This is finding one of the terms by the sum and the other term.

The original amount is called reduced, known term - deductible, and the result (i.e., the desired term) is called difference.

Number subtraction properties

1. a - (b + c) = (a - b) - c = (a - c) - b ;

2. (a + b) - c = (a - c) + b = a + (b - c) ;

3. a - (b - c) = (a - b) + c .

For a visual representation of arithmetic operations (both addition and subtraction), you can use number line- this is a straight line, which consists of a point of origin (this point corresponds to zero) and two rays propagating from it, one of which corresponds to positive numbers, and the other to negative ones.

Example of subtraction on the number line

On this number line, you can see that the numbers to the left of 0 have negative meaning. Subtracting from a negative number (in this case-1) one three times, we get the number -1.

Subtracting from the positive number 4, the positive number 3 (or a negative number-1 three times), we get one

Example

4 - 3 = 1 ;	3 - 4 = - 1 ;
-1 -3 = - 4 ;

Subtraction of numbers by a column

Units are subtracted first, then tens, hundreds, and so on. The difference of each column is written below it. If necessary, from the adjacent left column (i.e. from the highest order) is engaged 1 .

Let's take a look at a few examples of columnar subtraction below.

An example of subtracting two-digit numbers by a column

Example of subtracting three-digit numbers in a column

The principle of subtracting three-digit numbers is similar to the method of subtracting two-digit numbers, in this case the numbers are no longer tens, but hundreds.

An example of subtracting four-digit numbers by a column

The principle of subtracting four-digit numbers is similar to the method of subtracting three-digit numbers, in this case the numbers are no longer hundreds, but thousands.

It is very important even in Everyday life. Subtraction can often come in handy when counting change in a store. For example, you have one thousand (1000) rubles with you, and your purchases amount to 870. You, without paying yet, will ask: “How much change will I have?”. So, 1000-870 will be 130. And there are many different such calculations and without mastering this topic, it will be difficult in real life. Subtraction is an arithmetic operation during which the second number is subtracted from the first number, and the result will be the third.

The addition formula is expressed as follows: a - b = c

a- Vasya initially had apples.

b- the number of apples given to Petya.

c- Vasya has apples after the transfer.

Substitute in the formula:

Subtraction of numbers

Subtracting numbers is easy for any first grader to master. For example, 5 must be subtracted from 6. 6-5=1, 6 is greater than 5 by one, which means that the answer will be one. You can add 1+5=6 to check. If you are not familiar with addition, you can read ours.

Big number is divided into parts, let's take the number 1234, and in it: 4-ones, 3-tens, 2-hundreds, 1-thousands. If you subtract units, then everything is easy and simple. But let's take an example: 14-7. In the number 14: 1 is ten, and 4 is units. 1 ten - 10 units. Then we get 10 + 4-7, let's do this: 10-7 + 4, 10 - 7 \u003d 3, and 3 + 4 \u003d 7. Correct answer found!

Let's consider an example 23 -16. The first number is 2 tens and 3 ones, and the second is 1 tens and 6 ones. Let's represent the number 23 as 10+10+3 and 16 as 10+6, then represent 23-16 as 10+10+3-10-6. Then 10-10=0, 10+3-6 remains, 10-6=4, then 4+3=7. Answer found!

Similarly, it is done with hundreds and thousands

Column subtraction

Answer: 3411.

Subtraction of fractions

Imagine a watermelon. A watermelon is one whole, and cutting in half, we get something less than one, right? Half unit. How to write it down?

½, so we denote half of one whole watermelon, and if we divide the watermelon into 4 equal parts, then each of them will be denoted ¼. And so on…

how to subtract fractions

Everything is simple. Subtract from 2/4 ¼-th. When subtracting, it is important that the denominator (4) of one fraction coincides with the denominator of the second. (1) and (2) are called numerators.

So let's subtract. Make sure the denominators are the same. Then we subtract the numerators (2-1)/4, so we get 1/4.

Subtraction limits

Subtracting limits is not difficult. Here, a simple formula is sufficient, which says that if the limit of the difference of functions tends to the number a, then this is equivalent to the difference of these functions, the limit of each of which tends to the number a.

Subtraction of mixed numbers

A mixed number is an integer with a fractional part. That is, if the numerator is less than the denominator, then the fraction is less than one, and if the numerator is greater than the denominator, then the fraction is greater than one. A mixed number is a fraction that is greater than one and has an integer part highlighted, let's use an example:

To subtract mixed numbers, you need:

Bring fractions to a common denominator.

