This article provides an overview properties of operations with rational numbers. First, the basic properties on which all other properties are based are announced. After this, some other frequently used properties of operations with rational numbers are given.
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Let's list basic properties of operations with rational numbers(a, b and c are arbitrary rational numbers):
- Commutative property of addition a+b=b+a.
- Matching property addition (a+b)+c=a+(b+c) .
- The existence of a neutral element by addition - zero, the addition of which with any number does not change this number, that is, a+0=a.
- For every rational number a there is an opposite number −a such that a+(−a)=0.
- Commutative property of multiplication of rational numbers a·b=b·a.
- Combinative property of multiplication (a·b)·c=a·(b·c) .
- The existence of a neutral element for multiplication is a unit, multiplication by which any number does not change this number, that is, a·1=a.
- For every non-zero rational number a there is an inverse number a −1 such that a·a −1 =1 .
- Finally, addition and multiplication of rational numbers are related by the distributive property of multiplication relative to addition: a·(b+c)=a·b+a·c.
The listed properties of operations with rational numbers are basic, since all other properties can be obtained from them.
Other important properties
In addition to the nine listed basic properties of operations with rational numbers, there are a number of very widely used properties. Let's give them a brief overview.
Let's start with the property, which is written using letters as a·(−b)=−(a·b) or by virtue of the commutative property of multiplication as (−a) b=−(a b). The rule for multiplying rational numbers with different signs directly follows from this property; its proof is also given in this article. Specified property explains the rule “plus multiplied by minus is minus, and minus multiplied by plus is minus.”
Here is the following property: (−a)·(−b)=a·b. This implies the rule for multiplying negative rational numbers; in this article you will also find a proof of the above equality. This property corresponds to the multiplication rule “minus times minus is plus.”
Undoubtedly, it is worth focusing on multiplying an arbitrary rational number a by zero: a·0=0 or 0 a=0. Let's prove this property. We know that 0=d+(−d) for any rational d, then a·0=a·(d+(−d)) . The distribution property allows the resulting expression to be rewritten as a·d+a·(−d) , and since a·(−d)=−(a·d) , then a·d+a·(−d)=a·d+(−(a·d)). So we came to the sum of two opposite numbers, equal to a·d and −(a·d), their sum gives zero, which proves the equality a·0=0.
It is easy to notice that above we listed only the properties of addition and multiplication, while not a word was said about the properties of subtraction and division. This is due to the fact that on the set of rational numbers, the actions of subtraction and division are specified as the inverse of addition and multiplication, respectively. That is, the difference a−b is the sum of a+(−b), and the quotient a:b is the product a·b−1 (b≠0).
Given these definitions of subtraction and division, as well as the basic properties of addition and multiplication, you can prove any properties of operations with rational numbers.
As an example, let’s prove the distribution property of multiplication relative to subtraction: a·(b−c)=a·b−a·c. The following chain of equalities holds: a·(b−c)=a·(b+(−c))= a·b+a·(−c)=a·b+(−(a·c))=a·b−a·c, which is the proof.
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Badamshinskaya high school №2
mathematics
in 6th grade
"Actions with rational numbers"
prepared
mathematic teacher
Babenko Larisa Grigorievna
With. Badamsha
2014
Lesson topic:« Operations with rational numbers».
Lesson type :
Lesson of generalization and systematization of knowledge.
Lesson objectives:
educational:
Summarize and systematize students’ knowledge about the rules of operations with positive and negative numbers;
Strengthen the ability to apply rules during exercises;
Develop independent work skills;
developing:
Develop logical thinking, math speech,computing skills; - develop the ability to apply acquired knowledge to solutions applied problems; - broadening your horizons;
raising:
Upbringing cognitive interest to the subject.
Equipment:
Sheets with texts of tasks, assignments for each student;
Mathematics. Textbook for 6th grade educational institutions/
N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S. I. Shvartsburd. – M., 2010.
Lesson plan:
Organizing time.
Work orally
Reviewing the rules for adding and subtracting numbers with different signs. Updating knowledge.
Solving tasks according to the textbook
Running the test
Summing up the lesson. Setting homework
Reflection
During the classes
Organizing time.
Greetings from teacher and students.
Report the topic of the lesson, the plan of work for the lesson.
Today we have unusual lesson. In this lesson we will remember all the rules of operations with rational numbers and the ability to perform addition, subtraction, multiplication and division operations.
