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. I.e, 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

Simultaneous 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.

Adding one number to another is pretty easy. Consider an example, 4+3=7. This expression means that three units were added to four units, and as a result, seven units were obtained.
The numbers 3 and 4 that we added together is called terms. And the result of adding the number 7 is called sum.

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 unchanged:

This property of terms is called commutative law of addition.

Commutative law of addition.

The sum does not change from changing the places of the terms.

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 we add 1 + 2, and then we add to the resulting sum of 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 amount. Our example will look like this:

1+(2+4)=7

The answer remains the same. For both types of addition of the same example, the answer is the same. We conclude:

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

This addition property is called associative law of addition.

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

Associative 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 associative 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 would 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

Addition property with zero.

When you add a number to zero, the result is the same number.

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

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

a+0=a
0+ a=a

Questions about the addition of natural numbers:
Addition table, compile and see how the property of the commutative law works?
An addition table from 1 to 10 might look like this:

The 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 \u003d c, what will be the sum?
Answer: The sum is the sum of the terms. a+b and c.

In the expression a + b \u003d c terms, what will be?
Answer: a and b. The terms are the numbers that we add.

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

How many terms must be in the example so that the associative law of addition can be applied?
Answer: from three terms and more.

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

Examples for tasks.
Example #1:
Write down the answer for the presented 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
Decision:
a) 5921+0 =5921
b) 0+5921=5921

So, in general, the subtraction of natural numbers does NOT have the commutative property. Let's write this statement in letters. If a and b are unequal natural numbers, then a−b≠b−a. For example, 45−21≠21−45 .

The property of subtracting the sum of two numbers from a natural number.

The next property is related to the subtraction of the sum of two numbers from a natural number. Let's look at an example that will give us an understanding of this property.

Imagine that we have 7 coins in our hands. We first decide to keep 2 coins, but thinking that this will not be enough, we decide to save one more coin. Based on the meaning of adding natural numbers, it can be argued that in this case we decided to save the number of coins, which is determined by the sum 2 + 1. So, we take two coins, add another coin to them and put them in a piggy bank. In this case, the number of coins left in our hands is determined by the difference 7−(2+1) .

Now let's imagine that we have 7 coins, and we put 2 coins in the piggy bank, and after that - another coin. Mathematically, this process is described by the following numerical expression: (7−2)−1 .

If we count the coins that remain in the hands, then in the first and second cases we have 4 coins. That is, 7−(2+1)=4 and (7−2)−1=4 , so 7−(2+1)=(7−2)−1 .

The considered example allows us to formulate the property of subtracting the sum of two numbers from a given natural number. To subtract from a given natural number a given sum of two natural numbers is the same as subtracting the first term of this sum from a given natural number, and then subtracting the second term from the resulting difference.

Recall that we gave meaning to the subtraction of natural numbers only for the case when the minuend is greater than the subtrahend, or equal to it. Therefore, we can subtract a given sum from a given natural number only if this sum is not greater than the natural number being reduced. Note that under this condition, each of the terms does not exceed the natural number from which the sum is subtracted.

Using letters, the property of subtracting the sum of two numbers from a given natural number is written as an equality a−(b+c)=(a−b)−c, where a , b and c are some natural numbers, and the conditions a>b+c or a=b+c are satisfied.

The considered property, as well as the associative property of addition of natural numbers, allow you to subtract the sum of three or more numbers from a given natural number.

The property of subtracting a natural number from the sum of two numbers.

We pass to the next property, which is related to the subtraction of a given natural number from a given sum of two natural numbers. Consider examples that will help us "see" this property of subtracting a natural number from the sum of two numbers.

Suppose we have 3 candies in the first pocket, and 5 candies in the second, and let us need to give 2 candies. We can do this in different ways. Let's take them in turn.

First, we can put all the candies in one pocket, then take out 2 candies from there and give them away. Let's describe these actions mathematically. After we put the candies in one pocket, their number will be determined by the sum of 3 + 5. Now, out of the total number of candies, we will give away 2 candies, while the remaining number of candies we have will be determined by the following difference (3+5)−2 .

