What does it mean to subtract from a number. Subtraction

In this article, we will talk about an action called subtraction. First, let's give a general idea of subtraction, after which, based on the meaning of subtraction, we will give meaning subtraction of natural numbers. Next, we introduce terminology and notation. In conclusion, consider the range of problems solved by subtraction.

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Subtraction is a general idea of this action.

Subtraction is the inverse of addition (see Addition for an overview of this operation). If addition is associated with the union of two sets into one, then subtraction is associated with the separation of a given set into two sets.

Let's add some specifics.

Suppose we have some set of objects. Let's take one or more items from this set and put it aside. At the same time, we can say that we taken away or subtracted several items from the initially given set. That is, the meaning of subtraction is to exclude a certain set of objects from a given set of objects.

The meaning of subtracting natural numbers.

We know that the meaning of adding natural numbers, which correspond to the quantities of items being added, is to obtain information about the total number of items. What is the meaning of subtracting two natural numbers?

The subtraction of two natural numbers can be considered from two equal positions. In this case, the meaning of subtracting two natural numbers will depend on what meaning to give to the subtracted number.

So, the result of subtracting two natural numbers indicates

or the number of objects that will remain if a given number of objects is removed from a given set of them,
or the number of items that need to be removed from a given set of them so that the required number of items remains.

Let's give an example for the first case. Let's say we have 7 apples. Subtraction allows us to find out how many apples we have left after we give someone, for example, 2 apples. In this case, we subtract (give away) 2 apples from 7 apples.

Let's illustrate the second case. Let's say we have 7 apples. Using subtraction, we can find out how many apples we need to give away so that we have, for example, 3 apples. In this case, the difference 7−3 will tell us the required number of apples to give away.

In the considered sense, the subtraction of natural numbers is possible only when the number from which they are subtracted is greater than or equal to the number that is being subtracted (we cannot give away more apples than we have). We will strictly adhere to this restriction in the further study of the subtraction of natural numbers.

It is clear that the result of subtracting two natural numbers is a natural number or zero (recall that zero means the absence of something). Moreover, zero is obtained only when the natural number from which they are subtracting is equal to the number that is being subtracted (if we give away all the items that we have, then we will not have a single item left).

Reduced, subtracted, difference, minus sign "-".

Let's define terminology and notation.

To denote subtraction in writing, we will use minus sign type "-". First, we will write down the natural number from which we are subtracting, after that - the minus sign, then - the natural number that we are subtracting. For example, the entry 9−5 (such entries are called) means that 5 is subtracted from 9.

Now let's introduce the necessary terms. Minuend is the number to be subtracted from. Subtrahend is the number that is subtracted from the minuend. Difference is the number that is the result of the subtraction.

Differences will also be called numerical expressions composed of the minuend and the subtrahend with a minus sign between them. For example, in the difference 3−1, the natural number 3 is reduced, and the number 1 is subtracted.

Phrases " find the difference», « calculate the difference», « subtract from the natural number 36 the number 3" and so on. we will understand this: it is required to determine the number that is the result of subtracting given natural numbers.

Let's discuss one more point concerning the record of the minuend, subtrahend and the result of subtraction in the form of equality. Let's say we found out that the natural number 11 is the result of subtracting the number 24 from the number 35. Then we will write this result as the equality 35−24=11 (we talked about the equal sign in the section equal natural numbers). This entry can be read in one of the following ways: “ subtract 24 from 35 is 11” or “ subtract 24 from 35 is 11”.

So, schematically, the subtraction of two natural numbers looks like this:
minuend − subtrahend = difference.

The main problems solved with the help of subtraction.

First, subtraction allows us to solve problems related to the quantities of objects before and after they are divided into two sets.

We have already considered an example of the task of finding the number of objects that remains after the removal of a certain number of them from the original set when we talked about the meaning of subtracting natural numbers.

Other problems of this type are problems of finding the number of items that need to be removed from a given set of them so that the required number of items remains.

Let us give an example of such a task. Let's say we have 8 apples. How many apples do we need to give away so that we have 6 apples left? The desired amount is equal to the difference between natural numbers 8 and 6.

