Now we will look at examples subtracting negative numbers, and you will see that it is very easy. You just need to remember the rule: two minuses next to each other give a plus.

Example 1: Subtracting a negative number from a positive number

56 – (–34) = 56 + 34 = 90

As you can see, in order to subtract a negative number from a positive number, you simply need to add their modules.

Example 2: Subtracting a negative number from a negative number

– 60 – (– 25) = – 60 + 25 = – 35

– 15 – (– 30) = – 15 + 30 = 15

Thus, when subtracting a negative number from a negative one, we follow the rule, and we can end up with both a positive and a negative number.

There is a single rule governing the subtraction of any numbers: both negative and positive, and it sounds like this:

Rule of signs

In order to get rid of extra parentheses when subtracting negative numbers, we can use the sign rule.This rule says:

For example:

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Adding and subtracting negative numbers

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Rule for adding negative numbers

If you remember the math lesson and the topic “Adding and subtracting numbers with different signs”, then to add two negative numbers you need:

• perform the addition of their modules;
• add a “–” sign to the received amount.

According to the addition rule, we can write:

$(−a)+(−b)=−(a+b)$.

The rule for adding negative numbers applies to negative integers, rational numbers, and real numbers.

Example 1

Add the negative numbers $−185$ and $−23\789.$

Solution.

Let's use the rule for adding negative numbers.

Let's find the modules of these numbers:

$|-23 \ 789|=23 \ 789$.

Let's add the resulting numbers:

$185+23 \ 789=23 \ 974$.

Let's put the $“–”$ sign in front of the found number and get $−23\974$.

Brief solution: $(−185)+(−23\789)=−(185+23\789)=−23\974$.

Answer: $−23 \ 974$.

When adding negative rational numbers, they must be converted to the form of natural numbers, ordinary or decimal fractions.

Example 2

Add the negative numbers $-\frac(1)(4)$ and $−7.15$.

Solution.

According to the rule for adding negative numbers, you first need to find the sum of the modules:

$|-\frac(1)(4)|=\frac(1)(4)$;

It is convenient to reduce the obtained values ​​to decimal fractions and perform their addition:

$\frac(1)(4)=0.25$;

$0,25+7,15=7,40$.

Let’s put the $“–”$ sign in front of the resulting value and get $–7.4$.

Brief summary of the solution:

$(-\frac(1)(4))+(−7.15)=−(\frac(1)(4)+7.15)=–(0.25+7.15)=−7, $4.

To add a positive and negative number you need:

1. calculate the modules of numbers;

compare the resulting numbers:

• if they are equal, then the original numbers are opposite and their sum is zero;
• if they are not equal, then you need to remember the sign of the number whose modulus is greater;

2. subtract the smaller one from the larger module;

3. Before the resulting value, put the sign of the number whose modulus is greater.

Adding numbers with opposite signs amounts to subtracting a smaller negative number from a larger positive number.

The rule for adding numbers with opposite signs applies to integers, rationals, and real numbers.

Example 3

Add the numbers $4$ and $−8$.

Solution.

You need to add numbers with opposite signs. Let's use the corresponding addition rule.

Let's find the modules of these numbers:

The modulus of the number $−8$ is greater than the modulus of the number $4$, i.e. remember the $“–”$ sign.

Let’s put the sign $“–”$, which we remembered, in front of the resulting number, and we get $−4.$

Brief summary of the solution:

$4+(–8) = –(8–4) = –4$.

Answer: $4+(−8)=−4$.

To add rational numbers with opposite signs, it is convenient to represent them in the form of ordinary or decimal fractions.

Subtracting numbers with different and negative signs

Rule for subtracting negative numbers:

To subtract a negative number $b$ from a number $a$, it is necessary to add the number $−b$ to the minuend $a$, which is the opposite of the subtrahend $b$.

According to the subtraction rule, we can write:

$a−b=a+(−b)$.