Enter the integer part into the numerator

Make a calculation

subtraction lesson

Subtraction is an arithmetic operation, during which the difference of 2 numbers is searched and the answers are the third. The addition formula is expressed as follows: a - b = c.

You can find examples and tasks below.

At fraction subtraction it should be remembered that:

Given a fraction 7/4, we get that 7 is greater than 4, which means that 7/4 is greater than 1. How to select the whole part? (4+3)/4, then we get the sum of fractions 4/4 + 3/4, 4:4 + 3/4=1 + 3/4. Outcome: one whole, three fourths.

Subtraction Grade 1

The first class is the beginning of the journey, the beginning of learning and learning the basics, including subtraction. Education should be conducted in the form of a game. Always in the first class, calculations begin with simple examples on apples, sweets, pears. This method is used not in vain, but because children are much more interested when they are played with. And this is not the only reason. Children saw apples, sweets and the like very often in their lives and dealt with the transfer and quantity, so it will not be difficult to teach the addition of such things.

Subtraction tasks for first graders can come up with a whole cloud, for example:

Task 1. In the morning, walking through the forest, the hedgehog found 4 mushrooms, and in the evening, when he came home, the hedgehog ate 2 mushrooms for dinner. How many mushrooms are left?

Task 2. Masha went to the store for bread. Mom gave Masha 10 rubles, and bread costs 7 rubles. How much money should Masha bring home?

Task 3. In the morning there were 7 kilograms of cheese on the counter in the store. Before lunch, visitors bought 5 kilograms. How many kilograms are left?

Task 4. Roma took out the sweets that his dad gave him into the yard. Roma had 9 candies, and he gave 4 to his friend Nikita. How many candies does Roma have left?

First-graders mostly solve problems in which the answer is a number from 1 to 10.

Subtraction Grade 2

The second class is already higher than the first, and, accordingly, examples for solving too. So let's get started:

Numerical assignments:

Single digits:

1. 10 - 5 =
2. 7 - 2 =
3. 8 - 6 =
4. 9 - 1 =
5. 9 - 3 - 4 =
6. 8 - 2 - 3 =
7. 9 - 9 - 0 =
8. 4 - 1 - 3 =

Double figures:

1. 10 - 10 =
2. 17 - 12 =
3. 19 - 7 =
4. 15 - 8 =
5. 13 - 7 =
6. 64 - 37 =
7. 55 - 53 =
8. 43 - 12 =
9. 34 - 25 =
10. 51 - 17 - 18 =
11. 47 - 12 - 19 =
12. 31 - 19 - 2 =
13. 99 - 55 - 33 =

Text tasks

Subtraction 3-4 grade

The essence of subtraction in grades 3-4 is subtraction in a column of large numbers.

Consider the example 4312-901. To begin with, let's write the numbers one under the other, so that from the number 901 the unit is under 2, 0 under 1, 9 under 3.

Then we subtract from right to left, that is, from the number 2, the number 1. We get the unit:

Subtracting nine from three, you need to borrow 1 ten. That is, subtract 1 ten from 4. 10+3-9=4.

And since 4 took 1, then 4-1 = 3

Answer: 3411.

Subtraction Grade 5

Fifth grade is the time to work on complex fractions with different denominators. Let's repeat the rules: 1. Numerators are subtracted, not denominators.

So let's subtract. Make sure the denominators are the same. Then we subtract the numerators (2-1)/4, so we get 1/4. When adding fractions, only the numerators are subtracted!

2. To subtract, make sure the denominators are equal.

If there is a difference between fractions, for example, 1/2 and 1/3, then you will have to multiply not one fraction, but both to bring to a common denominator. The easiest way to do this is to multiply the first fraction by the denominator of the second, and the second fraction by the denominator of the first, we get: 3/6 and 2/6. Add (3-2)/6 and get 1/6.

3. Reducing a fraction is done by dividing the numerator and denominator by the same number.

The fraction 2/4 can be reduced to the form ½. Why? What is a fraction? ½ \u003d 1: 2, and if you divide 2 by 4, then this is the same as dividing 1 by 2. Therefore, the fraction 2/4 \u003d 1/2.

4. If the fraction is greater than one, then you can select the whole part.

Given a fraction 7/4, we get that 7 is greater than 4, which means that 7/4 is greater than 1. How to select the whole part? (4+3)/4, then we get the sum of fractions 4/4 + 3/4, 4:4 + 3/4=1 + 3/4. Outcome: one whole, three fourths.