Secondly, subtraction allows solving problems related to changing the value of any measurements (length, area, volume, speed, mass, time, etc.).

Let's take an example. A piece of fabric with an area of 5 square meters was cut from a piece of fabric with an area of 9 square meters. The difference between natural numbers 9 and 5 shows how much fabric is left. Here is another example. Now the air temperature is 15 degrees Celsius, and an hour ago it was 21 degrees. If we subtract the number 15 from the number 21, then we will find out how many degrees the temperature has changed over the past hour.

Thirdly, subtraction allows you to find out the difference between the quantities of objects in two sets, as well as the difference between two measurements of some quantity (mass, time, volume, etc.).

Let, for example, the first motorcyclist traveled 100 kilometers, and the second - 80. If we subtract the number 80 from the number 100, then we will find out how many kilometers the paths of motorcyclists differ. Another example. 3,500 fish fry were released into the first pond, and 7,500 fish fry were released into the second pond. By subtracting 3500 from the number 7500, we find out how much the numbers of fish launched into these ponds differ.

Bibliography.

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

To subtract means to subtract one number from another.

Subtraction is an operation in which a smaller number is subtracted from a larger one. When subtracting integers, the larger number is reduced by as many units as there are in the smaller one. Subtracting one number from another means turn down one number to another, so there is a subtraction the reverse action of addition.

In subtraction, two given numbers are called reduced and subtracted , and the desired - difference .

A smaller number is called a larger number, from which another is subtracted. It decreases with subtraction.

The subtracted is the smaller number that is subtracted from the larger one.

The difference is the output obtained from subtraction. The difference determines how one number is greater than another or shows the difference between two numbers.

subtraction sign. The operation of subtraction is indicated by the - (minus) sign.

Single digit subtraction

To indicate that 6 must be subtracted from 9, these numbers are written side by side, separating them with a - (minus) sign:

The difference between these numbers will be 3, and the progress of the calculation is expressed verbally:

nine minus six equals three.

In writing:

A larger number 9 will be reduced, a smaller number 6 will be subtracted, the number 3 will be the remainder.

Subtraction methods

There are two ways to subtract one number from another:

or you can subtract as many units from the larger number as there are in the smaller one. So, subtracting 6 from 9 means subtracting 6 from 9. The number 3 will be the desired remainder;

or you can add one to a smaller number until you get a larger number. So, subtracting 6 from 9, we add 3 units to 6. The number of units that must be added to the smaller number to equalize it with the larger one determines the difference. The smaller number with the difference must be equal to the larger number, therefore, the smaller number and the difference are terms, and the larger one is their sum. Based on this another definition of subtraction:

Subtraction is an operation in which, given the sum and one term, another term is found.

In this case the given amount is the minuend, the given term is the deductible, and the claimand I difference- another term.

Multi-digit subtraction

Subtraction of multi-digit numbers is based on the property of numbers, according to which subtracting a number is the same as subtracting all its parts. From this property it can be seen that subtracting some number is the same as subtracting successively all its units, tens, hundreds, etc. To indicate that 3517 must be subtracted from the number 7228, they write:

and subtract separately units from units, tens from tens, etc.

To facilitate subtraction, they sign a smaller number under a large one so that units of the same order are in the same vertical column, draw a line, put a subtraction sign on the left - and sign the difference under the line.

The course of calculation is expressed verbally:

Starting subtraction with simple units: 8 minus 7 is 1; signed under units 1.

Subtract tens: 2 without 1 gives 1, we sign under tens 1.

Subtract hundreds. Five cannot be subtracted from 2, so we take one from the next higher order (thousands), which we denote by putting a dot over 7. Each order unit contains 10 units of the next lower order. Adding these 10 units to 2, we get 12; 12 without 5 is 7, we sign under hundreds 7. When one is taken from a higher order, this is indicated by putting a dot over the order from which they occupy.

Subtract thousands. Instead of 7, only 6 thousand remained, for one was taken. 6 minus 3 is 3; sign under thousands 3.

