Subtraction- this is the arithmetic operation inverse to addition, by means of which as many units are subtracted (subtracted) from one number as they are contained in another number.
The number to be subtracted from is called reduced, the number that specifies how many units to subtract from the first number, is called deductible. The number resulting from subtraction is called difference(or remainder).
Let's take subtraction as an example. There are 9 candies on the table, if you eat 5 candies, then there will be 4 of them. The number 9 is reduced, 5 is subtracted, and 4 is the remainder (difference):
The - (minus) sign is used to write subtraction. It is placed between the minuend and the subtrahend, while the minuend is written to the left of the minus sign, and the subtrahend is written to the right. For example, the entry 9 - 5 means that the number 5 is subtracted from the number 9. To the right of the subtraction entry, put the sign = (equal), after which the result of the subtraction is written. Thus, the complete subtraction entry looks like this:
This entry reads as follows: the difference between nine and five is four, or nine minus five is four.
In order to get a natural number or 0 as a result of subtraction, the minuend must be greater than or equal to the subtrahend.
Consider how, using the natural series, you can perform subtraction and find the difference of two natural numbers. For example, we need to calculate the difference between the numbers 9 and 6, mark the number 9 in the natural series and count 6 numbers to the left from it. We get the number 3:
Subtraction can also be used to compare two numbers. Wanting to compare two numbers with each other, we ask ourselves how many units one number is more or less than the other. To find out, you need to subtract the smaller number from the larger number. For example, to find out how much 10 is less than 25 (or how much 25 is more than 10), you need to subtract 10 from 25. Then we find that 10 is less than 25 (or 25 is more than 10) by 15 units.
Subtraction check
Consider the expression
where 15 is the minuend, 7 is the subtrahend, and 8 is the difference. To find out if the subtraction was performed correctly, you can:
- add the subtrahend with the difference, if it turns out to be reduced, then the subtraction was performed correctly:
- subtract the difference from the minuend, if the subtrahend is obtained, then the subtraction was performed correctly:
The article will introduce the reader to the concepts of "difference of numbers", "subtracted" and "reduced".
In arithmetic, there are only four basic operations, which we call addition, multiplication, subtraction and division. Such actions are the basis of all mathematics - they allow us to carry out all calculations, both simple and the most complex. by the most simple actions Addition and subtraction are considered to be opposite to each other. True, we also use the word "addition" in everyday life.
We may come across the phrase “add efforts”, for example, when we need to do some work together. But with the term "subtraction" the situation is a little more complicated, and in conversation it is less common. We rarely hear expressions like " minuend», « subtrahend», « difference". But in today's article we will talk about them in detail from the point of view of mathematics.
What does the number to be reduced, the number to be subtracted, and the difference of numbers mean?
What does the number to be reduced, the number to be subtracted, and the difference of numbers mean? As you know, many scientific terms and expressions are taken from other languages, more often Greek and Latin. But the words that will be discussed below have Russian origin, so it will be easier for us to disassemble them.
For example, what can be said about the difference between numbers? If we pay attention to the root of the word "difference", then we will see, for example, its cognate word "difference". And if we are talking about mathematics, then there is nothing to think about - the word "difference" means the difference between some numbers, or rather, two numbers. The difference shows us how much one value is greater than the other or, conversely, the second is less than the first. Strictly in mathematics, this looks like the result of subtraction.
Let's take an example right away. Suppose the barmaid carries eight pies on a tray. She distributed five of them to visitors. How many pies will the barmaid have left on her tray? If you subtract 5 from 8, you get - 3. Now let's write this mathematically:
- 8 – 5 = 3
So the difference between eight and five is three. Now we understand what the term "difference" is.
Attention: If two numbers are equal to each other, then there is no difference between them, it is equal to zero (8 - 8 = 0).
Now we should find out what is subtracted and reduced. Let us again present the meaning of the words according to their meaning. What can be the reduced number? The minuend is the number that decreases when subtracted. Subtract another number from this number. What is subtractive? The subtrahend is just the number that we subtract from the minuend.
Let's go back to the barmaid example. We remember how five was taken away from eight, and we got three. We found out that three is the difference between these two numbers. Now it is not difficult for us to understand that 8 is the number to be reduced, and 5 is the number to be subtracted.
How to find minuend and subtract number?
