Subtraction of natural numbers. Properties of subtraction of natural numbers

If addition is associated with combining two sets into one, then subtraction is associated with separating a given set into two or more sets. Suppose we have a certain number of plastic sausages on a plate. Let's take one or more plastics from this set and put them aside, or better yet, eat them. We removed, that is, we took away several plastics from the initial set of sausage plastics, and the result on the plate changed downward. This is the meaning of subtraction.

Schematically subtracting two natural numbers as follows:

minuend − subtrahend = difference.

To indicate subtraction in writing, use the minus sign “−”.

First write down the minuend, then the minus sign, then the subtrahend. For example, writing 9 − 5 means that 5 is subtracted from 9.

Minuend is the number from which it is subtracted. In our example this is the number "9"

Subtrahend is the number that is subtracted from the minuend. In our example this is the number "5"

Difference is the number that is the result of subtraction.

Phrases "find the difference", "calculate the difference", “subtract the number 9 from the natural number 86” is understood as follows: you need to determine the number that is the result of subtracting these natural numbers.

PROPERTIES OF SUBTRACTING NATURAL NUMBERS

Property 1.

a − a = 0, where a is any natural number.

Property 2.

If a and b are unequal natural numbers, then a − b ≠ b − a

45 − 20 ≠ 20 − 45.

Property 3. Subtracting a given sum of two natural numbers from a given natural number is the same as subtracting the first term of a given sum from a given natural number, and then subtracting the second term from the resulting difference.

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.

10 - (2+1) = (10 - 2) - 1 = 7

Property 4. Subtracting a given natural number from a given sum of two numbers is the same as subtracting given number from one of the terms, then add the resulting difference and the other term. It should be noted that the number being subtracted must NOT be greater than the term from which this number is being subtracted.

In this lesson you will learn what direct and inverse operations are in mathematics. The teacher will talk about all the components of subtraction and also show two ways to subtract a sum from a number.

In life, we are constantly faced with direct and opposite actions. You can pour water into a mug, you can pour water out. You can go into the house, then leave the house. There are many such examples.

In mathematics, we can also easily find a pair of such opposite actions. This is addition and subtraction.

Rice. 1. Illustration of addition

Subtraction: there were 5 apples, 2 were taken away, 3 remained. The result was subtraction (Fig. 2).

Rice. 2. Subtraction

It is clear that adding and subtracting are opposite actions, thus addition and subtraction are mutually opposite actions.

To perform addition or subtraction, we do not take objects to help us and do not put them in one pile. We solve such a problem abstractly, using numbers and opposite operations.

For example, to subtract 2 from 5, we must understand what is left.

And to do this we need to imagine 5 as the sum of two parts.

And we understand that if we subtract 2, then 3 remains.

The same quantity can be represented and written in different ways. All these methods are equivalent: . We can always use the one that is convenient for us in this case. Now it is convenient for us to imagine that 5 is the sum of 3 and 2. Therefore, if we remove, subtract one part (2), then the second (3) will remain.

How to subtract 7 from 15?

We immediately imagine that . This means that after subtracting 7, 8 remains.

It becomes clear that subtraction is finding an unknown expansion number.

Let's look at the example again. To subtract 2 from the number 5, you need to represent 5 as two terms and find unknown term. This will be the result of the subtraction.

If you need to subtract a number from a number:

This means that the number must be represented as two terms and .

One term is unknown to us. We need to find him. This is the result of subtraction.

It’s clear what to take from the vase more apples what was there is impossible. Therefore, when we talk about subtracting natural numbers, we cannot subtract a larger number from a smaller number. Then there will be other numbers, not just natural ones, and subtracting a larger number from a smaller number will become possible.

Or here’s another reasoning: to subtract means to present it in the form of two terms, but the terms, the parts, cannot be greater than the whole.

But for now the agreement is as follows: from the number we subtract the number , only if not less than . The result will be a new number.