The progress of the calculation is expressed in writing:

Example. Subtract 6025 from 17004.

5 cannot be subtracted from 4. We borrow one from tens, the next highest order, but there are no ones in this order; we borrow from hundreds, and there are no hundreds; we borrow from thousands and denote this with a dot above the number 7.

The unit of the fourth has 10 units of the third order. Taking one of them for tens, we leave them in hundreds only 9. Adding 10 to 4, we have 14.

Subtracting, we get:

for units 14 - 5 = 9

for tens 9 - 2 = 7

for hundreds 9 - 0 = 9

for thousands 6 - 6 = 0

For tens of thousands, we have 1, because we transfer this figure of the reduced to the difference without change.

The course of calculation will be expressed in writing:

From the previous examples, we deduce subtraction rules:

To subtract whole numbers, you need to sign the subtrahend under the minuend so that units of the same order stand in the same vertical column, draw a line, under which you sign the difference.

Subtraction must begin with simple units, that is, from the first column, and then, moving to the next columns from the right hand to the left, subtract tens from tens, hundreds from hundreds, etc.

If the digit of the subtracted is less than the digit of the reduced, the difference is signed in the same column; if the digits are equal, the difference will be zero. If the digit of the subtrahend is greater than the corresponding digit of the reduced, take one from the next order of the reduced, marking this with a dot placed above the figure from which it is occupied, apply 10 to the digit of the reduced and subtract. The number with a dot is considered one less.

If, when subtracting, the digit of the minuend, from which they take, will be 0, followed by zeros in the minuend, then they take from the first significant digit, putting dots above it and all intermediate zeros. A digit with a dot is counted as one less, and zeros with a dot are counted as 9.

The subtraction is continued until the total difference is obtained.

The extra digits of the minuend are transferred to the difference.

Relationship between data and desired subtractions

From example 9 - 6 = 3, it can be seen that

The minuend is equal to the subtrahend, added to the difference: 9 = 6 + 3.

Subtrahend equals minuend without difference: 6 = 9 - 3.

The difference is equal to the minuend without the subtrahend: 3 = 9 - 6.

Arithmetic addition. The difference between a number and the nearest larger unit is called arithmetic complement. So, the arithmetic complements of the numbers 7, 79, 983 will be the numbers:

10 - 7 = 3
100 - 79 = 21
1000 - 983 = 17

Arithmetic addition is sometimes used to facilitate arithmetic calculations.

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 refers to various indicators, for example, blood sugar levels, atmospheric pressure, weather conditions. The concept of "difference", as a mathematical term, also exists.

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Arithmetic operations with numbers

The basic 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 - the result of multiplying numbers;
quotient is the result of division.

Explaining the concepts of sum, difference, product and quotient in mathematics in a simpler language, we can simply write them down only as phrases:

amount - add;
difference - take away;
product - multiply;
private - share.

Considering definitions, what is the difference of numbers in mathematics, this concept can be denoted in several ways:

And all these definitions are true.

How to find the difference in values

Let us take as a basis the notation of 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 for how to find the difference:

To find the difference, subtract the minuend from the minuend.

All clear. But at the same time, we got a few more mathematical terms. What do they mean?

Decreasing is a mathematical number from which it is subtracted and it decreases (becomes smaller).
The subtrahend is the mathematical number that is subtracted from the minuend.

Now it is clear that the difference consists of two numbers, which must be known in order to calculate it. And how to find them, we also use the definitions:

To find the minuend, add the difference to the minuend.
To find the subtrahend, you need to subtract the difference from the minuend.

Mathematical operations with the difference of numbers

Based on the derived rules, we can consider illustrative examples. Mathematics is an interesting science. Here we will take only the simplest numbers for solution. Having learned to subtract them, you will learn how to solve more complex values, three-digit, four-digit, integer, fractional, in powers, roots, others.

Simple examples

Example 1. Find the difference between two values.

20 - decreasing value,

15 - subtracted.

Solution: 20 - 15 = 5

Answer: 5 - the difference in values.

