List the properties of addition as they are read. Properties of addition, multiplication, subtraction and division of integers

Let's draw a rectangle on a piece of paper in a cage with sides of 5 cm and 3 cm. Let's break it into squares with a side of 1 cm ( fig. 143). Let's count the number of cells located in the rectangle. This can be done, for example, like this.

The number of squares with a side of 1 cm is 5 * 3. Each such square consists of four cells. Therefore, the total number of cells is (5 * 3 ) * 4 .

The same problem can be solved differently. Each of the five columns of the rectangle consists of three squares with a side of 1 cm. Therefore, one column contains 3 * 4 cells. Therefore, there will be 5 * (3 * 4 ) cells in total.

The cell count in Figure 143 illustrates in two ways associative property of multiplication for numbers 5, 3 and 4 . We have: (5 * 3 ) * 4 = 5 * (3 * 4 ).

To multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third numbers.

(ab)c = a(bc)

It follows from the commutative and associative properties of multiplication that when multiplying several numbers, factors can be interchanged and enclosed in brackets, thereby determining the order of calculations.

For example, the equalities are true:

abc=cba

17 * 2 * 3 * 5 = (17 * 3 ) * (2 * 5 ).

In figure 144, segment AB divides the rectangle considered above into a rectangle and a square.

We count the number of squares with a side of 1 cm in two ways.

On the one hand, there are 3 * 3 of them in the resulting square, and 3 * 2 in the rectangle. In total we get 3 * 3 + 3 * 2 squares. On the other hand, each of the three rows of this rectangle contains 3 + 2 squares. Then their total number is 3 * (3 + 2 ).

Equalsto 3 * (3 + 2 ) = 3 * 3 + 3 * 2 illustrates distributive property of multiplication with respect to addition.

To multiply a number by the sum of two numbers, you can multiply this number by each term and add the resulting products.

In literal form, this property is written as follows:

a(b + c) = ab + ac

It follows from the distributive property of multiplication with respect to addition that

ab + ac = a(b + c).

This equality allows the formula P = 2 a + 2 b to find the perimeter of a rectangle to be written as follows:

P = 2 (a + b).

Note that the distribution property is valid for three or more terms. For example:

a(m + n + p + q) = am + an + ap + aq.

The distributive property of multiplication with respect to subtraction also holds: if b > c or b = c, then

a(b − c) = ab − ac

Example 1 . Calculate in a convenient way:

1 ) 25 * 867 * 4 ;

2 ) 329 * 75 + 329 * 246 .

1) We use the commutative, and then the associative properties of multiplication:

25 * 867 * 4 = 867 * (25 * 4 ) = 867 * 100 = 86 700 .

2) We have:

329 * 754 + 329 * 246 = 329 * (754 + 246 ) = 329 * 1 000 = 329 000 .

Example 2 . Simplify the expression:

1) 4 a * 3 b;

2 ) 18m − 13m.

1) Using the commutative and associative properties of multiplication, we get:

4 a * 3 b \u003d (4 * 3) * ab \u003d 12 ab.

2) Using the distributive property of multiplication with respect to subtraction, we get:

18m - 13m = m(18 - 13 ) = m * 5 = 5m.

Example 3 . Write the expression 5 (2 m + 7) so that it does not contain brackets.

According to the distributive property of multiplication with respect to addition, we have:

5 (2 m + 7 ) = 5 * 2 m + 5 * 7 = 10 m + 35 .

Such a transformation is called opening brackets.

Example 4 . Calculate the value of the expression 125 * 24 * 283 in a convenient way.

Solution. We have:

125 * 24 * 283 = 125 * 8 * 3 * 283 = (125 * 8 ) * (3 * 283 ) = 1 000 * 849 = 849 000 .

Example 5 . Perform the multiplication: 3 days 18 hours * 6.