We have already figured out how to find the difference in numbers in mathematics. It's pretty simple. But can we find the minuend and subtrahend number if one number is unknown? Of course we can, since we will know the other two numbers. For example, how can we find a decreasing number? If we know the value of the difference and the subtrahend, then the sum of these two numbers is equal to the minuend:
- Y - 10 = 18, where Y is the number to be reduced
- So Y = 18 + 10
- 18 + 10 = 28
- Y=28
The subtrahend is just as easy. If we know the difference and the reduced, then we will get the subtrahend by subtracting the difference from the reduced number:
- 28 - B = 10, where B is the number to be subtracted
- So B = 28 - 10
- 28 – 10 = 18
- B=18
Video: Reduced, Subtracted, Difference
The difference or subtraction of integers is directly related to the topic of addition of integers. After all, knowing the sum and one of the terms, you can find the second term. Consider an example:
We have 10 apples in the basket. The first time 2 apples were added to the basket, how many apples were added to the basket the second time to end up with 10 apples?
Let x be the number of apples added a second time. If we add two apples to x, we get 10 apples. Mathematically, the entry will look like this:
to find the variable x, you need to remove 2 apples from the basket or subtract one known term 2 from the sum 10.
That is, the variable x=8.
Definition:
The difference of two integers is the integer that, when added to the subtrahend, gives the minuend.
The difference between integers a and b is denoted as a-b.
Differencea-b is the sum of the numbersa and opposite numberb.
a-b=a+(-b)
where b and –b are opposite numbers.
Example:
5-2=5+(-2)=3
Subtraction of positive integers in examples.
Example:
Subtract from the integer 12 the number 5.
Decision:
According to the rule of difference, we must replace the subtracted 5 with the opposite number, that is, -5 and execute.
Example:
From the number 37, subtract the number 56.
Decision:
It is necessary to replace the subtracted number 56 with the opposite number, that is, the number -56 and perform the addition of integers with different signs.
37-56=37+(-56)=-21
Example:
Subtract 7 from -4.
Decision:
We replace the subtracted number 7 with the opposite number -7 and add from according to the rule
4-7=-4+(-7)=-11
Subtraction of negative integers in examples.
Example:
Find the difference between the numbers 6 and -8.
Decision:
According to the rule of difference, you need to replace the subtracted -8 with the opposite number +8 or 8 and calculate the sum of integers. We get:
Subtract -10 from the integer -14.
It is necessary to replace the subtracted -10 with the opposite number +10 or 10 according to the rule for subtracting integers and then perform the addition.
14-(-10)=-14+10=-4
Subtract zero from integers.
If you subtract zero from an integer, then the number does not change..
Consider an example:
3-0=3+0=3
a-0=a
If we subtract zero from zero, we get zero.
Subtraction of identical integers.
Consider the problem:
Misha received 2 sweets from his mother and he immediately treated his friend Sasha with two sweets. How many sweets does Misha have left?
Decision:
Misha received 2 candies and gave away 2 candies, mathematically it can be written as follows:
Answer: Misha has 0 candies left.
That is, if you do Subtracting equal numbers results in zero.
Checking the result of subtraction.
How to check if you have found the difference of two integers correctly?
The answer is simple, it lies in the very definition of the difference of two integers. Need add the difference with the subtrahend, we get the minuend. The verbal formula would look like this:
Difference+Subtracted=Reduced
Example:
19-5=14
19 is our reduced;
5 - subtracted;
14 - difference.
Let's check:
We add the minuend to the difference, if the subtraction was done correctly, we get the minuend.
Another example:
Perform a subtraction test 12-23=-11
12 - reduced;
23 - subtracted;
-11 - difference.
Let's check the subtraction:
Difference+Subtracted=Reduced
There are four basic arithmetic operations: addition, subtraction, multiplication and division. They are the basis of mathematics, with their help all other, more complex calculations are performed. Addition and subtraction are the simplest of them and are mutually opposite. But with the terms used in addition, we often encounter in life.
We are talking about the "combination of efforts" in the effort to jointly obtain the desired result, about the "terms success" etc. The names associated with subtraction remain within the bounds of mathematics, rarely appearing in everyday speech. Therefore, the words "subtracted", "reduced", "difference" are less common. The rule of finding each of these components can be applied only if the meaning of these names is understood.
Unlike many scientific terms that are of Greek, Latin or Arabic origin, in this case words with Russian roots are used. So it is not difficult to understand their meaning, which means it is easy to remember what is denoted by what term.
