Addition of natural numbers: rules, examples and solutions. Lesson summary "addition of natural numbers and its properties"

Let's figure out how to use it to add tens with tens, hundreds with hundreds, etc.

Let's add 8 tens and 9 tens. From the addition table we find that 8+9=10+7. Therefore, if we add 8 tens and 9 tens, we get the sum of 10 tens and 7 tens, that is, the sum of 100 and 70. Thus, 80+90=100+70. The sum 100+70 is the sum of the digit terms of the number 170. It is convenient to write all these arguments in the form of a sequential chain of equalities: 80+90=100+70=170. Such notations mean that the values ​​of all expressions that are separated by equal signs are equal.

To consolidate the material, consider the solution to another example. Let's do the addition 4,000+7,000. The addition table gives us the equality 4+7=10+1. Thus, adding 4 thousand and 7 thousand is the same as adding 10 thousand and 1 thousand. Therefore, 4,000+7,000=10,000+1,000. The last sum is an expansion into digits of the natural number 11,000. We have 4 000+7 000=10 000+1 000=11 000 .

Addition of arbitrary natural numbers.

Before moving on to adding arbitrary natural numbers, we recommend that you thoroughly study the material in the article sum of digit terms so that you can, without hesitation, decompose any natural number into digits, and also, without hesitation, using a known decomposition, you can immediately write down the decomposed natural number. This will directly determine how easy it will be for you to add arbitrary natural numbers.

Let us describe the sequence of actions:

• we replace the terms with their expansions by digits;
• rearrange the terms so that the ones are next to the ones, the tens are next to the tens, the hundreds are next to the hundreds, and so on;
• we add units with units, then tens with tens, then hundreds with hundreds, etc.;
• all previous actions lead us to a sum, which is an expansion into the digits of a natural number;
• finally, we write down the required number by its expansion.

Let's look at the addition of two natural numbers using examples.

Example.

Perform addition 36+2.

Solution.

The decomposition of the number 36 into digits has the form 30+6, and the number 2 has the form 2. Then 36+2=30+6+2.

In this example, we do not need to rearrange the terms, since they are already in the order we need.

Now we add the units: 6+2=8. Therefore, 30+6+2=30+8.

We came to the sum 30+8, which is equal to 38.

Thus, the solution can be written as follows: 36+2=30+6+2=30+8=38.

Answer:

36+2=38 .

Example.

Add the numbers 57 and 17.

Solution.

Because 57=50+7, and 17=10+7, then 57+17=50+7+10+7.

After rearranging the terms, the sum will take the following form: 50+10+7+7.

Now we add the units (if you don’t remember by heart, then refer to the addition table): 7+7=10+4.

Thus, 50+10+7+7=50+10+10+4.

We move on to adding tens, that is, to finding the sum of three terms 50, 10 and 10. Let's first add 50 and 10, after which we add the remaining number 10 to the result. Let's go: 50+10=60, since 5+1=6, then 50+10+10=60+10=70, since 6+1=7.

We have, 50+10+10+4=70+4. The last sum is the digit decomposition of the number 74.

So, 57+17=50+7+10+7=50+10+7+7= 50+10+10+4=70+4=74 .

Answer:

57+17=74 .

Example.

Calculate the sum of the numbers 3007 and 200.

Solution.

The decomposition of the number 3007 into digits has the form 3000+7, and the number 200 has the form 200. Then 3 007+200=3 000+7+200=3 000+200+7 . We have obtained the digit expansion of the number 3207. Thus, 3,007+200=3,207.

Answer:

3 007+200=3 207 .

Example.

Add the numbers 28,301 and 73,745.

Solution.

Let's break down these numbers into digits: 28,301=20,000+8,000+300+1 and 73,745=70,000+3,000+700+40+5.

Then
28 301+73 745= 20 000+8 000+300+1+70 000+ 3 000+700+40+5= 20 000+70 000+8 000+ 3 000+300+700+40+1+5 .
(When moving equalities to the next line, the “=” sign is written again).

Add the units: 1+5=6. After this we have 20,000+70,000+8,000+ 3,000+300+700+40+1+5= 20,000+70,000+8,000+ 3,000+300+700+40+6.

There is no need to add tens.

We add hundreds: 300+700=1,000, since 3+7=10. At this stage we have 20,000+70,000+8,000+ 3,000+300+700+40+6= 20,000+70,000+8,000+ 3,000+1000+40+6.