This rule is valid for integers, rationals and real numbers. The rule can be used to subtract a negative number from a positive number, from a negative number, and from zero.

Example 4

Subtract the negative number $−5$ from the negative number $−28$.

Solution.

The opposite number for the number $–5$ is the number $5$.

According to the rule for subtracting negative numbers, we get:

$(−28)−(−5)=(−28)+5$.

Let's add numbers with opposite signs:

$(−28)+5=−(28−5)=−23$.

Answer: $(−28)−(−5)=−23$.

When subtracting negative fractions, you must convert the numbers to fractions, mixed numbers, or decimals.

Adding and subtracting numbers with different signs

The rule for subtracting numbers with opposite signs is the same as the rule for subtracting negative numbers.

Example 5

Subtract the positive number $7$ from the negative number $−11$.

Solution.

The opposite of $7$ is $–7$.

According to the rule for subtracting numbers with opposite signs, we get:

$(−11)−7=(–11)+(−7)$.

Let's add negative numbers:

$(−11)+(–7)=−(11+7)=−18$.

Brief solution: $(−28)−(−5)=(−28)+5=−(28−5)=−23$.

Answer: $(−11)−7=−18$.

When subtracting fractional numbers with different signs, it is necessary to convert the numbers to the form of ordinary or decimal fractions.

As you know, subtraction is the opposite of addition.

If “a” and “b” are positive numbers, then subtracting the number “b” from the number “a” means finding the number “c” which, when added “with” the number “b”, gives the number “a”.

The definition of subtraction holds true for all rational numbers. That is subtracting positive and negative numbers can be replaced by addition.

To subtract another from one number, you need to add the opposite number to the one being subtracted.

Or, in another way, we can say that subtracting the number “b” is the same as addition, but with the opposite number to the number “b”.

It is worth remembering the expressions below.

Rules for subtracting negative numbers

As can be seen from the examples above, subtracting the number “b” is addition with the number opposite to the number “b”.

This rule holds true not only when subtracting a smaller number from a larger number, but also allows you to subtract a larger number from a smaller number, that is, you can always find the difference of two numbers.

The difference can be a positive number, a negative number, or a zero number.

Examples of subtracting negative and positive numbers.

Convenient to remember rule of signs, which allows you to reduce the number of parentheses.

The plus sign does not change the sign of the number, so if there is a plus in front of the parenthesis, the sign in the parentheses does not change.

The minus sign in front of the parentheses reverses the sign of the number in the parentheses.

From the equalities it is clear that if there are identical signs before and inside the brackets, then we get “+”, and if the signs are different, then we get “−”.

The rule of signs also applies if the brackets contain not just one number, but an algebraic sum of numbers.

Please note that if there are several numbers in brackets and there is a minus sign in front of the brackets, then the signs in front of all numbers in these brackets must change.

To remember the rule of signs, you can create a table for determining the signs of a number.

Dividing negative numbers

How to perform division of negative numbers It's easy to understand by remembering that division is the inverse of multiplication.

If “a” and “b” are positive numbers, then dividing the number “a” by the number “b” means finding the number “c” that, when multiplied by “b,” gives the number “a”.

This definition of division applies to any rational numbers as long as the divisors are non-zero.

Therefore, for example, dividing the number “−15” by the number 5 means finding a number that, when multiplied by the number 5, gives the number “−15”. This number will be “−3”, since

Examples dividing rational numbers.

1. 10: 5 = 2, since 12 5 = 10
2. (−4) : (−2) = 2 since 2 · (−2) = −4
3. (−18) : 3 = −6 since (−6) 3 = −18
4. 12: (−4) = −3, since (−3) · (−4) = 12

From the examples it is clear that the quotient of two numbers with the same signs is a positive number (examples 1, 2), and the quotient of two numbers with different signs is a negative number (examples 3, 4).

Rules for dividing negative numbers

To find the modulus of a quotient, you need to divide the modulus of the dividend by the modulus of the divisor.

