Properties of addition. Combinative and distributive properties of multiplication How to read the combinative property of addition

a, b are the numbers on which the addition is performed, c is the result of the addition.

Addition of multi-digit numbers is done bitwise.

Example: 9067542 + 34981 = 9102523

Laws of addition.

1) commutative: a + b = b + a;

Example. 310 + 1454 = 1454 + 310. No matter how we add the result, the result will be 1764.

2) associative: (a + b) + c = a + (b + c);

Example: (329 + 85) + 120 = 329 + (85 + 120) = 329 + 205 =534;

3) the law of adding a number with zero: a + 0 = a.

Subtraction

a (minuend) - b (subtrahend) = c (difference)

Example: 42397 - 17963 = 24434

Properties of subtraction actions:

1) the law of subtracting a number from the sum:

(a + b) - c = (a - c) + b, if a > c or a = c;

2) the law of subtraction from a sum:

a - (b + c) = (a - b) - c;

3) the law of subtracting a number from a number:

4) the law of subtraction from zero:

5) the law of subtracting an amount from a sum:

(a + b) - (c + d) = ;

Problem as an example of addition and subtraction operations

Calculate in a convenient way:

1) (4981 - 2992) - 808;
2) (3975 + 5729) - (5729 + 975).

We apply the 2nd and 5th laws of subtraction:

1) (4981- 2992) - 808 = 4981 - (2992 + 808) = 4981 - 3800 = 1181;
2) (3975 + 5729) - (5729 + 975) = (3975 - 975) + (5729 - 5720)= 3000 + 0 = 3000

Multiplication

Multiplying the number a by the number b (b>1) means finding the sum of b terms (each term is equal to a).

a x b= a + a + ... + a

If b = 1, then a x 1 = a.

a (first factor) x b (second factor) = c (product)

For example: 57 + 57 + 57 + 34 + 34 = 57 x 3 + 34 x 2 = 171 + 68 + 239

Multiplication laws

1) commutative: a x b = b x a;

Example. 15 x 110 = 110 x 15.

2) associative: (a x b) x c = a x (b x c);

Example: (9 x 30) x 10= 9 x (30 x 10) = 9 x 300= 2700;

(65 x 25) x 44 = (25 x 65) x 44 = 25 x (65 X 44) = 25 x 2860 = 71500.

3) multiplication by zero: 0 x a = 0;

Example: 0 x 10 = 0.

4) distributive law of multiplication regarding the action of addition (subtraction):

a x (b + c) = a x b + a x c;

Problems as an example of the operation of multiplication

Task 1. Calculate in a convenient way:

1) (37 x 125) x 8;
2) 49 x 84 + 49 x 83 - 49 x 67.

1) (37 x 125) x 8 = 37 x (125 x 8) = 37 x 1000 = 37000;

2) 49 x 84 + 49 x 83 - 49 x 67 = 49 x (84 + 83 - 67) = 49 x 100 = 4900.

Task 2. 1 kW/h costs 12 rubles. An electric iron consumes 2 kW/h for 1 hour of operation. We ironed the clothes with an iron for two days: on the first day - 3 hours, on the second - 2 hours. How much does electricity cost for two days? Solve the problem yourself, and we will only give you the answers: for 3 hours - 72 rubles; for 2 hours - 48 rub.

Division

a (divisible) : b (divisor) = c (quotient)

Laws of division:

1) a: 1 = a, since a x 1 = a;
2) 0: a =0, since 0 x a = 0;
3) you cannot divide by 0!

2224222: 2222 = 1001

The law of dividing a sum (difference) by a number:

1) (a + b) : c = a: c + b: c, c is not equal to 0;
2) (a - b) : c = a: c -b: c, c is not equal to 0;

Example: (4800 + 9300) : 300 = 4800: 300 + 9300: 300 = 16 + 31 + 47.

The law of dividing a product by a number:

(a x b) :c = (a: c) x b = (b: c) x a, c is not equal to 0.

Let's draw a rectangle with sides 5 cm and 3 cm on a piece of checkered paper. Divide it into squares with sides 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.

Counting cells 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)

From the commutative and combinatory properties of multiplication it follows that when multiplying several numbers, the factors can be swapped and placed in parentheses, thereby determining the order of calculations.

For example, the following equalities are true:

abc = cba,

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

In Figure 144, segment AB divides the rectangle discussed above into a rectangle and a square.

Let's count the number of squares with a side of 1 cm in two ways.

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

Equal to 3 * (3 + 2 ) = 3 * 3 + 3 * 2 illustrates distributive property of multiplication relative to addition.

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

In literal form this property is written as follows:

a(b + c) = ab + ac

From the distributive property of multiplication relative to addition it follows 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 in this form:

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 relative to subtraction is also true: 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) 18 m − 13 m.

