develop the ability to open brackets, taking into account the sign in front of the brackets;
During the classes
I. Organizational moment.
Check it out buddy
Are you ready for class?
Is everything in place? Everything is fine?
Pen, book and notebook.
Is everyone sitting correctly?
Is everyone watching carefully?
I want to start the lesson with a question for you:
What do you think is the most valuable thing on Earth? (Children's answers.)
This question has worried humanity for thousands of years. This is the answer given by the famous scientist Al-Biruni: “Knowledge is the most excellent of possessions. Everyone strives for it, but it doesn’t come on its own.”
Let these words become the motto of our lesson.
II. Updating of previous knowledge, skills and abilities:
Verbal counting:
1.1. What is today's date?
2. Tell me what you know about the number 20?
3. Where is this number located on the coordinate line?
4. Give the opposite number.
5. Name the opposite number.
6. What is the name of the number 20?
7. What numbers are called opposites?
8. What numbers are called negative?
9. What is the modulus of the number 20? - 20?
10. What is the sum of opposite numbers?
2. Explain the following entries:
a) The brilliant ancient mathematician Archimedes was born in 0 287.
b) The brilliant Russian mathematician N.I. Lobachevsky was born in 1792.
c) The first Olympic Games took place in Greece in 776.
d) The first International Olympic Games took place in 1896.
e) The XXII Olympic Winter Games took place in 2014.
3. Find out what numbers are spinning on the “mathematical carousel” (all actions are performed orally).
II. Formation of new knowledge, skills and abilities.
You have learned how to perform various operations with integers. What will we do next? How will we solve examples and equations?
Let's find the meaning of these expressions
7 + (3 + 4) = -7 + 7 = 0
-7 + 3 + 4 = 0
What is the procedure in example 1? How much is in brackets? What is the procedure in the second example? The result of the first action? What can you say about these expressions?
Of course, the results of the first and second expressions are the same, which means you can put an equal sign between them: -7 + (3 + 4) = -7 + 3 + 4
What did we do with the brackets? (They lowered it.)
What do you think we will do in class today? (Children formulate the topic of the lesson.) In our example, what sign comes before the brackets. (Plus.)
And so we come to the next rule:
If there is a + sign in front of the parentheses, then you can omit the parentheses and this + sign, preserving the signs of the terms in the parentheses. If the first term in brackets is written without a sign, then it must be written with a + sign.
But what if there is a minus sign before the brackets?
In this case, you need to reason in the same way as when subtracting: you need to add the number opposite to the one being subtracted:
7 – (3 + 4) = -7 + (-7) = -7 + (-3) + (-4) = -7 – 3 – 4 = -14
- So, we opened the parentheses when there was a minus sign in front of them.
The rule for opening parentheses is when the parentheses are preceded by a “-“ sign.
To open parentheses preceded by a - sign, you need to replace this sign with +, changing the signs of all terms in the parentheses to the opposite, and then open the parentheses.
Let's listen to the rules for opening parentheses in poetry:
There is a plus before the parenthesis.
That's what he's talking about
Why do you omit the parentheses?
Let out all the signs!
Before the parenthesis the minus is strict
Will block our way
To remove brackets
We need to change the signs!
Yes, guys, the minus sign is very insidious, it is a “watchman” at the gate (brackets), it releases numbers and variables only when they change their “passports,” that is, their signs.
Why do you need to open the parentheses at all? (When there are parentheses, there is a moment of some element of incompleteness, some kind of mystery. It’s like closed door, behind which there is something interesting.) Today we learned this secret.
A short excursion into history:
Curly braces appear in the writings of Vieta (1593). Brackets became widely used only in the first half of the 18th century, thanks to Leibniz and even more so to Euler.
Physical education minute.
III. Consolidation of new knowledge, skills and abilities.
Work according to the textbook:
No. 1234 (open the brackets) – orally.
No. 1236 (open the brackets) – orally.
No. 1235 (find the meaning of the expression) - in writing.
No. 1238 (simplify the expressions) – work in pairs.
IV. Summing up the lesson.
1. Grades are announced.
2. Home. exercise. paragraph 39 No. 1254 (a, b, c), 1255 (a, b, c), 1259.
3. What have we learned today?
What new did you learn?