Subtraction presentation

The link to the presentation is below. The presentation covers the basics of sixth grade subtraction:Download Presentation

Presentation of addition and subtraction

Examples for addition and subtraction

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In order to subtract one number from another, we place the subtrahend under the minuend, as follows: units under units, tens under tens. For example, let's take a two-digit number as a minuend, and a single-digit number as a subtrahend.

7 – 5 = 2 we write the result under the units.

Now we subtract tens from tens, but the subtrahend does not have tens, so we omit the ten of the reduced in response.

27 – 5 = 22

Now let's take both two-digit numbers:

Subtract the units of the subtrahend from the units of the minuend:

6 – 4 = 2 write the result under the units

Now subtract the tens of the subtrahend from the tens of the minuend:

8 – 3 = 5 we write the result under tens.

As a result, we get the difference:

86 – 34 = 52

Subtraction with the transition through the ten

Let's try to find the difference between the following numbers:

Subtract units. It is impossible to subtract 9 from 7, we take one ten from the tens of the reduced one. In order not to forget, we put a dot over the tens.

17 – 9 = 8

Now subtract tens from tens. The subtrahend has no tens, but we borrowed one ten from the minuend:

2 tens - 1 tens = 1 tens

As a result, we get the difference:

27 – 9 = 18

Now, for example, take three-digit numbers:

Subtract units. 2 less 8 , so we take one ten of the tens of the reduced one: 2 + 10 = 12 (we write 10 above the ones). In order not to forget, we put a dot over the tens.

12 – 8 = 4 the result is written under the units.

We occupied one ten of the tens for units, which means that in the reduced one there are no longer three tens, but two ( 3 tens - 1 tens = 2 tens).

Two tens less than six, take one hundred or 10 tens out of hundreds ( 2 tens + 10 tens = 12 tens write 10 over the tens of the minuend), and in order not to forget, we put an end to the hundreds. Subtract tens:

12 tens - 6 tens = 6 tens The result is written under the tens.

We occupied one hundred out of hundreds reduced for tens, which means we don’t have 9 hundreds, and 8 hundreds ( 9 hundreds - 1 hundred = 8 hundreds). Subtract hundreds:

8 hundreds - 7 hundreds = 1 hundred . We write the result under hundreds.

As a result, we get:

932 – 768 = 164

Let's complicate the task. What to do if in the category from which you need to take ten, is equal to zero? For example:

We start with units. 2 less 8 , that is, it is necessary to take from tens. But for a decrease in tens 0 , which means that for tens you need to borrow from hundreds. In the hundreds place in the minuend too 0 , borrow from thousands. In order not to forget, we put a point over thousands.

In the hundreds of diminishing remains 9 , since we take one hundred for tens: 10 – 1 = 9 write 9 over hundreds.

Remains in the tens too 9 , since we took one ten for units: 10 – 1 = 9 write 9 over tens, and over units we write 10 .

Counting units:

12 – 8 = 4 write the result under the units.

Remaining in tens of minuends 9 , we consider:

9 – 6 = 3 write the result under tens.

Hundreds of diminishing left 9 , subtracted has no hundreds, omit 9 hundreds in response.

In the rank of thousands of diminished was 1 , we occupied it (dot over thousands), so there are no more thousands left. As a result, we get:

1002 – 68 = 934

So let's sum it up.

To find the difference between two numbers (column subtraction) :

1. we put the subtrahend under the minuend, we write units under units, tens under tens, and so on.
2. Subtract bit by bit.
3. If you need to take a ten from the next category, then put a dot over the category from which you borrowed. Above the category for which we occupy, we put 10.
4. If the digit from which we borrow is 0, then for it we borrow from the next digit of the reduced, over which we put a dot. Above the category for which they occupied, we put 9, since one ten was occupied.