Solution. We have:

3 days 18 hours * 6 = 18 days 108 hours = 22 days 12 hours

When solving the example, the distributive property of multiplication with respect to addition was used:

3 days 18 hours * 6 = (3 days + 18 hours) * 6 = 3 days * 6 + 18 hours * 6 = 18 days + 108 hours = 18 days + 96 hours + 12 hours = 18 days + 4 days + 12 hours = 22 days 12 hours

A number of results inherent in this action can be noted. These results are called properties of addition of natural numbers. In this article, we will analyze in detail the properties of the addition of natural numbers, write them using letters and give explanatory examples.

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Associative property of addition of natural numbers.

Now we give an example illustrating the associative property of addition of natural numbers.

Imagine a situation: 1 apple fell from the first apple tree, and 2 apples and 4 more apples fell from the second apple tree. Now consider the following situation: 1 apple and 2 more apples fell from the first apple tree, and 4 apples fell from the second apple tree. It is clear that the same number of apples will be on the ground in both the first and second cases (which can be checked recalculation). That is, the result of adding the number 1 to the sum of the numbers 2 and 4 is equal to the result of adding the sum of the numbers 1 and 2 to the number 4.

The considered example allows us to formulate the associative property of the addition of natural numbers: in order to add a given sum of two numbers to a given number, you can add the first term of this sum to this number and add the second term of this sum to the result obtained. This property can be written using letters like this: a+(b+c)=(a+b)+c, where a , b and c are arbitrary natural numbers.

Please note that in the equality a+(b+c)=(a+b)+c there are parentheses "(" and ")". Parentheses are used in expressions to indicate the order in which actions are performed - actions in brackets are performed first (more on this in the section). In other words, brackets enclose expressions whose values are evaluated first.

In conclusion of this section, we note that the associative property of addition allows us to uniquely determine addition of three, four or more natural numbers.

The property of adding zero and a natural number, the property of adding zero to zero.

We know that zero is NOT a natural number. So why did we decide to consider the addition property of zero and a natural number in this article? There are three reasons for this. First, this property is used when column addition of natural numbers. Second, this property is used when subtraction of natural numbers. Third: if we assume that zero means the absence of something, then the meaning of adding zero and a natural number is the same as sense of adding two natural numbers.

Let us carry out the reasoning that will help us formulate the addition property of zero and a natural number. Imagine that there are no items in the box (in other words, there are 0 items in the box), and a items are placed in it, where a is any natural number. That is, added 0 and a items. It is clear that after this action there are a items in the box. Therefore, the equality 0+a=a is true.

Similarly, if a box contains a items and 0 items are added to it (that is, no items are added), then after this action, a items will be in the box. So a+0=a .

Now we can state the property of addition of zero and a natural number: the sum of two numbers, one of which is zero, is equal to the second number. Mathematically, this property can be written as the following equality: 0+a=a or a+0=a, where a is an arbitrary natural number.

Separately, we pay attention to the fact that when adding a natural number and zero, the commutative property of addition remains true, that is, a+0=0+a .

Finally, we formulate the zero-zero addition property (it is quite obvious and does not need additional comments): the sum of two numbers that are each zero is zero. That is, 0+0=0 .

Now it's time to figure out how addition of natural numbers.

Bibliography.

Maths. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
Maths. Any textbooks for 5 classes of educational institutions.

The topic that this lesson is devoted to is “Properties of addition.” In it, you will get acquainted with the commutative and associative properties of addition, examining them with specific examples. Find out when you can use them to make the calculation process easier. Test cases will help determine how well you have learned the material.

Lesson: Addition Properties

Take a close look at the expression:

9 + 6 + 8 + 7 + 2 + 4 + 1 + 3

We need to find its value. Let's do it.

9 + 6 = 15
15 + 8 = 23
23 + 7 = 30
30 + 2 = 32
32 + 4 = 36
36 + 1 = 37
37 + 3 = 40

The result of the expression 9 + 6 + 8 + 7 + 2 + 4 + 1 + 3 = 40.
Tell me, was it convenient to calculate? Calculating was not very convenient. Look again at the numbers in this expression. Is it possible to swap them so that the calculations are more convenient?