We add up thousands. Since 8+3=10+1, then 8,000+3,000+1,000= 10,000+1,000+1,000= 10,000+2,000. At this stage we get
20 000+70 000+8 000+ 3 000+1 000+40+6= 20 000+70 000+10 000+2 000+40+6 .

Add up tens of thousands: 20,000+70,000+10,000= 90,000+10,000=100,000. Then 20 000+70 000+10 000+2 000+40+6= 100 000+2 000+40+6 .

The sum 100,000+2,000+40+6 is equal to the number 102,046.

Answer:

28 301+73 745=102 046 .

In conclusion of this point, we note that it is convenient to add multi-digit natural numbers in a column, so we recommend studying the material in the article adding natural numbers in a column.

Addition of natural numbers on a coordinate ray.

The purpose of this paragraph is to present a geometric interpretation of the operation of adding natural numbers. Will help us achieve this goal. We will assume that the coordinate beam is located horizontally and to the right.

On coordinate ray the addition of two natural numbers a and b is a sequence of the following actions. First we find the point with coordinate a. From this point, we lay out b unit segments one after another so that the distance from the origin occurs. This will take us to a point on the coordinate ray, the coordinate of which is a natural number, equal to the sum a+b . In other words, from a point with coordinate a we move to the right to a distance b, and at the same time we get to a point whose coordinate is equal to the sum of numbers a and b.

For clarity, let's give an example. Let's show what the addition of natural numbers 2 and 4 on a coordinate ray represents (see the figure below). From the point with coordinate 2 we plot 4 unit segments. After this we get to the point whose coordinate is the number 6. Thus, 2+4=6.

Checking the result of adding natural numbers by subtraction.

Checking the result of adding natural numbers using subtraction is based on a fairly obvious connection between addition and subtraction. It is easy to trace this connection by referring to the following example.

Let us have 7 apples and 2 pears. Let's add these fruits together, then the sum 7+2=9, due to the meaning of adding natural numbers, determines total fruit. It is clear that if 7 apples are put aside from the fruits put together (there are 9 in total), then 2 pears will remain on the other side. Due to the meaning of subtracting natural numbers, the described action corresponds to the equality 9−7=2. Similarly, if you put 2 pears from the fruits put together to one side, then 7 apples will remain on the other side. This action corresponds to the equality 9−2=7.

The considered example leads us to a rule, the formulation of which is as follows: if you subtract one of the terms from the sum of two natural numbers, the result will be the other term. This rule is written using letters as follows: if a+b=c subtraction of natural numbers.

Let's check the result of the addition. To do this, subtract the term 106 from the resulting sum 163 and see if we get a number equal to the second term 57. We have 163−106=57. Thus, the test was successful and we can say that the addition was performed correctly.

Answer:

106+57=163 .

Bibliography.

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

Column addition, or as they also say, column addition, is a method widely used for adding multi-digit natural numbers. The essence of this method is that the addition of two or more multi-digit numbers comes down to a few simple operations of adding single-digit numbers.

The article describes in detail how to perform the addition of two or more multi-digit natural numbers. The rule for adding numbers into a column and examples of solutions with an analysis of all the most typical situations that arise when adding numbers into a column are given.

Adding two numbers in a column: what do you need to know?

Before we move directly to the operation of column addition, let's look at some important points. To quickly master the material, it is advisable to:

1. Know and have a good understanding of the addition table. So, when carrying out intermediate calculations, you do not have to waste time and constantly refer to the addition table.
2. Remember the properties of addition of natural numbers. Especially properties related to adding zeros. Let us recall them briefly. If one of the two terms is equal to zero, then the sum is equal to the other term. The sum of two zeros is zero.
3. Know the rules for comparing natural numbers.
4. Know what the digit of a natural number is. Recall that the digit is the position and value of the digit in the notation of the number. The digit determines the meaning of a digit in a number - units, tens, hundreds, thousands, etc.

Let us describe the algorithm for adding numbers in a column using concrete example. Let us add the numbers 724980032 and 30095. First, you should write down these numbers according to the rules for writing addition in a column.

Numbers are written one below the other, the digits of each digit are located, respectively, one below the other. We put a plus sign on the left, and draw a horizontal line under the numbers.

Now we mentally divide the record into columns by digits.

All that remains to be done is to add up the single-digit numbers in each column.