How to subtract in a column

Subtraction of multi-digit numbers is usually performed in a column, writing the numbers one below the other (decreasing from above, subtracted from below) so that the digits of the same digits are one under the other (units under units, tens under tens, etc.). An action sign is placed between the numbers on the left. Draw a line under the subtrahend. The calculation begins with the discharge of units: units are subtracted from units, then from tens - tens, etc. The result of the subtraction is written under the line:

Consider an example when in some place the digit of the minuend is less than the digit of the subtrahend:

We cannot subtract 9 from 2, what should we do in this case? In the category of units, we have a shortage, but in the category of tens, the reduced one already has 7 tens, so we can transfer one of these tens to the category of units:

In the category of units, we had 2, we threw a dozen, it became 12 units. Now we can easily subtract 9 from 12. We write 3 under the line in the units place. In the tens place, we had 7 units, we threw one of them into simple units, 6 tens remained. We write under the line in the tens place 6. As a result, we got the number 63:

Subtraction by a column is usually not written down in such detail, instead, a dot is placed above the digit of the digit, from which the unit will be occupied, so as not to remember which digit will need to be additionally subtracted by the unit:

At the same time, they say this: you can’t subtract 9 from 2, we take a unit, we subtract 9 from 12 - we get 3, we write 3, we had 7 units in the tens place, we threw one, 6 left, we write 6.

Now consider column subtraction from numbers containing zeros:

Let's start subtracting. We subtract 3 from 7, write 4. We cannot subtract 5 from zero, so we are forced to take a unit in the highest digit, but we also have 0 in the highest digit, so for this digit we are also forced to take in a higher digit. We take a unit from the category of thousands, we get 10 hundreds:

We take one of the units of the hundreds digit to the least significant digit, we get 10 tens. Subtract 5 from 10, write 5:

In the hundreds place, we have 9 units left, so we subtract 6 from 9, write 3. In the thousands place, we had a unit, but we spent it on the lower digits, so zero remains here (you don’t need to write it down). As a result, we got the number 354:

Such a detailed record of the solution was given to make it easier to understand how subtraction by a column is performed from numbers containing zeros. As already mentioned, in practice the solution is usually written like this:

And all the mentioned actions are performed in the mind. To make subtraction easier, remember a simple rule:

If there is a dot above zero when subtracting, zero becomes 9.

Column Subtraction Calculator

This calculator will help you subtract numbers by a column. Just enter the minuend and subtrahend and click the Calculate button.

It is convenient to carry out a special method, which is called column subtraction or column subtraction. This method of subtraction justifies its name, since the minuend, the subtrahend and the difference are written in a column. Intermediate calculations are also carried out in columns corresponding to the digits of the numbers.

Convenience of subtraction natural numbers column is the simplicity of calculations. Calculations come down to using the addition table and applying the subtraction properties.

Let's see how column subtraction is performed. We will consider the subtraction process together with the solution of examples. So it will be clearer.

Page navigation.

What do you need to know to subtract by a column?

To subtract natural numbers in a column, you need to know, firstly, how subtraction is performed using the addition table.

Finally, it does not hurt to repeat the definition of the discharge of natural numbers.

Subtraction by a column on examples.

Let's start with the recording. The minuend is written first. Below the minuend is the subtrahend. Moreover, this is done in such a way that the numbers are one under the other, starting from the right. A minus sign is placed to the left of the recorded numbers, and a horizontal line is drawn below, under which the result will be recorded after the necessary actions have been taken.

Here are some examples of correct entries when subtracting by a column. Write down the difference in a column 56−9 , difference 3 004−1 670 , as well as 203 604 500−56 777 .

So, with the record sorted out.

We turn to the description of the process of subtraction by a column. Its essence lies in the sequential subtraction of the values ​​of the corresponding digits. First, the values ​​of the units digit are subtracted, then the values ​​of the tens digit, then the values ​​of the hundreds digit, and so on. The results are recorded under the horizontal line at the appropriate places. The number that is formed under the line after the completion of the process is the desired result of subtracting the two original natural numbers.

Imagine a diagram illustrating the process of subtraction by a column of natural numbers.

The above scheme gives a general picture of the subtraction of natural numbers by a column, but it does not reflect all the subtleties. We will deal with these subtleties when solving examples. Let's start with the simplest cases, and then we will gradually move towards more complex cases, until we figure out all the nuances that can occur when subtracting by a column.

Example.

First, subtract a column from the number 74 805 number 24 003 .

Solution.

Let's write these numbers as required by the column subtraction method:

We start by subtracting the values ​​​​of the digits of units, that is, we subtract from the number 5 number 3 . From the addition table we have 5−3=2 . We write the results obtained under the horizontal line in the same column in which the numbers are located 5 and 3 :

Now subtract the values ​​of the tens digit (in our example, they are equal to zero). We have 0−0=0 (we mentioned this property of subtraction in the previous paragraph). We write the resulting zero under the line in the same column:

Move on. Subtract the values ​​of the hundreds place: 8−0=8 (according to the property of subtraction, voiced in the previous paragraph). Now our entry will look like this:

Let's move on to subtracting the thousands place values: 4−4=0 (these are properties of subtraction of equal natural numbers). We have:

It remains to subtract the values ​​of the tens of thousands place: 7−2=5 . We write the resulting number under the line on Right place:

This completes the column subtraction. Number 50 802 , which turned out below, is the result of subtracting the original natural numbers 74 805 and 24 003 .