Secondly, we can give away 2 candies by taking them out of the first pocket. In this case, the difference 3−2 determines the remaining number of candies in the first pocket, and the total number of candies we have left will be determined by the sum (3−2)+5 .

Thirdly, we can give away 2 candies from the second pocket. Then the difference 5−2 will correspond to the number of remaining candies in the second pocket, and the total remaining number of candies will be determined by the sum 3+(5−2) .

It is clear that in all cases we will have the same number of sweets. Therefore, the equalities (3+5)−2=(3−2)+5=3+(5−2) are true.

If we had to give not 2, but 4 candies, then we could do it in two ways. First, give away 4 candies, having previously put them all in one pocket. In this case, the remaining number of sweets is determined by an expression like (3+5)−4 . Secondly, we could give away 4 candies from the second pocket. In this case, the total number of candies gives the following sum 3+(5−4) . It is clear that in the first and second cases we will have the same number of sweets, therefore, the equality (3+5)−4=3+(5−4) is true.

After analyzing the results obtained by solving the previous examples, we can formulate the property of subtracting a given natural number from a given sum of two numbers. Subtracting a given natural number from a given sum of two numbers is the same as subtracting a given number from one of the terms, and then adding the resulting difference and another term. It should be noted that the subtracted number should NOT be greater than the term from which this number is subtracted.

Let's write the property of subtracting a natural number from a sum using letters. Let a , b and c be some natural numbers. Then, provided that a is greater than or equal to c, then the equality (a+b)−c=(a−c)+b, and under the condition that b is greater than or equal to c , the equality (a+b)−c=a+(b−c). If both a and b are greater than or equal to c , then both last equalities are true, and they can be written as follows: (a+b)−c=(a−c)+b= a+(b−c) .

By analogy, one can formulate the property of subtracting a natural number from the sum of three or more numbers. In this case, this natural number can be subtracted from any term (of course, if it is greater than or equal to the number being subtracted), and the remaining terms can be added to the resulting difference.

To visualize the voiced property, we can imagine that we have many pockets, and they contain sweets. Suppose we need to give 1 candy. It is clear that we can give 1 candy from any pocket. At the same time, it doesn’t matter which pocket we give it from, since this does not affect the number of sweets that we have left.

Let's take an example. Let a , b , c and d be some natural numbers. If a>d or a=d , then the difference (a+b+c)−d is equal to the sum of (a−d)+b+c . If b>d or b=d , then (a+b+c)−d=a+(b−d)+c . If c>d or c=d , then the equality (a+b+c)−d=a+b+(c−d) is true.

It should be noted that the property of subtracting a natural number from the sum of three or more numbers is not a new property, since it follows from the properties of adding natural numbers and the property of subtracting a number from the sum of two numbers.

Bibliography.

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

Integers

The numbers used for counting are called natural numbers Number zero does not apply to natural numbers.

unambiguous numbers: 1,2,3,4,5,6,7,8,9 Double digits: 24.56, etc. Three-digit: 348,569 etc. polysemantic: 23,562,456789 etc.

Dividing a number into groups of 3 digits, starting from the right, is called classes: the first three digits are the class of units, the next three digits are the class of thousands, then millions, etc.

segment call the line drawn from point A to point B. Call AB or BA A B The length of the segment AB is called distance between points A and B.

Length units:

1) 10 cm = 1 dm

2) 100 cm = 1 m

3) 1cm=10mm

4) 1 km = 1000 m

Plane is a surface that has no edges, extending indefinitely in all directions. Straight has no beginning and no end. Two lines that have one common point intersect. Ray- this is a part of a straight line that has a beginning and no end (OA and OB). The rays into which a point divides a line are called additional each other.

Coordinate beam:

0 1 2 3 4 5 6 O E A B X O(0), E(1), A(2), B(3) – point coordinates. Of two natural numbers, the one that is called earlier when counting is smaller and the one that is called later when counting is greater. One is the smallest natural number. The result of comparing two numbers is written as an inequality: 5< 8, 5670 >368. The number 8 less than 28 and more than 5 can be written as a double inequality: 5< 8 < 28

Addition and subtraction of natural numbers

Addition

Numbers that add up are called terms. The result of addition is called the sum.