If we rearrange the numbers differently:

9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 = …
9 + 1 = 10
10 + 8 = 18
18 + 2 = 20
20 + 7 = 27
27 + 3 = 30
30 + 6 = 36
36 + 4 = 40

The final result of the expression is 9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 = 40.
We see that the results of the expressions are the same.

The terms can be interchanged if it is convenient for calculations, and the value of the sum will not change from this.

There is a law in mathematics: Commutative law of addition. It says that the sum does not change from the rearrangement of the terms.

Uncle Fyodor and Sharik argued. Sharik found the value of the expression as it was written, and Uncle Fyodor said that he knew another, more convenient way of calculating. Do you see a more convenient way to calculate?

The ball solved the expression as it is written. And Uncle Fyodor said that he knows the law that allows you to change the terms, and swapped the numbers 25 and 3.

37 + 25 + 3 = 65 37 + 25 = 62

37 + 3 + 25 = 65 37 + 3 = 40

We see that the result remains the same, but the calculation has become much easier.

Look at the following expressions and read them.

6 + (24 + 51) = 81 (to 6 add the sum of 24 and 51)
Is there a convenient way to calculate?
We see that if we add 6 and 24, we get a round number. It is always easier to add something to a round number. Take in parentheses the sum of the numbers 6 and 24.
(6 + 24) + 51 = …
(add 51 to the sum of numbers 6 and 24)

Let's calculate the value of the expression and see if the value of the expression has changed?

6 + 24 = 30
30 + 51 = 81

We see that the value of the expression remains the same.

Let's practice with one more example.

(27 + 19) + 1 = 47 (add 1 to the sum of the numbers 27 and 19)
What numbers can be conveniently grouped in such a way that a convenient way is obtained?
You guessed that these are the numbers 19 and 1. Let's take the sum of the numbers 19 and 1 in brackets.
27 + (19 + 1) = …
(to 27 add the sum of the numbers 19 and 1)
Let's find the value of this expression. We remember that the action in parentheses is performed first.
19 + 1 = 20
27 + 20 = 47

The meaning of our expression remains the same.

Associative law of addition: two adjacent terms can be replaced by their sum.

Now let's practice using both laws. We need to calculate the value of the expression:

38 + 14 + 2 + 6 = …

First, we use the commutative property of addition, which allows us to swap terms. Let's swap the terms 14 and 2.

38 + 14 + 2 + 6 = 38 + 2 + 14 + 6 = …

Now we use the associative property, which allows us to replace two neighboring terms by their sum.

38 + 14 + 2 + 6 = 38 + 2 + 14 + 6 = (38 + 2) + (14 + 6) =…

First, we find out the value of the sum of 38 and 2.

Now the sum is 14 and 6.

3. Festival of pedagogical ideas "Open Lesson" ().

do at home

1. Calculate the sum of the terms in different ways:

a) 5 + 3 + 5 b) 7 + 8 + 13 c) 24 + 9 + 16

2. Calculate the results of the expressions:

a) 19 + 4 + 16 + 1 b) 8 + 15 + 12 + 5 c) 20 + 9 + 30 + 1

3. Calculate the amount in a convenient way:

a) 10 + 12 + 8 + 20 b) 17 + 4 + 3 + 16 c) 9 + 7 + 21 + 13

We have defined addition, multiplication, subtraction and division of integers. These actions (operations) have a number of characteristic results, which are called properties. In this article, we will consider the basic properties of addition and multiplication of integers, from which all other properties of these operations follow, as well as the properties of subtraction and division of integers.

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Integer addition has several other very important properties.

One of them is related to the existence of zero. This property of integer addition states that adding zero to any whole number does not change that number. Let's write this property of addition using the letters: a+0=a and 0+a=a (this equality is valid due to the commutative property of addition), a is any integer. You may hear that the integer zero is called the neutral element in addition. Let's give a couple of examples. The sum of an integer −78 and zero is −78 ; if we add a positive integer 999 to zero, then we get the number 999 as a result.

We will now formulate another property of integer addition, which is related to the existence of an opposite number for any integer. The sum of any whole number with its opposite number is zero. Here is the literal form of this property: a+(−a)=0 , where a and −a are opposite integers. For example, the sum 901+(−901) is zero; similarly, the sum of the opposite integers −97 and 97 is zero.