The motto of our lesson will be a Chinese parable:
“Tell me and I will forget;
Show me and I will remember;
Let me do it and I’ll understand.”
I want to invite you on a journey.
In the middle of the space where the sunrise was clearly visible, stretched a narrow, uninhabited country - a number line. It is unknown where it began and it is unknown where it ended. And the first to populate this country were natural numbers. What numbers are called natural numbers and how are they designated?
Answer:
Numbers 1, 2, 3, 4,…..used to count objects or to indicate serial number one or another item among homogeneous objects, are called natural (N ).
Verbal counting
88-19 72:8 200-60
Answers: 134; 61; 2180.
There were an infinite number of them, but the country, although small in width, was infinite in length, so that everything from one to infinity fit in and formed the first state, a set of natural numbers.
Working on a task.
The country was extraordinarily beautiful. Magnificent gardens were located throughout its territory. These are cherry, apple, peach. We'll take a look at one of them now.
There are 20 percent more ripe cherries every three days. How many ripe fruits will this cherry have after 9 days, if at the beginning of observation there were 250 ripe cherries on it?
Answer: 432 ripe fruits will be on this cherry in 9 days (300; 360; 432).
Some new numbers began to settle on the territory of the first state, and these numbers, together with the natural ones, formed a new state, we will find out which one by solving the task.
The students have two sheets of paper on their desks:
1. Calculate:
1)-48+53 2)45-(-23) 3)-7.5:(-0.5) 4)-4x(-15)
1)56:(-8) 2)-3,3-4,7 3)-5,6:(-0,1) 4)9-12
1)48-54 2)37-(-37) 3)-52.7+42.7 4)-6x1/3
1)-12x(-6) 2)-90:(-15) 3)-25+45 4)6-(-10)
Exercise: Connect all the natural numbers in sequence without lifting your hand and name the resulting letter.
Answers to the test:
5 68 15 60
72 6 20 16
Question: What does this symbol mean? What numbers are called integers?
Answers: 1) To the left, from the territory of the first state, the number 0 settled, to the left of it -1, even further to the left -2, etc. to infinity. These numbers, together with the natural numbers, formed a new extended state, the set of integers.
2) Natural numbers, their opposite numbers and zero are called integers ( Z ).
Repetition of what has been learned.
1) The next page of our fairy tale is enchanted. Let's disenchant it, correcting mistakes.
27 · 4 0 -27 = 27 0 · (-27) = 0
63 3 0 · 40 (-6) · (-6) -625 124
50 · 8 27 -18: (-2)
Answers:
-27 4 27 0 (-27) = 0
-50 8 4 -36: 6
2) Let's continue listening to the story.
On free places fractions 2/5 were added to the number line; −4/5; 3.6; −2,2;... Fractions, together with the first settlers, formed the next expanded state - a set of rational numbers. ( Q)
1)What numbers are called rational?
2) Is any integer or decimal fraction a rational number?
3) Show that any integer, any decimal fraction is a rational number.
Task on the board: 8; 3 ; -6; - ; - 4,2; – 7,36; 0; .
Answers:
1) A number that can be written as a ratio , where a is an integer and n is a natural number, is called a rational number .
2) Yes.
3) .
You now know integers and fractions, positive and negative numbers, and also the number zero. All these numbers are called rational, which translated into Russian means “ subject to the mind."
positive zero negative
whole fractional whole fractional
In order to successfully study mathematics (and not only mathematics) in the future, you need to have a good knowledge of the rules of arithmetic operations with rational numbers, including the rules of signs. And they are so different! It won't take long to get confused.
Physical education minute.
Dynamic pause.
Teacher: Any work requires a break. Let's rest!
Let's do recovery exercises:
1) One, two, three, four, five -
Once! Get up, pull yourself up,
Two! Bend over, straighten up,
Three! Three claps of your hands,
Three nods of the head.
Four means wider hands.
Five - wave your arms. Six - sit quietly at your desk.
(Children perform movements following the teacher according to the content of the text.)
2) Blink quickly, close your eyes and sit there for a count of five. Repeat 5 times.
3) Close your eyes tightly, count to three, open them and look into the distance, counting to five. Repeat 5 times.
Historical page.
In life, as in fairy tales, people “discovered” rational numbers gradually. At first, when counting objects, natural numbers arose. At first there were few of them. At first, only the numbers 1 and 2 arose. The words “soloist”, “sun”, “solidarity” come from the Latin “solus” (one). Many tribes did not have other numerals. Instead of “3” they said “one-two”, instead of “4” they said “two-two”. And so on until six. And then came “a lot.” People came across fractions when dividing up spoils and when measuring quantities. To make it easier to work with fractions, they were invented decimals. They were introduced in Europe in 1585 by a Dutch mathematician.