So, to divide two numbers with the same signs, necessary:

Examples of dividing numbers with the same signs:

To divide two numbers with different signs, necessary:

Examples of dividing numbers with different signs:

You can also use the following table to determine the quotient sign.

Rule of signs for division

When calculating “long” expressions in which only multiplication and division appear, it is very convenient to use the sign rule. For example, to calculate a fraction

You may notice that the numerator has two minus signs, which when multiplied will give a plus. There are also three minus signs in the denominator, which when multiplied will give a minus sign. Therefore, in the end the result will turn out with a minus sign.

Reducing a fraction (further actions with the modules of numbers) is performed in the same way as before:

The quotient of zero divided by a number other than zero is zero.

You CANNOT divide by zero!

All previously known rules of division by one also apply to the set of rational numbers.

Where "a" is any rational number.

The relationships between the results of multiplication and division, known for positive numbers, remain the same for all rational numbers (except zero):

These dependencies are used to find the unknown factor, dividend and divisor (when solving equations), as well as to check the results of multiplication and division.

An example of finding the unknown.

Minus sign in fractions

Let's divide the number "−5" by "6" and the number "5" by "−6".

We remind you that the line in writing a common fraction is the same division sign, so you can write the quotient of each of these actions as a negative fraction.

Thus, the minus sign in a fraction can be:

• before a fraction;
• in the numerator;
• in the denominator.



When writing negative fractions, the minus sign can be placed in front of the fraction, transferred from the numerator to the denominator, or from the denominator to the numerator.

This is often used when working with fractions, making calculations easier.

Example. Please note that after placing the minus sign in front of the bracket, we subtract the smaller one from the larger module according to the rules for adding numbers with different signs.





Using the described property of sign transfer in fractions, you can act without finding out which of the given fractions has a greater modulus.

Fractions, fractions, definitions, notations, examples, operations with fractions.

This article is about common fractions. Here we will introduce the concept of a fraction of a whole, which will lead us to the definition of a common fraction. Next we will dwell on the accepted notation for ordinary fractions and give examples of fractions, let’s say about the numerator and denominator of a fraction. After this, we will give definitions of proper and improper, positive and negative fractions, and also consider the position of fractional numbers on the coordinate ray. In conclusion, we list the main operations with fractions.

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Shares of the whole

First we introduce concept of share.

Let's assume that we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange consisting of several equal slices. Each of these equal parts that make up the whole object is called parts of the whole or simply shares.

Note that the shares are different. Let's explain this. Let us have two apples. Cut the first apple into two equal parts, and the second into 6 equal parts. It is clear that the share of the first apple will be different from the share of the second apple.

Depending on the number of shares that make up the whole object, these shares have their own names. Let's sort it out names of beats. If an object consists of two parts, any of them is called one second part of the whole object; if an object consists of three parts, then any of them is called one third part, and so on.

One second share has a special name - half. One third is called third, and one quarter part - a quarter.

For the sake of brevity, the following were introduced: beat symbols. One second share is designated as or 1/2, one third share is designated as or 1/3; one fourth share - like or 1/4, and so on. Note that the notation with a horizontal bar is used more often. To reinforce the material, let’s give one more example: the entry denotes one hundred and sixty-seventh part of the whole.

The concept of share naturally extends from objects to quantities. For example, one of the measures of length is the meter. To measure lengths shorter than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. The shares of other quantities are applied similarly.

Common fractions, definition and examples of fractions

To describe the number of shares we use common fractions. Let us give an example that will allow us to approach the definition of ordinary fractions.

Let the orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . We denote two beats as , three beats as , and so on, 12 beats we denote as . Each of the given entries is called an ordinary fraction.

Now let's give a general definition of common fractions.

Common fractions– these are records of the form (or m/n), where m and n are any natural numbers.