And I want to end the lesson with wishes to each of you:
“Show your ability for mathematics,
Don't be lazy, but develop every day.
Multiply, divide, work, think,
Don’t forget to be friends with mathematics.”
Expanding parentheses is a type of expression transformation. In this section we will describe the rules for opening parentheses, and also look at the most common examples of problems.
Yandex.RTB R-A-339285-1
What is opening parentheses?
Parentheses are used to indicate the order in which actions are performed in numeric and literal expressions, as well as in expressions with variables. It is convenient to move from an expression with brackets to an identically equal expression without brackets. For example, replace the expression 2 · (3 + 4) with an expression of the form 2 3 + 2 4 without parentheses. This technique is called opening brackets.
Definition 1
Expanding parentheses refers to techniques for getting rid of parentheses and is usually considered in relation to expressions that may contain:
- signs “+” or “-” before parentheses containing sums or differences;
- the product of a number, letter or several letters and a sum or difference, which is placed in brackets.
This is how we are used to considering the process of opening brackets in the course school curriculum. However, no one is stopping us from looking at this action more broadly. We can call parenthesis opening the transition from an expression that contains negative numbers in parentheses to an expression that does not have parentheses. For example, we can go from 5 + (− 3) − (− 7) to 5 − 3 + 7. In fact, this is also an opening of parentheses.
In the same way, we can replace the product of expressions in brackets of the form (a + b) · (c + d) with the sum a · c + a · d + b · c + b · d. This technique also does not contradict the meaning of opening parentheses.
Here's another example. We can assume that any expressions can be used instead of numbers and variables in expressions. For example, the expression x 2 · 1 a - x + sin (b) will correspond to an expression without parentheses of the form x 2 · 1 a - x 2 · x + x 2 · sin (b).
One more point deserves special attention, which concerns the peculiarities of recording decisions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as an equality. For example, after expanding the parentheses instead of the expression 3 − (5 − 7) we get the expression 3 − 5 + 7 . We can write both of these expressions as the equality 3 − (5 − 7) = 3 − 5 + 7.
Carrying out actions with cumbersome expressions may require recording intermediate results. Then the solution will have the form of a chain of equalities. For example, 5 − (3 − (2 − 1)) = 5 − (3 − 2 + 1) = 5 − 3 + 2 − 1 or 5 − (3 − (2 − 1)) = 5 − 3 + (2 − 1) = 5 − 3 + 2 − 1 .
Rules for opening parentheses, examples
Let's start looking at the rules for opening parentheses.
For single numbers in brackets
Negative numbers in parentheses are often found in expressions. For example, (− 4) and 3 + (− 4) . Positive numbers in brackets also have a place.
Let us formulate a rule for opening parentheses containing single positive numbers. Let's assume that a is any positive number. Then we can replace (a) with a, + (a) with + a, - (a) with – a. If instead of a we take a specific number, then according to the rule: the number (5) will be written as 5 , expression 3 + (5) without brackets will take the form 3 + 5 , since + (5) is replaced by + 5 , and the expression 3 + (− 5) is equivalent to the expression 3 − 5 , because + (− 5) is replaced by − 5 .
Positive numbers are usually written without using parentheses, since parentheses are unnecessary in this case.
Now consider the rule for opening parentheses that contain a single negative number. + (− a) we replace with − a, − (− a) is replaced by + a. If the expression starts with a negative number (−a), which is written in brackets, then the brackets are omitted and instead (−a) remains − a.
Here are some examples: (− 5) can be written as − 5, (− 3) + 0, 5 becomes − 3 + 0, 5, 4 + (− 3) becomes 4 − 3 , and − (− 4) − (− 3) after opening the brackets takes the form 4 + 3, since − (− 4) and − (− 3) is replaced by + 4 and + 3 .
It should be understood that the expression 3 · (− 5) cannot be written as 3 · − 5. This will be discussed in the following paragraphs.
Let's see what the rules for opening parentheses are based on.
According to the rule, the difference a − b is equal to a + (− b) . Based on the properties of actions with numbers, we can create a chain of equalities (a + (− b)) + b = a + ((− b) + b) = a + 0 = a which will be fair. This chain of equalities, by virtue of the meaning of subtraction, proves that the expression a + (− b) is the difference a − b.