Rice. 3. Names of components when subtracting

The word "difference" is very similar to the word "difference". In fact, what is the difference, how different is the number 15 from the number 7, 15 apples from 7 apples? For 8 apples. That is, the difference between the numbers 15 and 7 is the difference between them.

Thus, on the one hand, the difference is the result of subtracting a smaller number from a larger number. On the other hand, this is how much one number differs from another, the difference between them.

Dad is 36 years old, and mom is 2 years younger. How old is mom?

Subtract 2 from 36.

This is the first type of problem that we solve using subtraction: we know one number, we need to find a second one that is smaller by a known amount. That is, we immediately know the minuend and subtrahend, numbers and .

There are 25 people in the class, 14 of them are girls. How many boys are there in the class?

It is clear that there are only 25 girls and boys. There are 14 girls, an unknown number of boys.

We need to find the unknown term. And searching for an unknown term is already a subtraction task. From 25 you need to subtract 14.

There are 11 boys in the class.

This is the second type of problem, when two numbers are added, one of them is known and the other is not. But the result, the amount, is known.

Known and are highlighted in blue. It is necessary to find the unknown term. But searching for an unknown term is subtraction.

My sister is 12 years old and my brother is 9. How old is my sister? older than brother?

My sister is 3 years older than my brother.

This is the third type of task - comparison task.

There were 17 apples in the vase. Petya took 4 apples, Masha took 3. How many apples are left in the vase?

Solution

Petya took 4, Masha - 3, they took a total of apples. To find how much is left, subtract:

If you write it in one line:

Let's count how many apples were left each time Petya and Masha took apples. Petya took 4, left. Masha took 3 more, left.

Or, in one line, .

There are 10 apples left in the vase.

Both methods are equivalent, the answer is the same. That is, subtracting an amount is the same as subtracting each term of this amount separately.

So, in general, subtracting natural numbers does NOT have the commutative property. Let's write this statement using 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 subtracting the sum of two numbers from a natural number. Let's look at an example that will give us an understanding of this property.

Let's 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 keep another 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 the piggy bank. In this case, the number of coins remaining in our hands is determined by the difference 7−(2+1) .

Now imagine that we have 7 coins, and we put 2 coins into 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 our hands, then in both the first and second cases we have 4 coins. That is, 7−(2+1)=4 and (7−2)−1=4, therefore, 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. Subtracting a given sum of two natural numbers from a given natural number is the same as subtracting the first term of a given sum from a given natural number, and then subtracting the second term from the resulting difference.

Let us 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 if this condition is met, 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 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 met.

The considered property, as well as the combinatory property of addition of natural numbers, make it possible 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.

Let's move on to the next property, which is associated with subtracting a given natural number from a given sum of two natural numbers. Let's look at examples that will help us “see” this property of subtracting a natural number from the sum of two numbers.

Let us have 3 candies in the first pocket, and 5 candies in the second, and let us need to give away 2 candies. We can do it different ways. Let's look at them one by one.

Firstly, we can put all the candies in one pocket, then take out 2 candies from there and give them away. Let us describe these actions mathematically. After we put the candies in one pocket, their number will be determined by the sum 3+5. Now, out of the total number of candies, we will give away 2 candies, while the remaining number of candies 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 remaining in our pocket 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 candies. Consequently, the equalities (3+5)−2=(3−2)+5=3+(5−2) are valid.

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

Having analyzed the results obtained from 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 the other term. It should be noted that the number being subtracted must NOT be greater than the term from which this number is being subtracted.

Let's write down 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, the equality is true (a+b)−c=(a−c)+b, and if the condition is met that b is greater than or equal to c, the equality is true (a+b)−c=a+(b−c). If both a and b are greater than or equal to c, then both of the last equalities are true, and they can be written as follows: (a+b)−c=(a−c)+b= a+(b−c) .