We start with the rightmost column (the units digit). We add up the numbers and write the value of the units under the line. If, when adding, the value of tens turns out to be different from zero, remember this number.

Add up the numbers in the second column. To the result we add the number of tens that we remembered in the previous step.

We repeat the entire process with each column, up to the far left.

This presentation is a simplified diagram of the algorithm for adding natural numbers in a column. Now that we understand the essence of the method, let's look at each step in detail.

First we add up the units, that is, the numbers in the right column. If we get a number less than 10, write it in the same column and move on to the next one. If the result of addition is greater than or equal to 10, then under the line in the first column we write down the value of the units place, and remember the value of the tens place. For example, the number turned out to be 17. Then we write down the number 7 - the value of units, and the value of tens - 1 - we remember. They usually say: “we write seven, one in mind.”

In our example, when adding the numbers in the first column, we get the number 7.

7 < 10 , поэтому записываем это число в разряд единиц результата, а запоминать нам ничего не нужно.

Next, we add the numbers in the next column, that is, in the tens place. We carry out the same actions, only we need to add the number that we kept in mind to the amount. If the amount is less than 10, simply write the number under the second column. If the result is greater than or equal to 10, we write down the value of the units of this number in the second column, and remember the number from the tens place.

In our case, we add the numbers 3 and 9, resulting in 3 + 9 = 12. We didn’t remember anything in the previous step, so we don’t need to add anything to this result.

12 > 10, so in the second column we write down the number 2 from the units place, and keep the number 1 from the tens place in mind. For convenience, you can write this number above the next column in a different color.

In the third column, the sum of the digits is zero (0 + 0 = 0). To this sum we add the number that we previously kept in mind, and we get 0 + 1 = 1. write down:

Moving on to the next column, we also add 0 + 0 = 0 and write the result as 0, since we did not remember anything in the previous step.

The next step gives 8 + 3 = 11. In the column we write the number 1 from the units digit. We keep the number 1 from the tens place in mind and move on to the next column.

This column contains only one number 9. If we didn't have the number 1 in memory, we would simply rewrite the number 9 under the horizontal line. However, given that we remembered the number 1 in the previous step, we need to add 9 + 1 and write down the result.

Therefore, under the horizontal line we write 0, and again keep one in mind.

Moving to the next column, add 4 and 1, write the result under the line.

The next column contains only the number 2. So in the previous step we didn’t remember anything, we just rewrote this number under the line.

We do the same with the last column containing the number 7.

There are no more columns, and there is also nothing in memory, so we can say that the column addition operation is over. The number written below the line is the result of adding the two upper numbers.

To understand all the possible nuances, let's look at a few more examples.

Example 1. Addition of natural numbers in a column

Let's add two natural numbers: 21 and 36.

First, let's write these numbers according to the rule for writing addition in a column:

Starting from the right column, we proceed to adding numbers.

Since 7< 10 , записываем 7 под чертой.

Add up the numbers in the second column.

Since 5< 10 , а в памяти с предыдущего шага ничего нет, записываем результат

There are no more numbers in the memory and in the next column, the addition is completed. 21 + 36 = 57

Example 2. Addition of natural numbers in a column

What is 47 + 38?

7 + 8 = 15, so let's write 5 in the first column under the line, and keep 1 in mind.

Now we add the values ​​from the tens place: 4 + 3 = 7. Don't forget about one and add it to the result:

7 + 1 = 8. We write the resulting number below the line.

This is the result of addition.

Example 3. Adding natural numbers in a column

Now let's take two three-digit numbers and add them.

3 + 9 = 12 ; 12 > 10

Write 2 below the line, keep 1 in mind.

8 + 5 = 13 ; 13 > 10

We add 13 and the memorized unit, we get:

13 + 1 = 14 ; 14 > 10

We write 4 below the line, keep 1 in mind.

Don't forget that in the previous step we remembered 1.

We write 0 below the line, keep 1 in mind.

In the last column we move the unit that we remembered earlier under the line and get the final result of the addition.

783 + 259 = 1042

Example 4. Addition of natural numbers in a column

Let's find the sum of the numbers 56927 and 90.

As always, first we write down the condition:

7 + 0 = 7 ; 7 < 10

2 + 9 = 11 ; 11 > 10

We write down 1 below the line, keep 1 in mind and move on to the next column.

We write 0 below the line, keep 1 in mind and move on to the next column.

The column contains one number 6. We add it with the remembered unit.