Basic properties of multiplication of integers

The multiplication of integers has all the properties of multiplication of natural numbers. We list the main of these properties.

Just as zero is a neutral integer with respect to addition, one is a neutral integer with respect to multiplication of integers. That is, multiplying any whole number by one does not change the number being multiplied. So 1·a=a , where a is any integer. The last equality can be rewritten as a 1=a , this allows us to make the commutative property of multiplication. Let's give two examples. The product of the integer 556 by 1 is 556; the product of one and a negative integer −78 is −78 .

The next property of integer multiplication is related to multiplication by zero. The result of multiplying any integer a by zero is zero, that is, a 0=0 . The equality 0·a=0 is also true due to the commutative property of multiplication of integers. In a particular case, when a=0, the product of zero and zero is equal to zero.

For the multiplication of integers, the property opposite to the previous one is also true. It claims that the product of two integers is equal to zero if at least one of the factors is equal to zero. In literal form, this property can be written as follows: a·b=0 , if either a=0 , or b=0 , or both a and b are equal to zero at the same time.

Distributive property of multiplication of integers with respect to addition

Together, the addition and multiplication of integers allows us to consider the distributive property of multiplication with respect to addition, which connects the two indicated actions. Using addition and multiplication together opens up additional possibilities that we would be missing if we considered addition separately from multiplication.

So, the distributive property of multiplication with respect to addition says that the product of an integer a and the sum of two integers a and b is equal to the sum of the products of a b and a c , that is, a (b+c)=a b+a c. The same property can be written in another form: (a+b) c=a c+b c .

The distributive property of multiplication of integers with respect to addition, together with the associative property of addition, makes it possible to determine the multiplication of an integer by the sum of three or more integers, and then the multiplication of the sum of integers by the sum.

Also note that all other properties of addition and multiplication of integers can be obtained from the properties we have indicated, that is, they are consequences of the above properties.

Integer subtraction properties

From the obtained equality, as well as from the properties of addition and multiplication of integers, the following properties of subtraction of integers follow (a, b and c are arbitrary integers):

Integer subtraction generally does NOT have the commutative property: a−b≠b−a .
The difference of equal integers is equal to zero: a−a=0 .
The property of subtracting the sum of two integers from a given integer: a−(b+c)=(a−b)−c .
The property of subtracting an integer from the sum of two integers: (a+b)−c=(a−c)+b=a+(b−c) .
The distributive property of multiplication with respect to subtraction: a (b−c)=a b−a c and (a−b) c=a c−b c.
And all other properties of integer subtraction.

Integer division properties

Arguing about the meaning of division of integers, we found out that the division of integers is the inverse of multiplication. We gave the following definition: division of integers is finding an unknown factor by a known product and a known factor. That is, we call the integer c the quotient of the integer a divided by the integer b when the product c·b is equal to a .

This definition, as well as all the properties of operations on integers considered above, allow us to establish the validity of the following properties of division of integers:

No integer can be divided by zero.
The property of dividing zero by an arbitrary non-zero integer a : 0:a=0 .
Property of dividing equal integers: a:a=1 , where a is any non-zero integer.
The property of dividing an arbitrary integer a by one: a:1=a .
In general, division of integers does NOT have the commutative property: a:b≠b:a .
The properties of dividing the sum and difference of two integers by an integer are: (a+b):c=a:c+b:c and (a−b):c=a:c−b:c , where a , b , and c are integers such that both a and b are divisible by c , and c is nonzero.
The property of dividing the product of two integers a and b by a nonzero integer c : (a b):c=(a:c) b if a is divisible by c ; (a b):c=a (b:c) if b is divisible by c ; (a b):c=(a:c) b=a (b:c) if both a and b are divisible by c .
The property of dividing an integer a by the product of two integers b and c (numbers a , b and c such that dividing a by b c is possible): a:(b c)=(a:b) c=(a :c) b .
Any other property of integer division.