By analogy, we can formulate the property of subtracting a natural number from the sum of three and 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 sounded property, you can imagine that we have many pockets and there are candies in them. Suppose we need to give away 1 candy. It is clear that we can give away 1 candy from any pocket. At the same time, it does not matter from which pocket we give it away, since this does not affect the amount of candy that we will have left.

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

Previously, we studied what natural numbers are and what properties exist in order to perform subtraction. This article presents the basic rules that will help us subtract natural numbers. To ensure that the information is clear and quickly remembered, we have provided theoretical material with detailed exercises and typical examples.

How are addition and subtraction related?

Addition and subtraction are closely related. Subtraction is the inverse of addition. To understand this information, consider a detailed example.

Let's imagine that as a result of adding objects c And b, we get item a . Based on the basics of addition of natural numbers, we can conclude that c + b = a. If we use the commutative property of addition, we can transform the resulting equality as b + c = a. We conclude that if we subtract from a b, then it will remain c. This equality a − b = c will be considered fair. By analogy, we find that by subtracting the number from a c, then it will remain b, that is, a − c = b.

Thanks to the example we looked at above, we can conclude that if the sum of the numbers c And b equal to a, then the number c is the difference of natural numbers b, and the number b– difference of numbers a And c. That is, c = a − b And b = a − c, If c + b = a.

Let's transform this statement and get an important rule.

Definition 1

If the sum of two numbers c And b equal to a, then the difference a−c equal to b, and the difference a − b equal to c.

We can now clearly see that addition and subtraction are inextricably linked. Based on this fact, the concept can be derived.

Definition 2

Subtraction is an action by which one term is found when the sum and the other term are known.

This definition is often used in various examples and tasks.

An addition table can often be used to find the sum of two numbers and to find one term if the sum and the other term are known.

Let's look at this statement with an example. Consider an exercise in which you need to find an unknown term if you know that the second term is equal to 5 , and the sum is equal 8 .

This can be done in two ways. Let's use a graphic illustration in which known numbers are highlighted in red and found numbers in blue.

Let's consider several ways.

First way. It is necessary to find a row in the table, the known term is located in the leftmost cell (take known number 5). After this, you need to find the column that intersects with the found row in the cell. This line must contain a known amount (according to the example, the number 8 ). The number we need to find is located in the top cell of the found column. We conclude that the number 3 – uh then this is the required term.

Second way. It is necessary to find a column in the addition table in the top cell of which the known term is located. We find a line that intersects with a known column in a cell that corresponds known amount. We conclude that the term that needs to be found is located in the leftmost cell of this row.

Since we know that addition and subtraction are closely related, this table can also be used to find the difference of natural numbers. Let's take a closer look this theory For example.

Imagine that you need to subtract the number 7 from the number 16 . We conclude that subtraction comes down to finding the number that adds up to the number 7 will give a number 16 . Let's use the table used above.

Subtracting from the number 16 number 7 , we get the required difference 9 .

In order to use this table, we recommend that you memorize the information and bring the process of finding numbers from the table to automaticity.

How to subtract digits of numbers

Using the addition table that we discussed above, you can subtract tens from tens, hundreds from hundreds, thousands from thousands. The way we can easily work with prime numbers, so, by analogy, you can subtract tens and hundreds. For example, 6 hundred minus 2 hundreds equals 4 hundreds, that is, 600 − 200 = 400 . We can also use the table in other cases.

If we remember that one hundred is 10 tens, one thousand is 10 hundreds, then we can calculate the difference of tens, hundreds, thousands and other numbers.

Let's look at an example.

Example 2

100 − 70 .

Convert numbers as tens. We get ten tens and seven tens. From the addition table we get 10 − 7 = 3 , then the difference 10 tens and 7 tens is equal 3 dozens, that is, 100 − 70 = 30 .

Example 3

It is necessary to calculate the difference 100 000 − 80 000 .

Because 100 000 - This 10 tens of thousands, and 80,000 is 8 tens of thousands, and 10 − 8 = 2 . We get that 100 000 − 80 000 = 20 000 .