6 + 1 = 7 ; 7 < 10

We write 7 under the line and move on to the next column.

The column contains one number 5​​​​​. We move it under the line and finish the addition operation.

56927 + 90 = 57017

We present the following example without intermediate results or explanations, as an example of writing column addition in practice.

Definition. The addition of natural numbers is an algebraic operation that has the following properties: "1) (a Î N)a + 1 = a", 2) "(a, b Î N)a + b" =(a +b)". Number a + b is called the sum of numbers a and b, and the numbers a and b themselves are terms. As is known, the sum of any two natural numbers is also a natural number, and for any natural numbers a and b the sum a + b is unique. In other words, the sum of natural numbers exists and is unique. The peculiarity of the definition is that it is not known in advance whether an algebraic operation possessing the indicated properties exists, and if it exists, is it unique? Therefore, when constructing the axiomatic theory of natural numbers, the following statement is proved: The addition of natural numbers exists and it is unique This theorem consists of two statements (two theorems): addition of natural numbers exists; addition of natural numbers is unique. The laws of addition are used to simplify calculations. For natural numbers there are two laws of addition: commutative and combinational. Rule: Changing the places of the terms does not change the sum (commutative law of addition). For example: 37 + 42 = 42 + 37 = 79.V general view: a + b = b + a. Rule. To add a third term to the sum of two terms, you can add the sum of the second and third terms to the first term (combinative law of addition). For example: (37 + 42)+ 13 = 37 + (42 + 13). In general form: (a + b) + c = a + (b + c). Often in examples, both addition laws are used for calculations at once. For example: 1,300 + 400 + 700 + 600 = (1,300 + 700) + (400 + 600) = 2,000 + 1,000 = 3,000.

Axiomatic definition of multiplication of natural numbers. Theorem on its existence and uniqueness with proof. Multiplication table.

Multiplication of natural numbers is an algebraic operation defined on the plural N of natural numbers, assigning to each pair (a, b) a number a * b, satisfying the properties (axioms): 1. (∀a є N)a∙1 = a ; 2. (∀ a,b є N) a∙b" = a∙b + a. The number a∙b is called the product of numbers a and b, and the numbers a and b themselves are factors. Theorem 1. Multiplication of natural numbers exists, and it is unique Using the definition of the multiplication operation, we will compile a multiplication table for single-digit natural numbers: a) 1×1=1; 2×1=2; 3×1=3; 4×1=4, etc. (based on the property 1); b)1×2=1×1'=1×1+1= 1+1=2; 2×2=2×1'= 2×1+1= 2+1=3; 3×2 =3×1'= 3×1+1= 3+1=4, etc. (based on property 2). Theorem 2. (∀a,b,c є N)(a+b)∙ c = a∙c + b∙c Proof: Let the natural numbers a and b be chosen arbitrarily, and c take different natural values. Let us denote by M the set of all those and only those natural numbers c for which the equality (a + b)c = a∙c + b∙c is true. Let us show that for c=1 the equality (a + b)∙1 = a∙1 + b∙1 is true. Indeed, (a + b)∙1 =a+b=a∙1 + b∙1. Let the distributive law be satisfied for an arbitrarily chosen number c, i.e. the equality (a+b)∙c = a∙c + b∙c is true. Based on the assumption, we will prove the validity of the equality: (a + b)∙c" = a∙c" + b∙c" for the number c". Let's consider the left side of the equality and show that it is equal to the right: (a + b)∙c" = (a + b)∙c + (a + b)=(a∙c+b∙c)+ (a+b) = (a∙c+a)+(b∙c+b)= a∙c'+b∙c' This equality (a + b)∙c = a∙c + b∙c is true for any natural number с, and since the numbers a and b were chosen arbitrarily, this equality is valid for any a and b. The left distributive law of multiplication is proved similarly: (∀а,b,с є N)а ∙(b+с)= а∙ b+а∙с. Theorem 3. (∀ а,b,с є N)(а∙b) ∙с= a∙(b ∙с).-associative. Theorem 4. (∀a,b є N) a ∙b = b∙a.- communicative. The multiplication operation satisfies two laws: ab = bа (commutative law of multiplication), a(bс) = (аb)с (associative law of multiplication). There is also a law connecting addition and multiplication: a (b + c) = ab + ac (distributive law) A multiplication table is a table where the rows and columns are headed by factors, and the table cells contain their product.The table is used to teach multiplication.