If you look closely at the name itself, it becomes noticeable that it is related to the words "different", "difference". From this it can be concluded that what is meant is the established difference between the quantities.
This concept in mathematics means:
- the difference between two numbers;
- it is a measure of how much one quantity is greater or less than another;
- this is the result obtained when subtracting - such a definition is offered by the school curriculum.
Note! If the quantities are equal to each other, then there is no difference between them. So their difference is zero.
What is minuend and subtrahend
As the name suggests, less is what is done less. And you can make the quantity smaller by subtracting a part from it. Thus, a diminished number is a number from which a part is taken away.
Subtracted, respectively, is the number that is subtracted from it.
| Minuend | Subtrahend | Difference | ||
| 18 | — | 11 | = | 7 |
| 14 | — | 5 | = | 9 |
| 26 | — | 22 | = | 4 |
Useful video: reduced, subtracted, difference
Rules for finding an unknown element
Having understood the terms, it is easy to establish by which rule each of the elements of subtraction is located.
Since the difference is the result of this arithmetic operation, it is found using this operation, no other rules are required here. But they are there in case the other term of the mathematical expression is unknown.
How to find the minuend
This term, as it was found out, refers to the amount from which the part was subtracted. But if one was subtracted, and the other remained in the end, therefore, the number consists of these two parts. It turns out that you can find the unknown reduced by adding two known elements.
So, in this case, to find the unknown, you should add the subtrahend and the difference:
Likewise in all such cases:
| ? | – | 5 | = | 9 |
| 9 | + | 5 | = | 14 |
| ? | – | 22 | = | 4 |
| 4 | + | 22 | = | 26 |
How to find subtrahend
If the whole consists of two parts (in this case, quantities), then subtracting one of them will result in the second. Thus, to find the unknown subtrahend, it is enough to subtract the difference from the whole instead.
Other similar examples are solved by the same rule.
| 14 | – | ? | = | 9 |
| 14 | – | 9 | = | 5 |
Definition: Subtraction is an operation by which the sum and one of the terms are used to find the second term.
For example:
if 55 + 35 = 90,
then 90 - 35 = 55.
AT general view:
if a + b = c,
then c - b = a.
Action subtraction checked by addition. The number from which we subtract is called the minuend, and the number that we subtract is called the subtrahend. The result of the subtraction action is the difference.
The subtrahend may not be one number, but the sum of several numbers, then the difference can also be determined according to the following rule, which is most often used in the calculation.
To calculate in a convenient way is to apply the laws of addition to specific numbers so that the very process of calculating the unknown is simplified (for example, use the table of additions to ten by digits, avoid when calculating the transition through ten, etc.).
Rule 1. To subtract the sum from a number, you can subtract one term from it, and subtract the second term from the result (difference).
For example:
126 - (56 + 30) = (126 - 56) - 30 = 40.
In general:
a - (b + c) \u003d (a - b) - c.
Rule 2. To subtract a number from the sum, you can subtract it from one of the terms and add the second term to the result.
Rule 2 can be used when calculating natural numbers only if one of the terms is greater than the number to be subtracted.
For example:
(71 + 7) - 51 \u003d (71 - 51) + 7 \u003d 20 + 7 \u003d 27, but (71 + 7) - 51 \u003d (7 - 51) + 71, since the difference (7 - 51) is not natural number.
In general terms: (a + b) - c \u003d (a - c) + b.
These difference properties are used to check the correctness of calculations during subtraction.
For example: 136 - 82 = 54.
Calculation check:
1) 54 + 82 = 136;
What is the difference between numbers in mathematics and how to find the difference between numbers
In this article, we will look at what a difference in numbers is in mathematics, and how a person interested in this science can find the difference in numbers.
What is the difference between numbers in mathematics
Subtraction is one of 4 arithmetic operations. It is used to designate mathematical sign"−" (minus). Subtraction is the opposite of the operation of addition.
The subtraction operation is generally written as follows:
Here the difference of numbers will be the number 4. Therefore, difference between any numbers A and B this is the number C, which, when added to B, will add up to A (4, when added to 2, gives 6 - so 4 is the difference between 6 and 2).
How to find the difference between numbers
From the definition itself, it follows how to calculate the difference between two numbers. With small numbers, you can do it in your mind. children in primary school are taught in the following way. Imagine that you have 5 apples and 3 of them are taken away. How much do you have left? That's right - 2 apples. Gradually, you will bring the calculations to automatism and will immediately give the answer.
However, for numbers above 50, this visual representation stops working. A large number of objects is hard to imagine in the mind, so another method comes to the rescue here:
Calculation of the difference in a column
Schoolchildren learn this method as part of a mathematics course, usually in the second or third grade. Adults who use a calculator often forget how to count in a column. However, a calculator is not always at hand. Refresh your school knowledge by watching this video.
Calculating the difference in a column - video
This method is also applicable when you need to subtract a larger number from a smaller one. AT real life this is usually not required, but may be useful when solving mathematical problems.
Suppose in the example "A − B = C" B is greater than A. Then C will be negative. To calculate the difference, "unfold" the example: calculate the value B − A. When you finish calculating this difference, you will get the number C, only with the opposite sign: it will be greater than zero. To complete the calculation, add a minus sign to the front of it. The result obtained is a negative number C, and will be the desired value of the difference A − B.
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What is the difference of numbers
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The difference of some numbers is the result of subtracting one number from another. In this case, the subtraction component from which it is subtracted is called the minuend, and the number that is subtracted is called the subtrahend.
For example, 29-13=16. Here 29 is the minuend, 13 is the subtrahend, and 16 is the difference.
Let's consider the simplest example.
Example.
Let's find the difference of numbers:
47-19=28.
Answer. 47-19=28.
You can find the difference not only of natural numbers, but also of integers, fractional, rational, irrational, etc.
Column subtraction is often used to find the difference between numbers.
To subtract in a column, it is necessary to write numbers in such a way that units are under units, tens under tens, etc. Subtraction is performed from right to left and from the top number is smaller.
The rule for finding the difference of rational fractions:
Rational fractions are preliminarily reduced to one denominator, written under the sign of one fraction, and the numerators are subtracted.
Example.
Let's find the difference of rational fractions.
Decision.
Let's use the rule of subtraction of rational fractions and reduce fractions to one denominator:
For subtraction mixed numbers they must first be converted to an improper fraction and then subtracted as rational fractions.
Example.
Let's find the difference between the numbers.
Decision.
Answer.
.
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How to Find the Difference of Numbers in Math
The basic arithmetic operations in mathematics are:
Each result of these actions also has its own name:
Considering definitions, what is the difference of numbers in mathematics, this concept can be denoted in several ways:
The difference between the numbers means how much one of them is greater than the other.Let us take as a basis the notation of the difference that the school curriculum offers us:
Once again resorting to school curriculum, we find the rule how to find the difference:
Answer: 5 - the difference in values.
32 - subtracted value.
- Example 3. Find the value to be subtracted.
- Example 4. Find the difference between three values.
Solution: 17 - 7 = 10
Answer: the subtracted value is 10.
More complex examples
In examples 1-3, actions with simple integers are considered. But in mathematics, the difference is calculated using not only two, but also several numbers, as well as integer, fractional, rational, irrational, etc.
Integer values are given: 56, 12, 4.
56 - decreasing value,
12 and 4 are subtracted values.
The solution can be done in two ways.
Method 1 (consecutive subtraction of subtracted values):
1) 56 - 12 = 44 (here 44 is the resulting difference between the first two values, which will be reduced in the second action);
Method 2 (subtracting two subtracted from the reduced sum, which in this case are called terms):
Answer: 40 is the difference of three values.
Given fractions with the same denominators, where
Let's go back to the rules:
7 - reduced value,
2) 2 * 3 = 6. Answer: 6 is the difference between the numbers 7 and 5.
Answer: - 11. This negative value is the difference between the two values, provided that the subtracted value is greater than the reduced one.
And even if at the beginning of the path the calculations are reduced to primitive examples, everything is ahead of you. And there is a lot to learn. We see that there are many actions with different values in mathematics. Therefore, in addition to the difference, it is necessary to study how to calculate the rest of the results of arithmetic operations:
The word difference can be used in many ways. It can also mean a difference in something, for example, opinions, views, interests. In some scientific, medical and other professional fields this term means different indicators, for example, blood sugar levels, atmospheric pressure, weather conditions. The concept of "difference", as a mathematical term, also exists.
Arithmetic operations with numbers
This is interesting: what is the modulus of a number?
More plain language explaining the concepts of sum, difference, product and quotient in mathematics, we can simply write them down only as phrases:
Difference in mathematics