Example 2. Find the minuend.

48 - difference,

32 - subtracted value.

Solution: 32 + 48 = 80

Example 3. Find the value to be subtracted.

7 - difference,

17 - reduced value.

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.

Example 4. Find the difference between three values.

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):

1) 12 + 4 = 16 (where 16 is the sum of two terms, which will be subtracted in the next step);

2) 56 - 16 = 40.

Answer: 40 is the difference of three values.

Example 5. Find the difference between rational fractional numbers.

Given fractions with the same denominators, where

4/5 - reduced fraction,

3/5 - subtracted.

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 deal with fractions that have different denominators. They must be able to bring to a common denominator.

Solution: 4/5 - 3/5 = (4 - 3)/5 = 1/5

Answer: 1/5.

Example 6. Triple the difference of numbers.

But how to execute such an example when you want to double or triple the difference?

Let's go back to the rules:

A double number is a value multiplied by two.
A triple number is a value multiplied by three.
The doubled difference is the difference in values multiplied by two.
A triple difference is the difference in values 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 7 and 18.

7 - reduced value;

18 - subtracted.

Everything seems to be clear. Stop! Is the subtrahend greater than the minuend?

And again, there is a rule applied for a specific case:

If the subtracted is greater than the minuend, the difference will be negative.

Answer: - 11. This negative value is the difference between the two values, provided that the subtracted value is greater than the reduced one.

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 in any mathematical calculations. All the calculations made on them are a great help for the hasty, uninquisitive, lazy. Math for Blondes is one such resource. And we all resort to it, regardless of hair color, gender and age.

At school, we were taught to calculate such actions with mathematical quantities in a column, and later on a calculator. The calculator is also a handy tool. But, for the development of thinking, intellect, outlook and other vital 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 needs to be pumped. So, without delay, start thinking.

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:

sum - by adding the terms;
product - by multiplying factors;
quotient - dividing the dividend by the divisor.

Here is some interesting math.

SUBTRACT

1. Subtract (one number from another), subtract (mat.). Subtract one number from another.

2. Withhold some amount of the money due for extradition. Deduct one percent from your salary.

Explanatory Dictionary of Ushakov. D.N. Ushakov. 1935-1940.

See what "Subtract" is in other dictionaries:

Subtract, calculate, hold, take away Dictionary of Russian synonyms. subtract 1. subtract; take away (colloquial) 2. see calculate the Dictionary of synonyms of the Russian language. Practical guide. M... Synonym dictionary

SUBTRACT, honor, honor; people, chla; read; honor; sov., what from what. 1. Withhold upon payment. B. debt. 2. Subtract one number from another. V. three out of five. | incompatibility subtract, ay, aesh. | noun deduction, ah, husband. (to 1 value).… … Explanatory dictionary of Ozhegov

subtract- (subtract, subtract) achori; subtract two toygala duerbe achori from five ... Russian-Nanai dictionary

Subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtract, subtracted, ... ... Word forms

subtract- subtract, subtract, subtract; past temp. in vychel, in vychla ... Russian spelling dictionary

Read, read; subtracted, chla, chlo; subtracted; deducted; ten, a, o; subtracting; St. what from what. 1. Subtract one number from another. B. seven out of ten. 2. Withhold part of the money intended for issuance. V. from the fee. ◁ Subtract, ayu, aesh; nsv ... encyclopedic Dictionary

subtract- Enlarge/reduce… Dictionary of synonyms of the Russian language

subtract- subtraction... Dictionary-thesaurus of synonyms of Russian speech

Books

The game "Learning to count" for children 5-7 years old,. The game contains 2 ecological acrylic dice with numbers from 1 to 12 on each and 1 dice with plus and minus signs. Throwing the dice on the table, the child must add or subtract the dropped numbers. For…
Add and Subtract (+ 100 Stickers), David Kirkby. What awaits you under the cover: ADDITION AND SUBTRACTION - a tutorial of the new series ʻI love to learn. Preparing for school`, along with the development of counting skills, instills literacy, introduces the child ...