Consider the following example.

Example.

Subtract a column from the number 5 777 number 5 751 .

Solution.

We do everything in the same way as in the previous example - we subtract the values ​​of the corresponding digits. After completing all the steps, the entry will look like this:

Under the line we got a number in the record of which there are numbers on the left 0 . If these numbers 0 discard, then we get the result of subtracting the original natural numbers. In our case, we discard two digits 0 obtained on the left. We have: difference 5 777−5 751 is equal to 26 .

Up to this point, we have subtracted natural numbers whose records consist of the same number of characters. Now, using an example, we will figure out how natural numbers are subtracted in a column when there are more signs in the record of the reduced than in the record of the subtrahend.

Example.

Subtract from the number 502 864 number 2 330 .

Solution.

We write the minuend and the subtrahend in a column:

Subtract the values ​​of the unit digit one by one: 4−0=4 ; followed by tens: 6−3=3 ; further - hundreds: 8−3=5 ; further - thousand: 2−2=0 . We get:

Now, to complete the column subtraction, we still need to subtract the values ​​of the tens of thousands place, and then the values ​​of the hundreds of thousands place. But from the values ​​of these digits (in our example, from the numbers 0 and 5 ) we have nothing to subtract (since the subtracted number 2 330 does not have digits in these digits). How to be? Very simple - the values ​​​​of these bits are simply rewritten under the horizontal line:

On this subtraction by a column of natural numbers 502 864 and 2 330 completed. The difference is 500 534 .

It remains to consider the cases when, at some step of column subtraction, the value of the digit of the reduced number is less than the value of the corresponding digit of the subtrahend. In these cases, you have to "borrow" from the senior ranks. Let's understand this with examples.

Example.

Subtract a column from the number 534 number 71 .

Solution.

At the first step, subtract from 4 number 1 , we get 3 . We have:

In the next step, we need to subtract the values ​​of the tens digit, that is, from the number 3 subtract the number 7 . Because 3<7 , then we cannot perform the subtraction of these natural numbers (the subtraction of natural numbers is defined only when the subtrahend is not greater than the minuend). What to do? In this case, we take 1 unit from the highest order and "exchange" it. In our example, "exchange" 1 a hundred per 10 tens. To visually reflect our actions, we put a thick dot over the number in the hundreds place, and over the number in the tens place we write the number 10 using a different color. The entry will look like this:

We add received after the "exchange" 10 tens to 3 available tens: 3+10=13 , and subtract from this number 7 . We have 13−7=6 . This number 6 write under the horizontal line in its place:

Let's move on to subtracting the values ​​of the hundreds place. Here we see a dot above the number 5, which means that from this number we took one “for exchange”. That is, now we have 5 , a 5−1=4 . From number 4 nothing else needs to be subtracted (since the original subtracted number 71 does not contain digits in the hundreds place). Thus, under the horizontal line we write the number 4 :

So the difference 534−71 is equal to 463 .

Sometimes, when subtracting by a column, you have to “exchange” units from the highest digits several times. In support of these words, we analyze the solution of the following example.

Example.

Subtract from natural number 1 632 number 947 column.

Solution.

In the first step, we need to subtract from the number 2 number 7 . Because 2<7 , then you immediately have to "exchange" 1 dozen on 10 units. After that, from the sum 10+2 subtract the number 7 , we get (10+2)−7=12−7=5 :

In the next step, we need to subtract the tens digit values. We see that over the number 3 worth a point, that is, we have not 3 , a 3−1=2 . And from this number 2 we need to subtract the number 4 . Because 2<4 , then again you have to resort to "exchange". But now we are exchanging 1 a hundred per 10 tens. In this case, we have (10+2)−4=12−4=8 :

Now we subtract the values ​​of the hundreds place. From the number 6 unit was occupied in the previous step, so we have 6−1=5 . From this number we need to subtract the number 9 . Because 5<9 , then we need to "exchange" 1 a thousand per 10 hundreds. We get (10+5)−9=15−9=6 :

The last step remains. From the one in the thousands place we borrowed in the previous step, so we have 1−1=0 . We do not need to subtract anything else from the resulting number. This number is written under the horizontal line: