Definition: Subtraction is an action that uses the sum and one of the terms to find the second term.
For example:
if 55 + 35 = 90,
then 90 - 35 = 55.
In general:
if a + b = c,
then c - b = a.
Action subtraction verified by addition. The number from which we subtract is called the minuend, and the number we subtract from is called the subtrahend. The result of the subtraction action is the difference.
The subtrahend can be not 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 calculations.
To calculate in a convenient way is to apply the laws of addition to specific numbers so that the process of calculating the unknown is simplified (for example, use the ten's complement table by digits, avoid crossing the ten when calculating, etc.).
Rule 1. To subtract a sum from a number, you can subtract one term from it, and subtract the second term from the resulting result (difference).
For example:
126 - (56 + 30) = (126 - 56) - 30 = 40.
In general:
a - (b + c) = (a - b) - c.
Rule 2. To subtract a number from a 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 being subtracted.
For example:
(71 + 7) - 51 = (71 - 51) + 7 = 20 + 7 = 27, but not (71 + 7) - 51 = (7 - 51) + 71, since the difference (7 - 51) is unnatural number.
In general terms: (a + b) - c = (a - c) + b.
These difference properties are used to check that subtraction calculations are correct.
For example: 136 - 82 = 54.
Checking calculations:
1) 54 + 82 = 136;
What is the difference of numbers in mathematics and how to find the difference of numbers
In this article we will look at what the difference of numbers is in mathematics, and how a person interested in this science can find the difference of numbers.
What is the difference between numbers in mathematics
Subtraction is one of the 4 arithmetic operations. It is designated by the mathematical sign “−” (minus). Subtraction is the opposite of addition.
The subtraction operation is generally written as follows:
Here the difference between the numbers will be the number 4. Therefore, difference between any numbers A and B this is the number C that, when added to B, will give a total of A (4 when added to 2 gives 6 - which means 4 is the difference between 6 and 2).
How to find the difference between numbers
Already from the definition itself it follows how to calculate the difference between two numbers. For small numbers, you can do this in your head. Children in primary school are taught as follows. 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 automation and will immediately give the answer.
However, for numbers above 50, this visual representation no longer works. It’s hard to imagine a large number of objects in your mind, so here another method comes to the rescue:
Column difference calculation
Students learn this technique as part of a math course, usually in 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. Brush up on 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. This is usually not required in real life, but can be useful when solving mathematical problems.
Let's say in the example "A − B = C" B is greater than A. Then C will be negative. To calculate the difference, “expand” the example: count 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, prefix it with a minus sign. 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 component of the subtraction from which it is subtracted is called the minuend, and the number that is being 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 look at a simple example.
Example.
Let's find the difference between the numbers:
47-19=28.
Answer. 47-19=28.
You can find the difference not only of natural numbers, but also of integers, fractions, rationals, irrationals, etc.
To find the difference between numbers, columnar subtraction is often used.
To subtract in a column, you need to write numbers so that the ones are under the ones, the tens are under the tens, etc. Subtraction is performed from right to left and from the top number the smaller one.
The rule for finding the difference of rational fractions:
Preliminary rational fractions are 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.
Solution.
Let's use the rule for subtracting rational fractions and reduce the fractions to one denominator:
To subtract mixed numbers, you must first convert them to improper fractions and then subtract them as rational fractions.
Example.
Let's find the difference between the numbers.
Solution.
Answer.
.
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How to find the difference between numbers in mathematics
The main arithmetic operations in mathematics are:
Each result of these actions also has its own name:
Looking at Definitions, what is the difference between numbers in mathematics, this concept can be defined in several ways:
The difference between numbers means how much more one of them is than the other.Let’s take as a basis the notation for the difference that the school curriculum offers us:
Once again resorting to the school curriculum, we find a rule on how to find the difference:
Answer: 5 - difference in values.
32 is the subtracted value.
- Example 3. Find the subtrahend value.
- Example 4. Find the difference between three values.
Solution: 17 - 7 = 10
Answer: Subtract value 10.
More complex examples
Examples 1-3 examine actions with simple integers. But in mathematics, the difference is calculated using not only two, but also several numbers, as well as integers, fractions, rational, irrational, etc.
The integer values are given: 56, 12, 4.
56 - value to be reduced,
12 and 4 are subtracted values.
The solution can be done in two ways.
Method 1 (sequential subtraction of subtracted values):
1) 56 - 12 = 44 (here 44 is the resulting difference of the first two quantities, which in the second action will be reduced);
Method 2 (subtracting two subtrahends from the sum being reduced, which in this case are called addends):
Answer: 40 is the difference of three values.
Given fractions with the same denominators, where
Let's use the rules again:
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 two quantities, provided that the quantity being subtracted is greater than the quantity being reduced.
And even though at the beginning of your journey the calculations are reduced to primitive examples, everything is ahead of you. And you will have to master a lot. We see that there are many operations with different quantities in mathematics. Therefore, in addition to the difference, it is necessary to study how to calculate the remaining results of arithmetic operations:
The word "difference" can have many meanings. This can also mean a difference in something, for example, opinions, views, interests. In some scientific, medical and other professional fields, this term refers to various indicators, for example, blood sugar levels, atmospheric pressure, and 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?
To explain in simpler language the concepts of sum, difference, product and quotient in mathematics, we can simply write them down only as phrases:
Difference in mathematics
DIFFERENCE, differences, women. 1. The number that makes up the remainder in a subtraction (mat.). The minuend is equal to the subtrahend plus the difference. 2. only units. distracted noun to different in 1 value; difference dissimilarity (book). Difference of views. Difference of characters. ❖ Various… Ushakov's Explanatory Dictionary
See the difference... Dictionary of Russian synonyms and similar expressions. under. ed. N. Abramova, M.: Russian Dictionaries, 1999. difference excess, difference; difference, difference, gap, dissimilarity; variety of sizes, difference, balance, margin, tension,... ... Synonym dictionary
- (difference) Change in the value of a variable between fixed points in time. If xt is the value of the variable x at time t, then the first difference is defined as Δxt=xt–xt–1. The second difference is equal to the first difference Δxt, minus the first... ... Economic dictionary
DIFFERENCE- (1) potentials (voltage (see (2))) a quantitative characteristic of the electric field of stationary electric charges () between two of its points, equal to the work of the electric field in moving a single positive charge from one... ... Big Polytechnic Encyclopedia
DIFFERENCE, difference, etc. see different. Dahl's Explanatory Dictionary. IN AND. Dahl. 1863 1866 … Dahl's Explanatory Dictionary
Subtraction result... Big Encyclopedic Dictionary
DIFFERENCE, and, female. 1. see different. 2. Result, the result of the subtraction. | adj. difference, oh, oh. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary
difference- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN differential ... Technical Translator's Guide
Difference is an ambiguous term: the result of subtraction. Difference (mineralogy) (for example, “medium-grained differences” or “chalk-like differences”) Potential difference ... Wikipedia
AND; and. 1. to Miscellaneous (1 digit); difference. R. beliefs, views. Discover r. in approaches to historical facts. // The difference between the two values being compared in numerical terms. R. altitudes above sea level. R. temperature. R. water levels. R. in... ... encyclopedic Dictionary
difference- ▲ magnitude difference difference magnitude of difference; subtraction result; quantitative difference. difference. differential (# pressure). increment. ▼ not by much, angle ↓ subtracted... Ideographic Dictionary of the Russian Language
Books
- Set of tables. Algebra. 7th grade. 15 tables + methodology, . The tables are printed on thick printed cardboard measuring 680 x 980 mm. The kit includes a brochure with teaching guidelines for teachers. Educational album of 15 sheets. Expressions...
- Time-distributed “difference-in-differences” using the example of assessing the impact of additional vocational training, A. V. Aistov. The paper presents an econometric model that describes the time distribution of the impact effect, built on the basis of the difference-in-differences methodology. The model allowed...
The word "difference" can have many meanings. This can also mean a difference in something, for example, opinions, views, interests. In some scientific, medical and other professional fields, this term refers to various indicators, for example, blood sugar levels, atmospheric pressure, and weather conditions. The concept of “difference” as a mathematical term also exists.
Arithmetic operations with numbers
The main arithmetic operations in mathematics are:
- addition;
- subtraction;
- multiplication;
- division.
Each result of these actions also has its own name:
- sum - the result obtained by adding numbers;
- difference - the result obtained by subtracting numbers;
- product is the result of multiplying numbers;
- the quotient is the result of division.
To explain in simpler language the concepts of sum, difference, product and quotient in mathematics, we can simply write them down only as phrases:
- amount - add;
- difference - subtract;
- product - multiply;
- private - to divide.
Looking at Definitions, what is the difference between numbers in mathematics, this concept can be defined in several ways:
And all these definitions are true.
How to find the difference between quantities
Let’s take as a basis the notation for the difference that the school curriculum offers us:
- The difference is the result of subtracting one number from another. The first of these numbers, from which the subtraction is carried out, is called the minuend, and the second, which is subtracted from the first, is called the subtrahend.
Once again resorting to the school curriculum, we find a rule on how to find the difference:
- To find the difference, you need to subtract the subtrahend from the minuend.
All clear. But at the same time we received several more mathematical terms. What do they mean?
- The minuend is a mathematical number from which it is subtracted and it decreases (becomes smaller).
- A subtrahend is a mathematical number that is subtracted from the minuend.
Now it is clear that the difference consists of two numbers that must be known to calculate it. And how to find them, we will also use the definitions:
- To find the minuend, you need to add the difference to the subtrahend.
- To find the subtrahend, you need to subtract the difference from the minuend.
Mathematical operations with number differences
Based on the derived rules, we can consider illustrative examples. Mathematics is an interesting science. Here we will take only the simplest numbers to solve. Having learned to subtract them, you will learn to solve more complex values, three-digit, four-digit, integer, fractional, powers, roots, etc.
Simple examples
- Example 1. Find the difference between two quantities.
20 - decreasing value,
15 - subtractable.
Solution: 20 - 15 = 5
Answer: 5 - difference in values.
- Example 2. Find the minuend.
48 - difference,
32 is the subtracted value.
Solution: 32 + 48 = 80
- Example 3. Find the subtrahend value.
7 - difference,
17 is the value being reduced.
Solution: 17 - 7 = 10
Answer: Subtract value 10.
More complex examples
Examples 1-3 examine actions with simple integers. But in mathematics, the difference is calculated using not only two, but also several numbers, as well as integers, fractions, rational, irrational, etc.
- Example 4. Find the difference between three values.
The integer values are given: 56, 12, 4.
56 - value to be reduced,
12 and 4 are subtracted values.
The solution can be done in two ways.
Method 1 (sequential subtraction of subtracted values):
1) 56 - 12 = 44 (here 44 is the resulting difference of the first two quantities, which in the second action will be reduced);
Method 2 (subtracting two subtrahends from the sum being reduced, which in this case are called addends):
1) 12 + 4 = 16 (where 16 is the sum of two terms, which will be subtracted in the next operation);
2) 56 - 16 = 40.
Answer: 40 is the difference of three values.
- Example 5. Find the difference between rational fractions.
Given fractions with the same denominators, where
4/5 is a fraction to be reduced,
3/5 - deductible.
To complete the solution, you need to repeat the actions with fractions. That is, you need to know how to subtract fractions with the same denominator. How to handle fractions that have different denominators. They must be able to bring them to a common denominator.
Solution: 4/5 - 3/5 = (4 - 3)/5 = 1/5
Answer: 1/5.
- Example 6. Triple the difference of numbers.
How to perform such an example when you need to double or triple the difference?
Let's use the rules again:
- Double a number is a value multiplied by two.
- Triple a number is a value multiplied by three.
- The double difference is the difference in magnitudes multiplied by two.
- A triple difference is a difference in magnitude multiplied by three.
7 - reduced value,
5 - subtracted value.
2) 2 * 3 = 6. Answer: 6 is the difference between the numbers 7 and 5.
- Example 7. Find the difference between values 7 and 18.
7 - reduced value;
18 - subtracted.
Everything seems clear. Stop! Is the subtrahend greater than the minuend?
And again there is a rule that applies to a specific case:
- If the subtrahend is greater than the minuend, the difference will be negative.
Answer: - 11. This negative value is the difference between two quantities, provided that the quantity being subtracted is greater than the quantity being reduced.
Math for blondes
On the World Wide Web you can find a lot of thematic sites that will answer any question. In the same way, online calculators for every taste will help you with any mathematical calculations. All the calculations made on them are an excellent help for the hasty, incurious, and lazy. Math for Blondes is one such resource. Moreover, we all resort to it, regardless of hair color, gender and age.
At school, we were taught to calculate such operations with mathematical quantities in a column, and later - on a calculator. The calculator is also a handy aid. But, for the development of thinking, intelligence, outlook and other life qualities, we advise you to perform arithmetic operations on paper or even in your mind. The beauty of the human body is the great achievement of the modern fitness plan. But the brain is also a muscle that sometimes requires pumping. So, without delay, start thinking.
And even though at the beginning of your journey the calculations are reduced to primitive examples, everything is ahead of you. And you will have to master a lot. We see that there are many operations with different quantities in mathematics. Therefore, in addition to the difference, it is necessary to study how to calculate the remaining results of arithmetic operations:
- the sum - by adding the terms;
- product - by multiplying factors;
- quotient - by dividing the dividend by the divisor.
This is some interesting arithmetic.
For many, hard sciences like mathematics are perceived as simpler than areas that require reasoning and involve a lot of variability. However, all subjects have their own difficulties, including technical ones.
Subtraction
In order to understand what the difference is, it is necessary to understand a number of mathematical terminology. First of all, you need to find out what subtraction is.
In another way, this concept is called reduction, and by this name it is somewhat easier to understand the meaning of the process. At its core, subtraction is a mathematical operation. What kind of operations are these? As a rule, they mean certain arithmetic or logical operations. A logical question arises: what is the essence of arithmetic operations?
The concept of arithmetic appeared quite a long time ago. It originated in ancient Greek, where it was translated as “number”. Today it is a branch of mathematics that studies numbers, their relationships to each other, and properties.