Working on Equations
You will find out the name of a mathematician by solving equations and using the coordinate line to find the letter corresponding to a given coordinate.
1) -2.5 + x = 3.5 2) -0.3 x = 0.6 3) y – 3.4 = -7.4
4) – 0.8: x = -0.4 5)a · (-8) =0 6)m + (- )=
E A T M I O V R N U S
-4 -3 -2 -1 0 1 2 3 4 5 6
Answers:
6 (C) 4)2 (B)
-2 (T) 5) 0 (I)
-4(E) 6)4(H)
STEVIN - Dutch mathematician and engineer (Simon Stevin)
Historical page.
Teacher:
Without knowing the past in the development of science, it is impossible to understand its present. People learned to perform operations with negative numbers even before our era. Indian mathematicians thought of positive numbers as “properties” and negative numbers as “debts.” This is how the Indian mathematician Brahmagupta (7th century) set out some rules for performing operations with positive and negative numbers:
"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,”
“The product of two assets or two debts is property,” “The product of assets and debt is debt.”
Guys, please translate the ancient Indian rules into modern language.
Teacher's message:
How can there be no life without sun heat,
Without winter snow and without flower leaves,
There are no operations without signs in mathematics!
The children are asked to guess which action sign is missing.
Exercise. Fill in the missing character.
− 1,2 1,4 = − 2,6
3,2 (− 8) = − 0,4
1 (− 1,7) = 2,7
− 4,5 (− 0,5) = 9
− 1,3 2,8 = 1,5
Answers: 1) + 2) ∙ 3) − 4) : 5) − 6) :
Independent work(write down the answers to the tasks on the sheet):
Compare numbers
find their modules
compare with zero
find their sum
find their difference
find the work
find the quotient
write the opposite numbers
find the distance between these numbers
10) how many integers are located between them
11) find the sum of all integers located between them.
Evaluation criteria: everything was solved correctly – “5”
1-2 errors - “4”
3-4 errors - “3”
more than 4 errors - “2”
Individual work by cards(additionally).
Card 1. Solve the equation: 8.4 – (x – 3.6) = 18
Card 2. Solve the equation: -0.2x · (-4) = -0,8
Card 3. Solve the equation: =
Answers to cards :
1) 6; 2) -1; 3) 4/15.
Game "Exam".
The inhabitants of the country lived happily, played games, solved problems, equations and invited us to play in order to sum up the results.
Students go to the board, take a card and answer the question written on the back.
Questions:
1. Which of two negative numbers is considered larger?
2. Formulate the rule for dividing negative numbers.
3. Formulate the rule for multiplying negative numbers.
4. Formulate a rule for multiplying numbers with different signs.
5. Formulate a rule for dividing numbers with different signs.
6. Formulate the rule for adding negative numbers.
7. Formulate a rule for adding numbers with different signs.
8.How to find the length of a segment on a coordinate line?
9.What numbers are called integers?
10. What numbers are called rational?
Summarizing.
Teacher: Today homework will be creative:
Prepare a message “Positive and negative numbers around us” or compose a fairy tale.
« Thank you for the lesson!!!"
Drawing. Arithmetic operations over rational numbers.
Text:
Rules for operations with rational numbers:
. when adding numbers with identical signs it is necessary to add up their modules and put them in front of the sum general sign;
. when adding two numbers with different signs, from a number with a larger modulus, subtract the number with a smaller modulus and put the sign of the number with a larger modulus in front of the resulting difference;
. When subtracting one number from another, you need to add to the minuend the number opposite to the one being subtracted: a - b = a + (-b)
. when multiplying two numbers with the same signs, their modules are multiplied and a plus sign is placed in front of the resulting product;
. when multiplying two numbers with different signs, their modules are multiplied and a minus sign is placed in front of the resulting product;
. when dividing numbers with the same signs, the module of the dividend is divided by the module of the divisor and a plus sign is placed in front of the resulting quotient;
. when dividing numbers with different signs, the module of the dividend is divided by the module of the divisor and a minus sign is placed in front of the resulting quotient;
. when dividing and multiplying zero by any number, do not equal to zero, it turns out zero:
. You can't divide by zero.