Subtracting a natural number from a sum of numbers

To find the difference between the sum of two numbers and a number, you must first calculate the sum from which the number is being subtracted. To simplify the subtraction process, you can use a certain property subtraction. Let's look at a few examples.

Example 4

Must be subtracted from the amount 50 + 8 natural number 20 .

Sum 50 + 8 – is the sum of the digit terms of the number 58 . We are looking for solutions. We use the above subtraction rule: since 20 < 50 , then the equality is true (50 + 8) − 20 = (50 − 20) + 8 . We can conclude that 50 − 20 = 30 ( 5 tens – 2 tens), then (50 − 20) + 8 = 30 + 8 . The required number is 38.

The solution can be represented as a chain of equalities: (50 + 8) − 20 = (50 − 20) + 8 = 30 + 8 = 38 .

Example 5

Must be subtracted from the amount 21 + 8 number 3 . Just like 3 < 21 And 3 < 8 , then the equalities (21 + 8) − 3 = (21 − 3) + 8 and (21 + 8) − 3 = 21 + (8 − 3) are valid.

Let's choose the most suitable calculation option. Subtract from the smaller number. In the example 8 < 21 . So, (21 + 8) − 3 = 21 + (8 − 3) = 21 + 5 = 26 .

Let's complicate the example. It is necessary to calculate the difference of the number 20 from the amount 20 000 + 6 000 + 300 + 50 + 1 . Let's use the property of subtraction that we learned above.

Calculating the difference is quite easy: (20,000 + 6,000 + 300 + 50 + 1) − 20 = 20,000 + 6,000 + 300 + (50 − 20) + 1 = = 20,000 + 6,000 + 300 + 30 + 1 = 26,331.

Let's look at the solution to another example: (107 + 42 + 9) − 3 = 107 + 42 + (9 − 3) = 107 + 42 + 6 = 155 .

Subtracting the sum of numbers from a natural number

To subtract the amount two numbers from a natural number, you need to calculate the sum, and then carry out the subtraction.

You can use the subtraction property given above. Let's look at a few examples.

Example 6

It is necessary to subtract from the number 100 amount 90 + 8 .

According to the property, we get: 100 − (90 + 8) = (100 − 90) − 8 . We find 100 − 90 = 10 .

Let's imagine the calculation as: (100 − 90) − 8 = 10 − 8 = 2 .

Example 7

It is necessary to find the difference of the number 17 and sums of numbers 8 And 4 .

We get that: 17 − (8 + 4) = (17 − 8) − 4 . We use the table and find that 17 − 8 = 9, then (17 − 8) − 4 = 9 − 4 = 5 . The solution can be briefly written as: 17 − (8 + 4) = (17 − 8) − 4 = 9 − 4 = 5 .

Right side of equality a − (b + c) = (a − b) - c sometimes written as a − (b + c) = a − b − c. In this case it is implied that a − b − c = (a − b) − c. Difference 15 − (7 + 2) one can imagine how 15 − 7 − 2 . Calculate the difference - subtract the number from 15 7. Subtract 2 from the result obtained.

Thus, 15 − (7 + 2) = 15 − 7 − 2 = 8 − 2 = 6 .

Using the property of subtraction and the combinatory property of addition, you can find the difference between the sum of two, three or more numbers.

Example 8

You need to subtract from a number 1 000 sum of three numbers of the form 900 + 90 + 1 .

Amount 900 + 90 + 1 let's imagine how 900 And 90 + 1 , that is, 900 + 90 + 1 = 900 + (90 + 1) (refer to the appropriate section for better understanding). We use the subtraction property learned above: 1 000 − (900 + (90 + 1)) = (1 000 − 900) − (90 + 1) . Since 1,000 − 900 = 100, then (1,000 − 900) − (90 + 1) = 100 − (90 + 1). Subtract the amount from the number: 100 − (90 + 1) = (100 − 90) − 1 = 10 − 1 = 9 .