So, subtraction - these are number operations related to binary. The essence of binary operations is that they use two arguments (parameters) and produce one result.
It is worth considering how to find the difference of a number. First of all, two arguments are needed, that is, two numbers. Then you need to reduce the value of the first number by the value of the second. When this operation is expressed in writing, a minus sign is used. It looks like this: a – b = c, where a is the first numerical value, b is the second, and c is the difference between the numbers.
Properties and Features
As a rule, students have much more problems with subtraction than with addition. This is partly due to the properties of these mathematical operations. Everyone knows that changing the places of the terms does not change the value of the sum. In subtraction, everything is much more complicated. If you swap the numbers, you get a completely different result. A similar property in addition and decrease is that the zero element does not change the original number.
In subtraction, everything is relatively simple if the first number is greater than the second, but in school we will also consider counterexamples. In this case, the concept of a negative number arises.
For example, if you need to subtract the number 2 from 5, then everything is not difficult. 5-2=3, so the difference of the number will be 3. However, what if you need to calculate how much two minus five is?
In expression 2-5, the difference will go into minus, that is, into a negative value. You can easily subtract a two from a two, thus getting a zero, but from a five there are still three left. Thus, the result of this expression will be negative three. That is, 2-5=-3.
Features of subtracting negative numbers
There are also situations where the second number is, in fact, less than the first, but is negative. For example, consider the expression 7-(-4). The easiest way to understand this operation is by turning the combination –(- into a regular plus. The signs even superficially resemble it. In this regard, the result of the expression, that is, the difference in numbers, will be 11.
If both numbers are negative, then the subtraction will occur as follows.
6-(-7): the minus of the first number will remain, and the combination of the two subsequent minuses will turn into a plus. Thus, you need to understand how much -6+7 will be. The difference is not difficult to find - it is equal to one.
If you need to subtract a positive number from a negative one, then the expression can be represented as a simple addition, and then add a minus to the result. For example, -3-4 (4 is a positive number) will result in -7.
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 we come across terms used in addition more often in life.
We talk about the “addition of efforts” when trying together to obtain the desired result, about the “components of achieved success,” etc. The names associated with subtraction remain within the boundaries of mathematics, rarely appearing in everyday speech. Therefore, the words “subtracted”, “reduced”, “difference” are less common. The rule for finding each of these components can only be applied if you understand the meaning of these names.
Unlike many scientific terms that have Greek, Latin or Arabic origin, in this case words with Russian roots are used. So it’s not difficult to understand their meaning, which means it’s easy to remember what is meant by which term.
If you look closely at the name itself, it becomes noticeable that it has to do with the words “different”, “difference”. From this we can conclude that what is meant is an established difference between quantities.
This concept in mathematics means:
- difference between two numbers;
- it is a measure of how much more or less one quantity is than another;
- this is the result obtained when performing a subtraction - this is the definition offered by the school curriculum.
Note! If the quantities are equal to each other, then there is no difference between them. This means their difference is zero.
What are minuend and subtrahend?
As the name suggests, diminished is something that is done less. And you can make the quantity smaller by subtracting a part from it. Thus, the minuend is a number from which a part is subtracted.
Subtracted, accordingly, is the number that is subtracted from it.
| Minuend | Subtrahend | Difference | ||
| 18 | — | 11 | = | 7 |
| 14 | — | 5 | = | 9 |
| 26 | — | 22 | = | 4 |
Useful video: minuend, subtrahend, difference
Rules for finding an unknown element
Having understood the terms, it is easy to establish by what rule each of the elements of subtraction is found.
Since the difference is the result of a given arithmetic operation, it is found using this action; no other rules are required here. But they are there in case the other term of the mathematical expression is unknown.
How to find a minuend
This term, as it was found out, refers to the quantity from which a part has been 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 an unknown minuend by adding two known elements.
So, in this case, to find the unknown, you should add the subtrahend and the difference:
The same is true in all similar cases:
| ? | – | 5 | = | 9 |
| 9 | + | 5 | = | 14 |
From the example it is clear that a certain value was subtracted from 18, and what was left was 7. To find this value, you need to subtract 7 from 18.
| 26 | – | ? | = | 4 |
| 26 | – | 4 | = | 22 |
Thus, knowing the exact meaning of the names, you can easily guess what rule should be used to search for each unknown element.
Useful video: how to find an unknown minuend
Conclusion
The four basic arithmetic operations are the basis on which all mathematical calculations are based, from simple to the most complex. Of course, in our time, when people strive to entrust everything to technology, including the thought process, it is more common and faster to make calculations using a calculator. But any skill increases a person’s independence – from technical means, from others. It is not necessary to make mathematics your specialty, but having at least minimal knowledge and skills means having additional support for your own confidence.