The voiced definition of ordinary fractions allows us to give examples of common fractions: 5/10, , 21/1, 9/4, . And here are the records do not fit the stated definition of ordinary fractions, that is, they are not ordinary fractions.

Numerator and denominator

For convenience, ordinary fractions are distinguished numerator and denominator.

Numerator common fraction (m/n) is a natural number m.

Denominator common fraction (m/n) is a natural number n.

So, the numerator is located above the fraction line (to the left of the slash), and the denominator is located below the fraction line (to the right of the slash). For example, let's take the common fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.

It remains to discuss the meaning contained in the numerator and denominator of an ordinary fraction. The denominator of a fraction shows how many parts one object consists of, and the numerator, in turn, indicates the number of such parts. For example, the denominator 5 of the fraction 12/5 means that one object consists of five shares, and the numerator 12 means that 12 such shares are taken.

Natural number as a fraction with denominator 1

The denominator of a common fraction can be equal to one. In this case, we can consider that the object is indivisible, in other words, it represents something whole. The numerator of such a fraction indicates how many whole objects are taken. Thus, an ordinary fraction of the form m/1 has the meaning of a natural number m. This is how we substantiated the validity of the equality m/1=m.

Let's rewrite the last equality as follows: m=m/1. This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103,498 is equal to the fraction 103,498/1.

So, any natural number m can be represented as an ordinary fraction with a denominator of 1 as m/1, and any ordinary fraction of the form m/1 can be replaced by a natural number m.

Fraction bar as a division sign

Representing the original object in the form of n shares is nothing more than division into n equal parts. After an item is divided into n shares, we can divide it equally among n people - each will receive one share.

If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects between n people, giving each person one share from each of the m objects. In this case, each person will have m shares of 1/n, and m shares of 1/n gives the common fraction m/n. Thus, the common fraction m/n can be used to denote the division of m items between n people.

This is how we got an explicit connection between ordinary fractions and division (see the general idea of ​​​​dividing natural numbers). This connection is expressed as follows: the fraction line can be understood as a division sign, that is, m/n=m:n .

Using an ordinary fraction, you can write the result of dividing two natural numbers for which a whole division cannot be performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, everyone will get five-eighths of an apple: 5:8 = 5/8.

Equal and unequal fractions, comparison of fractions

A fairly natural action is comparing fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as another 1/6 of this apple.

As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or unequal. In the first case we have equal common fractions, and in the second – unequal ordinary fractions. Let us give a definition of equal and unequal ordinary fractions.

Two common fractions a/b and c/d equal, if the equality a·d=b·c is true.

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Lesson 3. How a computer works

For successful “communication” with a computer, it is harmful to perceive it as a black box that is about to produce something unexpected. To understand the computer's reaction to your actions, you need to know how it works and how it works.

In that In the IT lesson we will learn how most computing devices (which include not only personal computers) work.

In the second lesson, we figured out that a computer is needed to process information, store and transmit it. Let's see how information is processed.

How information is stored on a computer

The computer stores, transmits and processes information in the form zeros "0" And units "1", that is, it is used binary code and the binary number system.

For example, the decimal number " 9 "he sees it as a binary number" 1001 ».

Stored in the form of zeros and ones all data that need to be processed and that's it programs, which guide the processing process.

For example, the computer sees a photograph like this (only the first two lines of a file of 527 lines):

This is how a person sees the image:

The computer sees a set of "0" and "1"

(first two lines of the file):



And the text for a computer looks like this:

A person sees the text:

The computer again sees a set of “0s” and “1s”:

Today we will not understand the intricacies of calculations and transformations, but will look at the process in general.

Where is the information stored?

When information is entered into a computer (recorded), it is stored on a special device - data storage device. Typically the data storage device is HDD (Winchester).

This device is called a hard drive because of its design. Inside its body there is one or more solid pancakes (metal or glass), on which all data is stored(text documents, photographs, films, etc.) and installed programs(operating system, application programs such as Word, Excel, etc.).