A short summary of the solution is: 1,000 − (900 + 90 + 1) = (1,000 − 900) − (90 + 1) = 100 − (90 + 1) = (100 − 90) − 1 = 10 − 1 = 9

Difference 1 000 − (900 + 90 + 1) may also look like ((1 000 − 900) − 90) − 1 . Another way to write this is as 1 000 − 900 − 90 − 1 . In these cases, the difference of the first two numbers is first found, then the third number is subtracted from the result obtained, and so on.

Example 9

It is necessary to subtract from the number 20 the sum of the numbers 10, 4, 3 and 1 . We get that: 20 − (10 + 4 + 3 + 1) = 20 − 10 − 4 − 3 − 1 = 10 − 4 − 3 − 1 = 6 − 3 − 1 = 3 − 1 = 2 .

Subtracting units from tens, hundreds, thousands

From the number 10 Any number from 1 before 9 . We use the table presented above. But what to do in other cases? It is necessary to represent the minuend as the sum of two terms, one of which is equal 10 , then subtract it from the amount. Let's consolidate our knowledge of the material with an example:

Example 10

Must be subtracted from 60 number 5 .

Number 60 represent it as the sum of two numbers, one of which is equal 10 . We find the second number by subtracting from 60 number 10 . Because 60 − 10 = 50 , That 60 = 50 + 10 . We will replace 60 amount 50 + 10 , getting 60 − 5 = (50 + 10) − 5 . We get that: (50 + 10) − 5 = 50 + (10 − 5) = 50 + 5 = 55 .

Having looked at subtracting ones from tens, let's move on to subtracting ones from hundreds.

To from 100 subtract a number from 1 before 10 need to 100 imagine how 90+10 90 + 10 and use the rule.

Example 11

We need to find the difference 100 − 7 .

Let's imagine 100 How 90 + 10 and execute: 100 − 7 = (90 + 10) − 7 = 90 + (10 − 7) = 90 + 3 = 93 . Let's complicate the example. Subtract from the number 500 number 3 . Let's imagine 500 as a sum. Second term = 500 − 100, that is, 400 . We have 500 = 400 + 100 . 100 = 90 + 10 , 500 = 400 + 90 + 10 .

Thus, 500 − 3 = (400 + 90 + 10) − 3 .

Let's finish the calculation: (400 + 90 + 10) − 3 = 400 + 90 + (10 − 3) = 400 + 90 + 7 = 497.

Let's move on to subtracting units from thousands.

Example 12

It is necessary to calculate the difference 1,000 − 8.

Because 1 000 = 900 + 100 , A 100 = 90 + 10 , That 1 000 = 900 + 90 + 10 .

Then 1 000 − 8 = (900 + 90 + 10) − 8 = 900 + 90 + (10 − 8) = 900 + 90 + 2 = 992 .

Example 13

Must be subtracted from 7 000 unit.

7 000 let's write it as 7 000 = 6 000 + 1 000 = 6 000 + 900 + 100 = 6 000 + 900 + 90 + 10 .

We conclude:
7 000 − 1 = (6 000 + 900 + 90 + 10) − 1 = 6 000 + 900 + 90 + (10 − 1) = 6 000 + 900 + 90 + 9 = 6 999 .

Example 14

It is necessary to calculate the difference 100 000 − 4 .

Because
100 000 = 90 000 + 10 000 = 90 000 + 9 000 + 1 000 = = 90 000 + 9 000 + 900 + 100 = 90 000 + 9 000 + 900 + 90 + 10
That
100 000 − 4 = (90 000 + 9 000 + 900 + 90 + 10) − 4 = = 90 000 + 9 000 + 900 + 90 + (10 − 4) = 90 000 + 9 000 + 900 + 90 + 6 = 99 996 .

Example 15

Must be subtracted from 4 000 000 number 5 .