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

4 a * 3 b = (4 * 3 ) * ab = 12 ab.

2) Using the distributive property of multiplication relative to subtraction, we obtain:

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

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

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

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

This transformation is called opening parentheses.

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 relative 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.

Adding one number to another is quite simple. Let's look at an example, 4+3=7. This expression means that three units were added to four units and the result was seven units.
The numbers 3 and 4 that we added are called terms. And the result of adding the number 7 is called amount.

Sum is the addition of numbers. Plus sign “+”.
In literal form, this example would look like this:

a+b=c

Addition components:
a- term, b- terms, c- sum.
If we add 4 units to 3 units, then as a result of addition we will get the same result; it will be equal to 7.

From this example we conclude that no matter how we swap the terms, the answer remains the same:

This property of terms is called commutative law of addition.

Commutative law of addition.

Changing the places of the terms does not change the sum.

In literal notation, the commutative law looks like this:

a+b=b+a

If we consider three terms, for example, take the numbers 1, 2 and 4. And we perform the addition in this order, first add 1 + 2, and then add to the resulting sum 4, we get the expression:

(1+2)+4=7

We can do the opposite, first add 2+4, and then add 1 to the resulting sum. Our example will look like this:

1+(2+4)=7

The answer remains the same. Both types of addition for the same example have the same answer. We conclude:

(1+2)+4=1+(2+4)

This property of addition is called associative law of addition.

The commutative and associative law of addition works for all non-negative numbers.

Combination law of addition.

To add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

(a+b)+c=a+(b+c)

The combination law works for any number of terms. We use this law when we need to add numbers in a convenient order. For example, let's add three numbers 12, 6, 8 and 4. It will be more convenient to first add 12 and 8, and then add the sum of two numbers 6 and 4 to the resulting sum.
(12+8)+(6+4)=30

Property of addition with zero.

When you add a number with zero, the resulting sum will be the same number.

3+0=3
0+3=3
3+0=0+3

In a literal expression, addition with zero will look like this:

a+0=a
0+ a=a

Questions on the topic of addition of natural numbers:
Make an addition table and see how the property of the commutative law works?
An addition table from 1 to 10 might look like this:

Second version of the addition table.

If we look at the addition tables, we can see how the commutative law works.

In the expression a+b=c, what will be the sum?
Answer: the sum is the result of adding the terms. a+b and c.

In the expression a+b=c terms, what will be?
Answer: a and b. Addends are numbers that we add together.

What happens to a number if you add 0 to it?
Answer: nothing, the number will not change. When adding with zero, the number remains the same, because zero is the absence of ones.

How many terms should there be in the example so that the combinational law of addition can be applied?
Answer: from three terms or more.

Write down the commutative law in literal terms?
Answer: a+b=b+a

Examples for tasks.
Example #1:
Write down the answer to the given expressions: a) 15+7 b) 7+15
Answer: a) 22 b) 22

Example #2:
Apply the combination law to the terms: 1+3+5+2+9
1+3+5+2+9=(1+9)+(5+2)+3=10+7+3=10+(7+3)=10+10=20
Answer: 20.

Example #3:
Solve the expression:
a) 5921+0 b) 0+5921
Solution:
a) 5921+0 =5921
b) 0+5921=5921

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 adding natural numbers, write them using letters and give explanatory examples.

Page navigation.

Combinative property of addition of natural numbers.

Now let's give an example illustrating the associative property of adding natural numbers.

Let's 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 this 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 there will be the same number of apples on the ground in both the first and second cases (which can be verified by recalculation). That is, the result of adding the number 1 with the sum of numbers 2 and 4 is equal to the result of adding the sum of numbers 1 and 2 with the number 4.

The considered example allows us to formulate the combinatory property of adding natural numbers: in order to add a given sum of two numbers to a given number, we can add the first term of the given sum to this number and add the second term of the given sum to the resulting result. 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 the equality a+(b+c)=(a+b)+c contains parentheses “(” and “)”. Parentheses are used in expressions to indicate the order in which actions are performed - the actions in parentheses are performed first (more about this is written in the section). In other words, expressions whose values are evaluated first are placed in parentheses.

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

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

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

Let us carry out some reasoning that will help us formulate the property of adding zero and a natural number. Let's imagine that there are no objects in the box (in other words, there are 0 objects in the box), and a objects are placed in it, where a is any natural number. That is, we added 0 and a objects. It is clear that after this action there are a objects 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 there will be a items in the box. So a+0=a .

Now we can give the formulation of the property of adding 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, let us 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, let us formulate the property of adding zero to zero (it is quite obvious and does not need additional comments): the sum of two numbers, each equal to zero, is equal to zero. That is, 0+0=0 .

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

Bibliography.

Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
Mathematics. Any textbooks for 5th grade of general education institutions.