Hard disk (data storage) stores programs and data

Information on the hard drive is stored even after the computer is turned off.

We will learn more about the design of a hard drive in one of the following IT lessons.

What processes all the information in a computer?

The main task of a computer is process information, that is, perform calculations. Most of the calculations are performed by a special device - CPU. This is a complex microcircuit containing hundreds of millions of elements (transistors).



Processor - processes information

The program tells the processor what to do at a given time; it indicates what data needs to be processed and what needs to be done with it.



Data processing scheme

Programs and data are loaded from the storage device (hard drive).

But HDD – relatively slow device, and if the processor waited until the information was read, and then written back after processing, it would remain idle for a long time.

Let's not leave the processor idle

Therefore, a faster storage device was installed between the processor and the hard drive - RAM(random access memory, RAM). This is a small printed circuit board that contains fast memory chips.



RAM – speeds up processor access to programs and data

All necessary programs and data are read from the hard drive into the RAM in advance. During work processor accesses RAM, reads the commands of the program, which tells what data needs to be taken and how exactly to process it.

When you turn off your computer, the contents of the RAM are not saved there (unlike the hard drive).

Information Processing Process

So now we know which devices are involved in information processing. Let's now look at the entire calculation process.



Animation of the process of information processing by a computer (IT-uroki.ru)

When the computer is turned off, all programs and data are stored on the hard drive. When you turn on the computer and starting the program, the following happens:

1. The program from the hard drive is entered into RAM and tells the processor what data to load into RAM.

2. The processor alternately executes program commands, processing data in portions, taking them from RAM.

3. When the data is processed, the processor returns the calculation result to RAM and takes the next portion of data.

4. The result of the program is returned to the hard drive and saved.

The described steps are shown with red arrows in the animation (exclusively from the site IT-uroki.ru).

Input and output of information

In order for the computer to receive information for processing, it must be entered. For this purpose they are used input devices:

• Keyboard(using it we enter text and control the computer);
• Mouse(we use the mouse to control the computer);
• Scanner(put the image into the computer);
• Microphone(record sound), etc.

To display the result of information processing, we use data output devices:

• Monitor(display the image on the screen);
• Printer(we display the text and image on paper);
• Acustic systems or “speakers” (listening to sounds and music);

In addition, we can input and output data to other devices using:

• External drives(from them we copy existing data to the computer):
  • flash drive,
  • compact disc (CD or DVD),
  • Portable Hard Drive,
  • diskette;
• Computer network(we receive data from other computers via Internet or city network).

If we add input/output devices to our circuit, we get the following diagram:



Data input, processing and output

That is the computer works with ones and zeroes, and when the information arrives at the output device, it translated into familiar images(image, sound).

Let's sum it up

So, today we, together with the site IT-uroki.ru, found out how does a computer work. In short, the computer receives data from input devices (keyboard, mouse, etc.), stores it on the hard drive, then transfers it to RAM and processes it using the processor. The processing result is returned first to RAM, then either to the hard drive or directly to output devices (for example, a monitor).

If you have any questions, you can ask them in the comments to this article.

You can learn more about all the devices listed in today's lesson in subsequent lessons on the IT lessons website. In order not to miss new lessons, subscribe to the site news.

Copying prohibited

Let me remind you that the IT lessons website has constantly updated reference books:

Video supplement

Today is a short educational video about the production of processors.


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What 1st grade students should be able to do in mathematics by the end of the school year. The final test in mathematics is designed to test the knowledge, skills and abilities acquired by students by the end of the first year of study.

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Let's start with a simple example. Let's determine what the expression 2-5 is equal to. From point +2 we will put down five divisions, two to zero and three below zero. Let's stop at point -3. That is, 2-5=-3. Now notice that 2-5 is not at all equal to 5-2. If in the case of adding numbers their order does not matter, then in the case of subtraction everything is different. The order of the numbers matters.