Because
4 000 000 = 3 000 000 + 1 000 000 = 3 000 000 + 900 000 + 100 000 = = 3 000 000 + 900 000 + 90 000 + 10 000 = 3 000 000 + 900 000 + 90 000 + 9 000 + 1 000 = = 3 000 000 + 900 000 + 90 000 + 9 000 + 900 + 100 = = 3 000 000 + 900 000 + 90 000 + 9 000 + 900 + 90 + 10
That
4 000 000 − 5 = (3 000 000 + 900 000 + 90 000 + 9 000 + 900 + 90 + 10) − 5 = = 3 000 000 + 900 000 + 90 000 + 9 000 + 900 + 90 + (10 − 5) = = 3 000 000 + 900 000 + 90 000 + 9 000 + 900 + 90 + 5 = 3 999 995 .

Subtracting units from arbitrary numbers

Definition 3

To subtract from such a number single digit number, you need to decompose the minuend into digits, and then subtract the number from the sum.

Let's look at typical examples that will help you understand the material.

Example 16

It is necessary to determine the difference between numbers 46 And 2 .

Number 46 present how 40 + 6 , Then 46 − 2 = (40 + 6) − 2 = 40 + (6 − 2) = 40 + 4 = 44 . To make the task more difficult, let's find the difference 46 And 8 . We have 46 − 8 = (40 + 6) − 8. Because 8 more than 6 , That: ( 40 + 6) − 8 = (40 − 8) + 6. We calculate 40 − 8 using the example: 40 − 8 = (30 + 10) − 8 = 30 + (10 − 8) = 30 + 2 = 32 . Then (40 − 8) + 6 = 32 + 6 = 38 . Now let's subtract from 6 047 number 5 . Lay out 6 047 and subtract the number from the sum: 6 047 − 5 = (6 000 + 40 + 7) − 5 = 6 000 + 40 + (7 − 5) = 6 000 + 40 + 2 = 6 042

Let's reinforce our skills with one more example.

Example 17

It is necessary to subtract from the number 2 503 number 8 .

We expand and get: 2 503 − 8 = (2 000 + 500 + 3) − 8 . Because 8 more than 3 , but less than 500 , That (2 000 + 500 + 3) − 8 = 2 000 + (500 − 8) + 3 . Let's calculate the difference 500 − 8 , for this we represent the number 500 as a sum 400 + 100 = 400 + 90 + 10 (if necessary, return to the previous paragraph of this article) and perform the necessary calculations:
500 − 8 = (400 + 90 + 10) − 8 = 400 + 90 + (10 − 8) = 400 + 90 + 2 = 492 . 2 000 + (500 − 8) + 3 = 2 000 + 492 + 3 = 2 495 .

Subtraction from arbitrary natural numbers

To subtract tens and hundreds from a number, you need to represent the minuend as a sum and perform the subtraction. Let's look at this process using several examples.

Example 18

Let's find the difference 400 and 70 .

Let's expand 400 as 300 + 100 . Then 400 − 70 = (300 + 100) − 70 . According to the property, we get: (300 + 100) − 70 = 300 + (100 − 70) = 300 + 30 = 330 . We can also subtract from the number 1 000 number 40 . Let's imagine that 1 000 − 40 = (900 + 100) − 40 = 900 + (100 − 40) = 900 + 60 = 960 .

According to the rule, (7 000 + 900 + 100) − 10 = 7 000 + 900 + (100 − 10) = 7 000 + 900 + 90 = 7 990 .

We use this rule in similar cases.

Example 19

We'll find 400 000 − 70 .

400 000 let's expand it as 300 000 + 90 000 + 9 000 + 900 + 100 , Then
400 000 − 70 = (300 000 + 90 000 + 9 000 + 900 + 100) − 70 = 300 000 + 90 000 + 9 000 + + 900 + (100 − 70) = 300 000 + 90 000 + 9 000 + 900 + 30 = 399 993

Let's use similar principles to calculate hundreds, thousands, and others.

