for dividing negative numbers
To find the modulus of a quotient, you need to divide the modulus of the dividend by the modulus of the divisor.
So, to divide two numbers with the same signs, you need to:

. Place a “+” sign in front of the result.
Examples of dividing numbers with the same signs:
. (- 9) : (- 3) = + 3
. 6: 3 = 2

To divide two numbers with different signs, you need to:
. divide the module of the dividend by the module of the divisor;
. Place a “-” sign in front of the result.
Examples of dividing numbers with different signs:
. (- 5) : 2 = - 2,5
. 28: (- 2) = - 14
You can also use the following table to determine the quotient sign.
Rule of signs for division

When calculating “long” expressions in which only multiplication and division appear, it is very convenient to use the sign rule. For example, to calculate a fraction

Please note that the numerator has 2 minus signs, which when multiplied will give a plus. There are also three minus signs in the denominator, which when multiplied will give a minus sign. Therefore, in the end the result will turn out with a minus sign.

Reducing a fraction (further actions with the modules of numbers) is performed in the same way as before:

The quotient of zero divided by a number other than zero is zero.
0: a = 0, a ≠ 0
You CANNOT divide by zero!

All previously known rules of division by one also apply to the set of rational numbers.
. a: 1 = a
. a: (- 1) = - a
. a: a = 1
, where a is any rational number.

The relationships between the results of multiplication and division, known for positive numbers, remain the same for all rational numbers (except zero):
. if a . b = c; a = c: b; b = c: a;
. if a: b = c; a = c. b; b = a: c
These dependencies are used to find the unknown factor, dividend and divisor (when solving equations), as well as to check the results of multiplication and division.

An example of finding the unknown.
x. (- 5) = 10

x = 10: (- 5)

x = - 2

Minus sign in fractions
Divide the number (- 5) by 6 and the number 5 by (- 6).

We remind you that the line in the notation of an ordinary fraction is the same division sign, and we write the quotient of each of these actions in the form of a negative fraction.

Thus, the minus sign in a fraction can be:
. before a fraction;
. in the numerator;
. in the denominator.

When writing negative fractions, the minus sign can be placed in front of the fraction, transferred from the numerator to the denominator, or from the denominator to the numerator.

This is often used when working with fractions, making calculations easier.

Example. Please note that after placing the minus sign in front of the bracket, we subtract the smaller one from the larger module according to the rules for adding numbers with different signs.

Using the described property of sign transfer in fractions, you can act without finding out which of the given fractions has a greater modulus.

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.

Page navigation.

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 understand rule for adding negative numbers. To add two negative numbers, you need:

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 −18,007.

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 whole solution in short form: (−304)+(−18 007)= −(304+18 007)=−18 311 .

Answer:

−18 311 .

The 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 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

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 look at specific examples of solving such 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 a debt, a 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) . Since, by 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 the rule for adding negative numbers We also proved it.

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 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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Goals and objectives of the lesson:

Summarize and systematize students’ knowledge on this topic.
Develop subject and general academic skills and abilities, the ability to use acquired knowledge to achieve a goal; establish patterns of diversity of connections to achieve a level of systematic knowledge.
Developing self-control and mutual control skills; develop desires and needs to generalize the facts received; develop independence and interest in the subject.

Lesson plan:

I. Teacher's opening speech.

II. Checking homework.

III. Reviewing the rules for adding and subtracting numbers with different signs. Updating knowledge.

IV. Solving tasks using cards

V. Independent work according to options.

VI. Summing up the lesson. Setting homework.

Lesson progress

I. Organizational moment

Students, under the guidance of the teacher, check the presence of a diary, workbook, tools, mark those missing, check the readiness of the class for the lesson, and the teacher psychologically prepares the children for work in the lesson.

Popular wisdom tells us “repetition is the mother of learning.”

Today we will teach you the final lesson on the topic of addition and subtraction of positive and negative numbers.

The purpose of our lesson is to review the material on this topic and prepare for test work.

And the motto of our lesson, I think, should be the statement: “We will learn to add and subtract with “5”!”

II. Checking homework

№1114. Fill in the blanks of the table:

№1116. The album contains 1105 stamps, the number of foreign stamps amounted to 30% of the number of Russian stamps. How many foreign and how many Russian stamps were in the album?