Now let's go to negative area scales. Suppose we need to add +5 to -2. (From now on, we will put "+" signs in front of positive numbers and enclose both positive and negative numbers in parentheses so as not to confuse the signs in front of numbers with addition and subtraction signs.) Now our problem can be written as (-2)+ (+5). To solve it, we go up five divisions from point -2 and end up at point +3.

Is there any practical meaning to this task? Of course have. Let's say you have $2 in debt and you earned $5. This way, after you pay off the debt, you will have $3 left.

You can also move down the negative area of ​​the scale. Suppose you need to subtract 5 from -2, or (-2)-(+5). From point -2 on the scale, move down five divisions and end up at point -7. What is the practical meaning of this task? Let's say you owed $2 and had to borrow $5 more. You now owe $7.

We see that with negative numbers we can carry out the same addition and subtraction operations, as with the positive ones.

True, we have not yet mastered all operations. We only added to negative numbers and subtracted only positive ones from negative numbers. What should you do if you need to add negative numbers or subtract negative numbers from negative numbers?

In practice, this is similar to debt transactions. Let's say you were charged $5 in debt, it means the same thing as if you received $5. On the other hand, if I somehow force you to accept responsibility for someone else's $5 debt, that would be the same as taking that $5 away from you. That is, subtracting -5 is the same as adding +5. And adding -5 is the same as subtracting +5.

This allows us to get rid of the subtraction operation. Indeed, “5-2” is the same as (+5)-(+2) or according to our rule (+5)+(-2). In both cases we get the same result. From point +5 on the scale we need to go down two divisions and we get +3. In the case of 5-2 this is obvious, because subtraction is a downward movement.

In the case of (+5)+(-2) this is less obvious. We add a number, which means we move up the scale, but we add a negative number, which means we do the opposite, and these two factors taken together mean that we don't have to move up the scale, but in the opposite direction, that is down.

Thus, we again get the answer +3.

Why, exactly, is it necessary? replace subtraction with addition? Why move up “in the opposite sense”? Isn't it easier to just move down? The reason is that in the case of addition the order of the terms does not matter, but in the case of subtraction it is very important.

We already found out earlier that (+5)-(+2) is not at all the same as (+2)-(+5). In the first case the answer is +3, and in the second -3. On the other hand, (-2)+(+5) and (+5)+(-2) result in +3. Thus, by switching to addition and abandoning subtraction operations, we can avoid random errors associated with rearranging addends.

You can do the same when subtracting a negative. (+5)-(-2) is the same as (+5)+(+2). In both cases we get the answer +7. We start at point +5 and move “down in the opposite direction,” that is, up. We would act in exactly the same way when solving the expression (+5)+(+2).

Students actively use replacing subtraction with addition when they begin to study algebra, and therefore this operation is called "algebraic addition". In fact, this is not entirely fair, since such an operation is obviously arithmetic and not at all algebraic.

This knowledge is unchanged for everyone, so even if you receive education in Austria through www.salls.ru, although studying abroad is valued more highly, you will be able to apply these rules there too.

SUBTRACTION

Mathematics, 6th grade

(N.Ya.Vilenkin)

mathematics teacher of municipal educational institution "Upshinskaya basic"

comprehensive school" Orsha district of the Republic of Mari El




The meaning of subtraction

Task. A pedestrian walked 9 km in 2 hours. How many kilometers did he walk in the first hour if his distance in the second hour is 4 km?

In this problem the number 9 - amount two terms, one of which is equal 4 , and the other is unknown.

An action that uses the sum and one of the terms to find another term is called by subtraction.




The meaning of subtraction

Since 5 + 4 = 9,

then the required term is equal to 5.

They write 9 – 4 = 5

9 – 4 = 5

difference

subtrahend

minuend




The meaning of subtraction

– 5 + 14 = 9

9 – 14 = ?

? + 14 = 9

9 – 14 = –5

– 9 – 14 = ?