Based on the properties of opposite numbers and the rules for subtracting negative numbers, we can state that − (− a) = a, a − (− b) = a + b.
There are expressions that are made up of a number, minus signs and several pairs of parentheses. Using the above rules allows you to sequentially get rid of brackets, moving from inner to outer brackets or in the opposite direction. An example of such an expression would be − (− ((− (5)))) . Let's open the brackets, moving from inside to outside: − (− ((− (5)))) = − (− ((− 5))) = − (− (− 5)) = − (5) = − 5 . This example can also be analyzed in the opposite direction: − (− ((− (5)))) = ((− (5))) = (− (5)) = − (5) = − 5 .
Under a and b can be understood not only as numbers, but also as arbitrary numeric or alphabetic expressions with a "+" sign in front that are not sums or differences. In all these cases, you can apply the rules in the same way as we did for single numbers in parentheses.
For example, after opening the parentheses the expression − (− 2 x) − (x 2) + (− 1 x) − (2 x y 2: z) will take the form 2 · x − x 2 − 1 x − 2 · x · y 2: z . How did we do it? We know that − (− 2 x) is + 2 x, and since this expression comes first, then + 2 x can be written as 2 x, − (x 2) = − x 2, + (− 1 x) = − 1 x and − (2 x y 2: z) = − 2 x y 2: z.
In products of two numbers
Let's start with the rule for opening parentheses in the product of two numbers.
Let's pretend that a and b are two positive numbers. In this case, the product of two negative numbers − a and − b of the form (− a) · (− b) we can replace with (a · b) , and the products of two numbers with opposite signs of the form (− a) · b and a · (− b) can be replaced with (− a b). Multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus gives a minus.
The correctness of the first part of the written rule is confirmed by the rule for multiplying negative numbers. To confirm the second part of the rule, we can use the rules for multiplying numbers with different signs.
Let's look at a few examples.
Example 1
Let's consider an algorithm for opening parentheses in the product of two negative numbers - 4 3 5 and - 2, of the form (- 2) · - 4 3 5. To do this, replace the original expression with 2 · 4 3 5 . Let's open the brackets and get 2 · 4 3 5 .
And if we take the quotient of negative numbers (− 4) : (− 2), then the entry after opening the brackets will look like 4: 2
In place of negative numbers − a and − b can be any expressions with a minus sign in front that are not sums or differences. For example, these can be products, quotients, fractions, powers, roots, logarithms, trigonometric functions and so on.
Let's open the brackets in the expression - 3 · x x 2 + 1 · x · (- ln 5) . According to the rule, we can make the following transformations: - 3 x x 2 + 1 x (- ln 5) = - 3 x x 2 + 1 x ln 5 = 3 x x 2 + 1 x ln 5.
Expression (− 3) 2 can be converted into the expression (− 3 2) . After this you can expand the brackets: − 3 2.
2 3 · - 4 5 = - 2 3 · 4 5 = - 2 3 · 4 5
Dividing numbers with different signs may also require preliminary expansion of parentheses: (− 5) : 2 = (− 5: 2) = − 5: 2 and 2 3 4: (- 3, 5) = - 2 3 4: 3, 5 = - 2 3 4: 3, 5.
The rule can be used to perform multiplication and division of expressions with different signs. Let's give two examples.
1 x + 1: x - 3 = - 1 x + 1: x - 3 = - 1 x + 1: x - 3
sin (x) (- x 2) = (- sin (x) x 2) = - sin (x) x 2
In products of three or more numbers
Let's move on to products and quotients, which contain a larger number of numbers. To expand the parentheses will work here next rule. If there is an even number of negative numbers, you can omit the parentheses and replace the numbers with their opposites. After this, you need to enclose the resulting expression in new parentheses. If there are an odd number of negative numbers, omit the parentheses and replace the numbers with their opposites. After this, the resulting expression must be placed in new brackets and a minus sign must be placed in front of it.
Example 2
For example, take the expression 5 · (− 3) · (− 2) , which is the product of three numbers. There are two negative numbers, therefore we can write the expression as (5 · 3 · 2) and then finally open the brackets, obtaining the expression 5 · 3 · 2.