III. Reviewing the rules for adding and subtracting numbers with different signs. Updating knowledge.

Students repeat: the rule for adding negative numbers, the rule for adding numbers with different signs, the rule for subtracting numbers with different signs. Then solve examples to apply each of these rules. (Slides 4-10)

Updating students' knowledge of finding the length of a segment on a coordinate line using the known coordinates of its ends:

4)Task “Guess the word”

On globe Birds live - the unmistakable “compilers” of the weather forecast for the summer. The name of these birds is encrypted on the card.

After completing all the tasks, the student receives a key word, and the answers are checked using a projector.

Key FLAMINGOS build nests in the form of a cone: high - to rainy summer; low – to dry. (Show students the model Slides 14-16)

IV. Solving tasks using cards.

V. Independent work on options.

Each student has an individual card.

Option 1.

Mandatory part.

1. Compare the numbers:

a) –24 and 15;

b) –2 and –6.

2. Write down the opposite number:

3. Follow these steps:

4. Find the meaning of the expression:

VI. Summing up the lesson. Setting homework.

The questions are projected on the screen.

The number that corresponds to a point on a coordinate line...
Of two numbers on a coordinate line, the number that is located...
A number that is neither negative nor positive...
The distance from the number to the origin on the number line...
Natural numbers, their opposites and zero...

Setting homework:

prepare for the test:
review the rules for adding and subtracting positive and negative numbers;
solve No. 1096 (k, l, m) No. 1117

Lesson summary.

A sage was walking, and three people met him, carrying carts with stones for construction under the hot sun. The sage stopped and asked each one a question. The first one asked: “What have you been doing all day?” And he answered with a grin that he had been carrying the damned stones all day. The sage asked the second: “What did you do all day?” And he replied: “And I did my job conscientiously.” And the third smiled, his face lit up with joy and pleasure: “And I took part in the construction of the temple.”

Guys! Let's try to evaluate everyone's work for the lesson.

Whoever worked like the first person picks up the blue squares.

Those who worked conscientiously raise green squares.

Those who took part in the construction of the Temple of Knowledge raise red squares.

Reflection- Do your knowledge and skills correspond to the motto of the lesson?

What knowledge did you need today?

Lesson and presentation on the topic: "Examples of adding and subtracting negative numbers"

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Educational aids and simulators in the Integral online store for 6th grade
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Interactive simulator for the textbook by Vilenkin N.Ya.

Guys, let's review the material we covered.

Addition- this is a mathematical operation, after which we obtain the sum of the original numbers (the first term and the second term).

Number modulus- this is the distance on the coordinate line from the origin to any point.
The number module has certain properties:
1. The modulus of the number zero is zero.
2. The modulus of a positive number, for example, five, is the number five itself.
3. The modulus of a negative number, for example, minus seven is the positive number seven.

Adding two negative numbers

When adding two negative numbers, you can use the concept of modulus. Then you can discard the signs of the numbers and add their modules, and assign the sum negative sign, since both numbers were initially negative.

For example, you need to add the numbers: - 5 + (-23) =?
We discard the signs and add the modules of numbers. We get: 5 + 23 = 28.
Now we assign the resulting amount a minus sign.
Answer: -28.

More examples of addition.

39 + (-45) = - 84
-193 + (-205) = -398

When adding fractions, you can use the same method.

Example: -0.12 + (-3.4) = -3.52

Addition of positive and negative numbers

Adding numbers with different signs is slightly different from adding numbers with the same signs.

Let's look at an example: 14 + (-29) =?
Solution.
1. We discard the signs, we get the numbers 14 and 29.
2. Subtract the smaller number from the larger number: 29 - 14.
3. Before the difference we put the sign of the number whose modulus is greater. In our example, this is the number -29.

14 + (-29) = -15

Answer: -15.

Adding numbers using the number line

If you have difficulty adding negative numbers, you can use the number line method. It is visual and convenient for small numbers.
For example, let's add two numbers: -6 and +8. Mark the point -6 on the number line.

Then we move the point representing the number -6 eight positions to the right, because the second term is equal to +8 and we will get to the point indicating the number +2.

Answer: +2.

Example 2.
Let's add two negative numbers: -2 and (-4).
Mark the point -2 on the number line.

Then move it four positions to the left, because the second term is equal to -4 and we get to point -6.

The answer is -6.