Example 20

We'll find 5 000 − 800 .

Let's imagine 5 000 How 4 000 + 1 000 . Then 5 000 − 800 = (4 000 + 1 000) − 800 . We use the property: (4 000 + 1 000) − 800 = 4 000 + (1 000 − 800) . Since a thousand is ten hundred, then 1 000 − 800 = 200 . Thus, 4,000 + (1,000 − 800) = 4,000 + 200 = 4,200.

This rule can be used for calculations. Remember it, it will be useful to you more than once.

Example 21

Let's find the difference 140 and 40 .

Because 140 = 100 + 40 , That 140 − 40 = (100 + 40) − 40 . We get: (100 + 40) − 40 = 100 + (40 − 40) = 100 + 0 = 100 (40 − 40) = 0 due to properties, and 100 + 0 = 100 .

We'll find 140 – 60 . We have 140 − 60 = (100 + 40) − 60 . Since 60 is more than 40 , That: (100 + 40) − 60 = (100 − 60) + 40 = 40 + 40 = 80 .

Subtracting arbitrary numbers

Let's consider the rule when the subtrahend is decomposed into digits. After representing a number as a sum of digit terms, the subtraction property described above is used. Subtraction begins with units, then tens, hundreds, and so on.

Example 22

Let's calculate 45 − 32 .

Let's break down 32 into digits: 32 = 30 + 2 . We have 45 − 32 = 45 − (30 + 2) . Let's imagine how 45 − (30 + 2) = 45 − (2 + 30) . Now we apply the property of subtracting a sum from a number: 45 − (2 + 30) = (45 − 2) − 30 . It remains to calculate 45 − 2 , then subtract the number 30 .

Once you have mastered the previous rules, you will be able to do this easily.

So, 45 − 2 = (40 + 5) − 2 = 40 + (5 − 2) = 40 + 3 = 43 . Then (45 − 2) − 30 = 43 − 30 . It remains to represent the minuend as a sum of bit terms and complete the calculations: 43 − 30 = (40 + 3) − 30 = (40 − 30) + 3 = 10 + 3 = 13

It is convenient to write the entire solution in the form of a chain of equalities:
45 − 32 = 45 − (2 + 30) = (45 − 2) − 30 = ((40 + 5) − 2) − 30 = = (40 + (5 − 2)) − 30 = (40 + 3) − 30 = (40 − 30) + 3 = 10 + 3 = 13

Let's complicate the example a little.

Subtract the number from 85 18 .

We sort the number into digits 18 , and we get 18 = 10 + 8 . Swap the terms: 10 + 8 = 8 + 10. Now we subtract the resulting sum of bit terms from the number 85 and apply the property of subtracting a sum from a number: 85 − 18 = 85 − (8 + 10) = (85 − 8) − 10 . We calculate the difference in brackets:
85 − 8 = (80 + 5) − 8 = (80 − 8) + 5 = ((70 + 10) − 8) + 5 = (70 + (10 − 8)) + 5 = (70 + 2) + 5 = 70 + 7 = 77

Then (85 − 8) − 10 = 77 − 10 = (70 + 7) − 10 = (70 − 10) + 7 = 60 + 7 = 67

To consolidate the material, we will analyze the solution to another example.

Example 23

Subtract from the number 23 555 number 715 .

Because 715 = 700 + 10 + 5 = 5 + 10 + 700 = 5 + (10 + 700) , then 23,555 − 715 = 23,555 − (5 + 10 + 700) . Subtract the amount from the number as follows: 23 555 − (5 + (10 + 700)) = (23 555 − 5) − (10 + 700) .

Let's calculate the difference in brackets:
23 555 − 5 = (20 000 + 3 000 + 500 + 50 + 5) − 5 = 20 000 + 3 000 + 500 + 50 + (5 − 5) = = 20 000 + 3 000 + 500 + 50 + 0 = 20 000 + 3 000 + 500 + 50 = 23 550 .