In this lesson we will recall the basic properties of operations with numbers. We will not only review the basic properties, but also learn how to apply them to rational numbers. We will consolidate all the knowledge gained by solving examples.
Basic properties of operations with numbers:
The first two properties are properties of addition, the next two are properties of multiplication. The fifth property applies to both operations.
There is nothing new in these properties. They were valid for both natural and integer numbers. They are also true for rational numbers and will be true for the numbers we will study next (for example, irrational numbers).
Permutation properties:
Rearranging the terms or factors does not change the result.
Combination properties:, .
Adding or multiplying multiple numbers can be done in any order.
Distribution property:.
The property connects both operations - addition and multiplication. Also, if it is read from left to right, then it is called the rule for opening parentheses, and if in reverse side- rule of adjudication common multiplier out of brackets.
The following two properties describe neutral elements for addition and multiplication: adding zero and multiplying by one does not change the original number.
Two more properties that describe symmetrical elements for addition and multiplication, the sum of opposite numbers is zero; work reciprocal numbers equals one.
Next property: . If a number is multiplied by zero, the result will always be zero.
The last property we'll look at is: .
Multiplying a number by , we get the opposite number. This property has a special feature. All other properties considered could not be proven using the others. The same property can be proven using the previous ones.
Multiplying by
Let's prove that if we multiply a number by , we get the opposite number. For this we use the distribution property: .
This is true for any numbers. Let's substitute and instead of the number:
On the left in parentheses is the sum of mutually opposite numbers. Their sum is zero (we have such a property). On the left now. On the right, we get: 
.
Now we have zero on the left, and the sum of two numbers on the right. But if the sum of two numbers is zero, then these numbers are mutually opposite. But the number has only one opposite number: . So, this is what it is: .
The property has been proven.
Such a property, which can be proven using previous properties, is called theorem
Why are there no subtraction and division properties here? For example, one could write the distributive property for subtraction: .
But since:
- Subtracting any number can be equivalently written as addition by replacing the number with its opposite:
- Division can be written as multiplication by its reciprocal:
This means that the properties of addition and multiplication can be applied to subtraction and division. As a result, the list of properties that need to be remembered is shorter.
All the properties we have considered are not exclusively properties of rational numbers. Other numbers, for example, irrational ones, also obey all these rules. For example, the sum of its opposite number is zero: .
Now we will move on to the practical part, solving several examples.
Rational numbers in life
Those properties of objects that we can describe quantitatively, designate with some number, are called values: length, weight, temperature, quantity.
The same quantity can be denoted by both an integer and a fractional number, positive or negative.
For example, your height is m - a fractional number. But we can say that it is equal to cm - this is already an integer (Fig. 1).
Rice. 1. Illustration for example
One more example. A negative temperature on the Celsius scale will be positive on the Kelvin scale (Fig. 2).
Rice. 2. Illustration for example
When building the wall of a house, one person can measure the width and height in meters. He succeeds fractional values. He will carry out all further calculations with fractional (rational) numbers. Another person can measure everything in the number of bricks in width and height. Having received only integer values, he will carry out calculations with integers.
The quantities themselves are neither integer nor fractional, neither negative nor positive. But the number with which we describe the value of a quantity is already quite specific (for example, negative and fractional). It depends on the measurement scale. And when we move from real values to mathematical model, then we work with a specific type of numbers
Let's start with addition. The terms can be rearranged in any way that is convenient for us, and the actions can be performed in any order. If terms of different signs end in the same digit, then it is convenient to perform operations with them first. To do this, let's swap the terms. For example:
Common fractions with same denominators easy to fold.
Opposite numbers add up to zero. Numbers with the same decimal tails are easy to subtract. Using these properties, as well as the commutative law of addition, you can make it easier to calculate the value of, for example, the following expression:
Numbers with complementary decimal tails are easy to add. With whole and in fractional parts mixed numbers convenient to work separately. We use these properties when calculating the value of the following expression:
Let's move on to multiplication. There are pairs of numbers that are easy to multiply. Using the commutative property, you can rearrange the factors so that they are adjacent. The number of minuses in a product can be counted immediately and a conclusion can be drawn about the sign of the result.
Consider this example:
If from the factors equal to zero, then the product is equal to zero, for example: .
The product of reciprocal numbers is equal to one, and multiplication by one does not change the value of the product. Consider this example:
Let's look at an example using the distributive property. If you open the parentheses, then each multiplication is easy.