– 23 + 14 = –9

? + 14 = –9

– 9 – 14 = – 23




The meaning of subtraction

Subtracting negative numbers has the same meaning: The action by which the sum and one of the terms is used to find another term is called subtraction.

9 – (–14) = ?

23 + (–14) = 9

? + (–14) = 9

9 – (–14) = 23

Find the unknown term

– 9 – (–14) = ?

5 + (–14) = –9

? + (–14) = –9

– 9 – (–14) = 5




9 – (–14) = 23

9 – 14 = –5

9 + (–14) = –5

9 + 14 = 23

– 9 – (–14) = 5

– 9 – 14 = – 23

– 9 + (–14) = – 23

– 9 + 14 = 5

Think about how to replace subtraction with addition.

RULE. To subtract another from a given number, you need to add to the minuend the number opposite to the subtracted one.




SUBTRACTION

A – b =a + ( –b )

15 – 18 = 15 + ( –18 ) =

15 – ( –18 ) = 15 + 18 =




SUBTRACTION

Replace subtraction with addition and find the value of the expression:

12 – 20 =

3,4 – 10 =

– 10 – ( –13 ) =

– 1,2 – ( –1,3 ) =

17 – ( –13 ) =

2,3 – ( –3,5 ) =

– 21 – 13 =

– 5,1 – 4,9 =




SUBTRACTION

5 – 10 = 5 + ( – 10 )

RULE. Any expression containing only addition and subtraction signs can be considered as a sum

Name each term in the sum:

5 – 10 + 7 –15 –23 =

– n + y – 9 + b – c – 1 =




CALCULATE:

– 10 + 7 – 15 =

12 – 17 – 11 =

12 + 23 – 41 =

– 2 – 33 + 20 =

24 – 75 + 20 =


6 – 2 –5 RULE. The difference between two numbers is positive if the minuend is greater than the subtrahend. "width="640"

8 – 6 =

2

minuend

subtrahend

difference

– 2 – ( –5 ) =

3

minuend

difference

subtrahend

When is the difference between two numbers positive?

8 6

– 2 –5

RULE. The difference of two numbers is positive if minuend is greater than subtrahend .




10 – 15 =

– 5

minuend

subtrahend

difference

– 8 – ( –6 ) =

– 2

minuend

difference

subtrahend

Compare minuend and subtrahend in the examples.

When is the difference between two numbers negative?

10 15

– 8 –6

RULE. The difference of two numbers is negative if minuend is less than subtrahend .




Think about when the difference of two numbers is 0. Give examples.

0

minuend

difference

subtrahend

Determine the sign of the difference without performing calculations:

– 12 – ( –13 ) =

3,4 – 10 =

15 – ( –11 ) =

2,3 – ( –3,5 ) =

– 5,1 – 4,9 =

– 31 – 23 =




Finding the length of a segment

X

A (–3)

– 3 + x = 4

x = 4 – (–3) = 7

AT 4)

AB - ?

AB = 7 units.

RULE.




Finding the length of a segment

A (–1)

AB = –1 – (–5) = 4 units.

AT 5)

AB - ?

AB = 4 units.

RULE. To find the length of a segment on a coordinate line, you need to subtract the coordinate of its left end from the coordinate of its right end.




Questions for consolidation:

• What does subtracting negative numbers mean?
• How to replace subtraction with addition?
• When is the difference between two numbers positive?
• When is the difference between two numbers negative?
• When is the difference between two numbers equal to zero?
• How to find the length of a segment on a coordinate line?

primary school teacher MAOU Lyceum No. 21, Ivanovo




A LITTLE HISTORY

Indian mathematicians thought of positive numbers as "property" , and negative numbers are like "debts"

Rules for addition and subtraction as stated by Brahmagupta:

• “The sum of two properties is property.”
• "The sum of two debts is a debt"
• “The sum of property and debt is equal to their difference”

Brahmagupta, Indian mathematician and astronomer.