Numbers. Addition of natural numbers. Properties of addition of natural numbers. Addition of natural numbers A program is an algorithm written in a programming language that is used to perform any action. Translator

The result of adding two or more numbers is called sum, and the numbers themselves are terms.

The sum of two negative numbers... Add up the numbers, similarly to the positive ones, write down the result with a minus sign. For example, (-6) + (- 5.3) \u003d - (6 + 5.3) \u003d - 11.3.

The sum does not change from the permutation of the places of the terms a + b \u003d b + a.

Subtracting numbers

The result of the action is called difference... The numbers themselves - minuend and subtrahend.

Adding positive and negative numbers - this is nothing more than subtraction! Few people think that the subtraction of 7-2 can be represented as 7 + (- 2), they got the addition of a negative and a positive number. In order to add two numbers with opposite signs, it is necessary to subtract the smaller from the larger number, and the sign of the sum must coincide with the sign of the larger number.

For example, - 8+3=- (8-3)=- five; or -7 + 45=+ (45-7)=+ 38=38.

Multiplication of numbers

The result of multiplying two or more numbers is called work, and the numbers themselves are multipliers.

Multiply number and on the b - means to find the amount b terms, each of which is equal to a.

For example,

The product of two numbers of the same sign is a positive number. For example,

The product of two numbers with different signs is a negative number. For example,

Permutation of the factors does not change the value of the product ab \u003d ba.

1) For any natural numbers a and b equality is true a + b \u003d b + a... This property is called the displacement (commutative) law of addition, which is formulated as follows: the value of the sum does not change from the permutation of the terms.

2) For any natural a, b and c equality is true (a + b) + c \u003d a + (b + c). This property is called the combinational (associative) law of addition, which is formulated as follows: the value of the sum will not change if any group of terms is replaced by their sum.

1) For any natural numbers a and b equality is true ab \u003d ba... This property is called the displacement law of multiplication, which is formulated as follows: the value of the product does not change from the permutation of the factors.

2) For any natural a, b and c equality is true (ab) c \u003d a (bc). This property is called the combination law of multiplication, which is formulated as follows: the value of the product will not change if any group of factors is replaced by a product.

3) For any values a, b and c equality is true (a + b) c \u003d ac + bc. This property is called the distributional (distributive) law of multiplication (with respect to addition), which is formulated as follows: to multiply the sum by a number, it is enough to multiply each term by this number and add the resulting products. Similarly, you can write: (a-b) c \u003d ac-bc.

This is an action on two numbers, the result of which is a new natural number obtained by increasing the value of one number by the value of another number.

Add two natural numbers - means to count as many units to the first number as they are contained in the second number.

Example 1. Mom brought home several apples in two bags. One package contained 3 apples, and the second - 2. How many apples did mom bring home?

To answer this question, when you take apples out of bags, you must count them at the same time, for example, putting apples from the first bag, say: one, two, three, and then, taking apples out of the second bag, continue: four, five. So there are only 5 apples.

When listing the apples, we added the number of apples from the second to the number of apples from the first package and got the total number of all apples, i.e. 5.

Example 2. Add two numbers: 4 and 2.

Decision:

Let's count all the units of the second to the first number: add one more to four units, you get five units, add one to five, you get six. Thus, from the two given numbers 4 and 2, we received a new number 6, containing four units of the first number and two units of the second, that is, as many units as there were in both numbers.

The numbers to be added are called terms, and the result of addition, that is, the number resulting from addition, is called sum.

To record addition, the + (plus) sign is used. It is placed between the terms. For example, the record 2 + 5 means that the numbers 2 and 5 are added. To the right of the addition record, they put the \u003d (equal) sign, after which the sum is written:

Addition is an action that is always doable, that is, no matter what natural numbers we take as terms, you can always find their sum.

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Based on the addition of 2 natural numbers. Adding 3 or more numbers looks like sequential addition of 2 numbers. In addition, due to transposable and, the numbers that are added can be swapped and any 2 of the added numbers can be replaced with their sum.

Combination property of addition proves that the result of the addition of 3 numbers a, b and c does not depend on the place of the parentheses. Thus, the amounts a + (b + c)and (a + b) + c can be written as a + b + c... This expression is called sumand the numbers a, b and c - terms.

Similarly, due to combination property of addition, are equal to the sums (a + b) + (c + d), (a + (b + c)) + d, ((a + b) + c) + d, a + (b + (c + d)) and a + ((b + c) + d). That is, the result of the addition of 4 natural numbers a, b, cand d does not depend on the location of the brackets. In this case, the amount is written as: a + b + c + d.

If the expression does not have parentheses, and it consists of more than two terms, you yourself can arrange the parentheses as you like and add 2 numbers in succession to get the answer. That is, the process of adding 3 or more numbers is reduced to the sequential replacement of 2 adjacent terms with their sum.

For example, let's calculate the sum 1+3+2+1+5 ... Let's consider 2 methods from a large number of existing ones.

First way. At each step, we replace the first 2 terms with the sum.