Addition properties:

1. Displacement property: The sum of the numbers does not change when the terms are rearranged: a + b = b + a(a and b are any natural numbers and 0) 2. Associative property: To add the sum of two numbers to a number, you can first add the first term, and then the second term to the resulting sum: a + (b + c) = (a + b) + c = a + b + c(a, b and c are any natural numbers and 0).

3. Addition with zero: Adding zero does not change the number:

a + 0 = 0 + a = a(a is any natural number).

The sum of the lengths of the sides of a polygon is called the perimeter of this polygon.

Subtraction

The action by which the sum and one of the terms find another term is called subtraction.

The number to be subtracted from is called reduced, the number that is being subtracted is called deductible, the result of subtraction is called difference. The difference between two numbers shows how much first number more second or how much second number smaller first.

Subtraction properties:

1. The property of subtracting a sum from a number: In order to subtract the sum from a number, you can first subtract the first term from this number, and then subtract the second term from the resulting difference:

a - (b + c) = (a - b) -with= a – b –with(b + c > a or b + c = a).

2. The property of subtracting a number from a sum: To subtract a number from the sum, you can subtract it from one term, and add another term to the resulting difference

(a + b) - c \u003d a + (b - c), if with< b или с = b

(a + b) - c \u003d (a - c) + b, if with< a или с = a.

3. Zero subtraction property: If you subtract zero from a number, then it will not change:

a - 0 = a(a is any natural number)

4. The property of subtraction from a number of the same number: If you subtract this number from a number, you get zero:

a - a = 0(a is any natural number).

Numeric and alphabetic expressions

Action records are called numeric expressions. The number obtained as a result of performing all these actions is called the value of the expression.

Multiplication and division of natural numbers

Multiplication of natural numbers and its properties

To multiply a number m by a natural number n means to find the sum of n terms, each of which is equal to m.

The expression m · n and the value of this expression are called the product of the numbers m and n. The numbers m and n are called factors.

Multiplication Properties:

1. Commutative property of multiplication: The product of two numbers does not change when the factors are rearranged:

a b = b a

2. Associative property of multiplication: To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor:

a (b c) = (a b) c.

3. Property of multiplication by one: The sum of n terms, each of which is equal to 1, is equal to n:

1 n = n

4. Property of multiplication by zero: The sum of n terms, each of which is equal to zero, is equal to zero:

0 n = 0

The multiplication sign can be omitted: 8 x = 8x,

or a b = ab,

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

Division

The action by which the product and one of the factors find another factor is called division.

The number that is being divided is called divisible; the number by which it is divided is called divider, the result of division is called private.

The quotient shows how many times the dividend is greater than the divisor.

You can't divide by zero!

Division properties:

1. When dividing any number by 1, the same number is obtained:

a: 1 = a.

2. When dividing a number by the same number, a unit is obtained:

a: a = 1.

3. When you divide zero by a number, you get zero:

0: a = 0.

To find the unknown factor, you need to divide the product by another factor. 5x = 45 x = 45: 5 x = 9

To find the unknown dividend, you need to multiply the quotient by the divisor. x: 15 = 3 x = 3 15 x = 45

To find the unknown divisor, divide the dividend by the quotient. 48: x = 4 x = 48: 4 x = 12

Division with remainder

The remainder is always less than the divisor.

If the remainder is zero, then we say that the dividend is divisible by the divisor without a remainder or, otherwise, completely. To find the dividend a when dividing with a remainder, it is necessary to multiply the incomplete quotient c by the divisor b and add the remainder d to the resulting product.

a = c b + d

Expression simplification

Multiplication properties:

1. Distributive property of multiplication with respect to addition: To multiply a sum by a number, you can multiply each term by this number and add the resulting products:

(a + b)c = ac + bc.

2. Distributive property of multiplication with respect to subtraction: To multiply the difference by a number, you can multiply the minuend and subtrahend by this number and subtract the second from the first product:

(a - b)c \u003d ac - bc.

