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.

Page navigation.

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 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 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 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 the addition of natural numbers is performed.

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

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 in 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.

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