Then (23 555 − 5) − (10 + 700) = 23 550 − (10 + 700) .

Once again we turn to the property of subtracting a natural number from a sum: 23 550 − (10 + 700) = (23 550 − 10) − 700 .
(23 550 − 10) − 700 = 23 540 − 700 = (20 000 + 3 000 + 500 + 40) − 700 = = 20 000 + (3 000 − 700) + 500 + 40

Subtract 700 from 3,000 and: 3 000 − 700 = (2 000 + 1 000) − 700 = 2 000 + (1 000 − 700) = 2 000 + 300 = 2 300 , Then 20 000 + (3 000 − 700) + 500 + 40 = 20 000 + 2 300 + 500 + 40 = 22 840 .

Let's look at what geometric point of view subtraction is. We use a coordinate beam. Subtracting the number b from a on the coordinate ray is found as follows: we determine the point, the coordinate is a. Set aside in the direction of the point O single segments in an amount determined by the subtrahend b. So we will find a point on the coordinate ray, the coordinate is equal to the difference a − b. In other words, this is a movement to the left from a point with coordinate a to a distance b, hitting the point with coordinate a − b.

Let's look at subtraction on a coordinate ray using a picture. So we get to the point with coordinate 2 so that 6 − 4 = 2 .

Checking the result of subtraction by addition

Testing the result of subtracting two natural numbers is based on the relationship between subtraction and addition. There we found out that if c + b = a, That a − b = c And a − c = b. If a − b = c, That c + b = a; If a − c = b, That b + c = a. Let us prove the validity of these equalities.

Let from a be put aside b, after which it remains c. This action corresponds to the equality a − b = c. We will return deferred b in place, then we pay a. Then we can talk about the fairness of equality c + b = a.

Now we can formulate a rule that allows us to check the result of subtraction by addition: we need to add the subtrahend to the resulting difference, and the result should be a number equal to the minuend. If the resulting number is not equal to the one being reduced, then an error was made during the subtraction.

All that remains is to analyze the solutions of several examples in which the result of subtraction is checked using addition.

Example 24

50 was subtracted 42 and it was received 6 . Was the subtraction done correctly?

Let's check the resulting subtraction result. To do this, add the subtrahend to the resulting difference: 6 + 42 = 48 (If necessary, study other paragraphs on this topic). Since we received a number that is not equal to the minuend 50 , then it can be argued that the subtraction was carried out incorrectly. It was a mistake.

Example 25

It is necessary to determine the difference 1 024 − 11 and check the result.

We calculate the difference: 1 024 − 11 = 1 024 − (1 + 10) = (1 024 − 1) − 10 = 1 023 − 10 = 1 013 .

Now let's check:

1 013 + 11 = (1 000 + 10 + 3) + (10 + 1) = = 1 000 + 10 + 10 + 3 + 1 = 1 000 + 20 + 4 = 1 024

We received a number equal to the one being reduced, therefore, the difference was calculated correctly. 1 024 − 11 = 1 023 .

Checking the result of a subtraction by subtraction

The correctness of the result of subtracting natural numbers can be checked not only using addition, but also using subtraction. To do this, you need to subtract the found difference from the minuend. This should result in a number equal to the one being subtracted. Otherwise, an error was made in the calculations.

Let's consider this rule in more detail. This will allow you to check the result of subtracting numbers by subtraction. Let's imagine that we have a fruits, including b apples and c pears If we put aside the apples, we will only have c pears, and we have a − b = c. If we put all the pears aside, we would only have b apples, while a − c = b.

Example 26

A number was subtracted from the number 543 343 , the result was the number 200 .

Perform the test.

Let's remember the connection between subtraction and addition: 200 + 343 = 543 . From the minuend 543 we subtract the difference 200 , we get 543 − 200 = (500 + 43) − 200 = (500 − 200) + 43 = 30 + 43 = 343 .

This number is equal to the one being subtracted, the subtraction is done correctly.

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