3a + 7a = (3 + 7)a = 10a

Order of actions

Addition and subtraction of numbers are called the actions of the first step, and multiplication and division of numbers are the actions of the second step.

Rules for the order of actions:

1. If there are no brackets in the expression and it contains actions of only one stage, then they are performed in order from left to right.

2. If the expression contains the actions of the first and second steps and there are no brackets in it, then the actions of the second step are performed first, then the actions of the first step.

3. If the expression contains brackets, then first perform the actions in brackets (taking into account rules 1 and 2)

Each expression specifies the program of its calculation. It is made up of commands.

Degree of. Square and cube numbers

A product in which all factors are equal to each other is written shorter: a · a · a · a · a · a = a6 Read: a to the sixth power. The number a is called the base of the degree, the number 6 is the exponent, and the expression a6 is called the degree.

The product of n and n is called the square of n and denoted by n2 (en squared):

n2 = n n

The product n n n is called the cube of the number n and is denoted by n3 (en cubed): n3 = n n n

The first power of a number is equal to the number itself. If the numerical expression includes powers of numbers, then their values ​​are calculated before performing other actions.

Areas and volumes

Writing a rule using letters is called a formula. Path formula:

s = vt, where s is the path, v is the speed, t is the time.

v=s:t

t=s:v

Square. The formula for the area of ​​a rectangle.

To find the area of ​​a rectangle, multiply its length by its width. S=ab, where S is the area, a is the length, b is the width

Two figures are called equal if one of them can be superimposed on the second so that these figures coincide. The areas of equal figures are equal. The perimeters of congruent figures are equal.

The area of ​​the whole figure is equal to the sum of the areas of its parts. The area of ​​each triangle is half the area of ​​the entire rectangle.

Square is a rectangle with equal sides.

The area of ​​a square is equal to the square of its side:

Area units

Square millimeter - mm2

Square centimeter - cm2

Square decimeter - dm2

Square meter -m2

Square kilometer - km2

Field areas are measured in hectares (ha). A hectare is the area of ​​a square with a side of 100 m.

The areas of small plots of land are measured in ares (a).

Ar (weaving) - the area of ​​​​a square with a side of 10 m.

1 ha = 10,000 m2

1 dm2 = 100 cm2

1 m2 = 100 dm2 = 10,000 cm2

If the length and width of the rectangle are measured in different units, then they must be expressed in the same units to calculate the area.

cuboid

The surface of a cuboid consists of 6 rectangles, each of which is called a face.

Opposite faces of a cuboid are equal.

The sides of the faces are called parallelepiped edges, and the vertices of the faces the vertices of the parallelepiped.

A cuboid has 12 edges and 8 vertices.

A cuboid has three dimensions length, width and height

Cube is a rectangular parallelepiped with the same dimensions. The surface of a cube consists of 6 equal squares.

Volume of a cuboid: To find the volume of a cuboid, multiply its length by its width by its height.

V=abc, V – volume, a length, b – width, c – height

Cube volume:

Volume units:

Cubic millimeter - mm3

Cubic centimeter - cm3

Cubic decimeter - dm3

Cubic meter - mm3

Cubic kilometer - km3

1 m3 = 1000 dm3 = 1000 l

1 l = 1 dm3 = 1000 cm3

1 cm3 = 1000 mm3 1 km3 = 1,000,000,000 m3

Circle and circle

A closed line that is at the same distance from a given point is called a circle.

The part of the plane that lies inside the circle is called a circle.

This point is called the center of both the circle and the circle.

A line segment that connects the center of the circle to any point on the circle is called circle radius.

A line segment that joins two points on a circle and passes through its center is called circle diameter.

The diameter is equal to two radii.

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. And 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 verified by 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 paragraph, we note that the associative property of addition allows us to uniquely determine the addition of three, four and 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 adding natural numbers in a column. Second: this property is used when subtracting natural numbers. Third: if we consider 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 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, let us 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. I.e, 0+0=0 .

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

Bibliography.

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