Because sum of numbers 1 and 3 equals 4 means:

1+3+2+1+5=4+2+1+5 (we replaced the sum 1 + 3 with 4).

Because the sum of 4 + 2 is equal to 6, then:

4+2+1+5=6+1+5.

Because the sum of numbers 6 and 1 is 7, then:

6+1+5=7+5

And the last step 7+5=12 ... Thus:

1+3+2+1+5=12

We did the addition by placing the parentheses as follows: (((1+3)+2)+1)+5.

Second way.Let's arrange the brackets like this: ((1+3)+(2+1))+5 .

As 1+3=4 , and 2+1=3 , then:

((1+3)+(2+1))+5=(4+3)+5

The sum of 4 and 3 is 7, which means:

(4+3)+5=7+5.

And the last step: 7+5=12.

On the result of adding 2, 3, 4, etc. numbers are not affected not only by the arrangement of brackets, but also by the order of writing the terms. Thus, when summing natural numbers, you can change the places of the terms. This sometimes results in a more streamlined decision process.

Properties of addition of natural numbers.

• To get the number following the natural, add one to it.

For example: 3 + 1 \u003d 4; 39 + 1 \u003d 40.

• When rearranging the places of the terms, the sum does not change:

3 + 4 = 4 + 3 = 7 .

This addition property is called travel law.

• The sum of 3 or more terms will not change from changing the order of addition of numbers.

For example: 3 + (7 + 2) \u003d (3 + 7) + 2 \u003d 12;

means: a + (b + c) \u003d (a + b) + c.

Therefore, instead of 3 + (7 + 2) write 3 + 7 + 2 and add the numbers in order, from left to right.

This property of addition is called combination law of addition.

• When adding 0 to a number, the sum is equal to the number itself.

3 + 0 = 3 .

Conversely, when adding a number to zero, the sum is equal to the number.

0 + 3 = 3;

means: a + 0 \u003d a; 0 + a \u003d a.

• If point C separates the segment AB, then the sum of the lengths of the segments ACand CB equal to the length of the segment AB.

AB \u003d AC + CB.

If a AC \u003d 2cmand CB \u003d 3 cm,

then AB \u003d 2 + 3 \u003d 5 cm.

"Addition and subtraction of numbers" - Auxiliary memorization techniques. Combination law of multiplication. Results of the topic "Addition and subtraction". The travel law of addition. Grade 3? guide route. Distribution law. 2nd quarter. Acquaintance with three-digit numbers. Computing in grade 3. Consciously performing computations. Discharge composition.

"Number as a result of measuring a value" - "Number as a result of measuring a value" math lesson in grade 1. Measuring the length of a segment using a measure.

"Tolstoy Two Brothers" - We will be lost for nothing - we will be lost in vain We will remain with nothing - we will remain with nothing. To warm up. Epic's Fable Fairy Tale Play. Without looking back, very quickly. He opened a school for peasant children in Yasnaya Polyana in 1859. Work on the 2nd part of the tale. L.N. Tolstoy 1828-1910. Story. My memory is strong. Nearby (near).

"Adding negative numbers" - The sum of two negative numbers is always greater than each of the terms. The sum of two negative numbers is always positive. Example: -8.7 + (-3.5) \u003d - (8.7 + 3.5) \u003d - 12.2. Blitz - poll. Lesson Addition of negative numbers. Physical education. Rene Descartes. The history of negative numbers. The sum of two negative numbers is always negative.

"Addition of numbers 1 class" - Consolidation of the studied. Make and solve the problem: Before you are a series of numbers: 10 11 13 16. How much is 16, more than 10? Educational: to teach students the technique of addition with the transition through a dozen in "parts". "A general technique for adding single-digit numbers with a transition through ten." "Chain". Try to understand everything And count carefully!

"Two frosts" - Whistled, clicked - and ran. Frost shook his head - Blue nose and said: - Eh, you are young, brother, and stupid. Run after the merchant. How can we have fun - freeze people? The elder brother, Frost - Blue Nose, chuckles, and pats the mitten with his mitten. Let him how he dresses, let him know what Frost is - Red nose.

Addition - an arithmetic operation, which is performed on two numbers and consists in finding a number that means the amount that corresponds to these two original numbers, if taken together. The number that is the result of the operation of adding two numbers is called the sum of these numbers.

Addition is indicated by a "+" (plus) sign, which is placed between the two operands. For example, the notation "A + B" means "conclude A and B" or "the sum of A and B". The notation "A + B \u003d C" means: the number C is the sum of the numbers A and B.

Addition is simply illustrated at the everyday level. For example, you can imagine that two numbers correspond to the number of inhabitants of a two-story house. Then the sum of these numbers indicates the number of inhabitants of the whole house.

Formally, the operation of adding natural numbers can be defined as follows:

• x + 1 \u003d S (x)
• x + S (y) \u003d S (x + y)

where S (x) is the number following x.

In accordance with this, the result of the addition (sum) of two single-digit numbers is determined as follows:

	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18