In the product (− 2, 5) · (− 3) : (− 2) · 4: (− 1, 25) : (− 1) five numbers are negative. therefore (− 2, 5) · (− 3) : (− 2) · 4: (− 1, 25) : (− 1) = (− 2, 5 · 3: 2 · 4: 1, 25: 1) . Having finally opened the brackets, we get −2.5 3:2 4:1.25:1.
The above rule can be justified as follows. Firstly, we can rewrite such expressions as a product, replacing division by multiplication by the reciprocal number. We represent each negative number as the product of a multiplying number and - 1 or - 1 is replaced by (− 1) a.
Using the commutative property of multiplication, we swap factors and transfer all factors equal to − 1 , to the beginning of the expression. The product of an even number minus one is equal to 1, and the product of an odd number is equal to − 1 , which allows us to use the minus sign.
If we did not use the rule, then the chain of actions to open the parentheses in the expression - 2 3: (- 2) · 4: - 6 7 would look like this:
2 3: (- 2) 4: - 6 7 = - 2 3 - 1 2 4 - 7 6 = = (- 1) 2 3 (- 1) 1 2 4 (- 1 ) · 7 6 = = (- 1) · (- 1) · (- 1) · 2 3 · 1 2 · 4 · 7 6 = (- 1) · 2 3 · 1 2 · 4 · 7 6 = = - 2 3 1 2 4 7 6
The above rule can be used when opening parentheses in expressions that represent products and quotients with a minus sign that are not sums or differences. Let's take for example the expression
x 2 · (- x) : (- 1 x) · x - 3: 2 .
It can be reduced to the expression without parentheses x 2 · x: 1 x · x - 3: 2.
Expanding parentheses preceded by a + sign
Consider a rule that can be applied to expand parentheses that are preceded by a plus sign, and the “contents” of those parentheses are not multiplied or divided by any number or expression.
According to the rule, the brackets, together with the sign in front of them, are omitted, while the signs of all terms in the brackets are preserved. If there is no sign before the first term in parentheses, then you need to put a plus sign.
Example 3
For example, we give the expression (12 − 3 , 5) − 7 . By omitting the parentheses, we keep the signs of the terms in parentheses and put a plus sign in front of the first term. The entry will look like (12 − 3, 5) − 7 = + 12 − 3, 5 − 7. In the example given, it is not necessary to place a sign in front of the first term, since + 12 − 3, 5 − 7 = 12 − 3, 5 − 7.
Example 4
Let's look at another example. Let's take the expression x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x and carry out the actions with it x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x = = x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x
Here's another example of expanding parentheses:
Example 5
2 + x 2 + 1 x - x y z + 2 x - 1 + (- 1 + x - x 2) = = 2 + x 2 + 1 x - x y z + 2 x - 1 - 1 + x + x 2
How are parentheses preceded by a minus sign expanded?
Let's consider cases where there is a minus sign in front of the parentheses, and which are not multiplied (or divided) by any number or expression. According to the rule for opening brackets preceded by a “-” sign, brackets with a “-” sign are omitted, and the signs of all terms inside the brackets are reversed.
Example 6
Eg:
1 2 = 1 2 , - 1 x + 1 = - 1 x + 1 , - (- x 2) = x 2
Expressions with variables can be converted using the same rule:
X + x 3 - 3 - - 2 x 2 + 3 x 3 x + 1 x - 1 - x + 2,
we get x - x 3 - 3 + 2 · x 2 - 3 · x 3 · x + 1 x - 1 - x + 2 .
Opening parentheses when multiplying a number by a parenthesis, expressions by a parenthesis
Here we will look at cases where you need to expand parentheses that are multiplied or divided by some number or expression. Formulas of the form (a 1 ± a 2 ± … ± a n) b = (a 1 b ± a 2 b ± … ± a n b) or b · (a 1 ± a 2 ± … ± a n) = (b · a 1 ± b · a 2 ± … ± b · a n), Where a 1 , a 2 , … , a n and b are some numbers or expressions.
Example 7
For example, let's expand the parentheses in the expression (3 − 7) 2. According to the rule, we can carry out the following transformations: (3 − 7) · 2 = (3 · 2 − 7 · 2) . We get 3 · 2 − 7 · 2 .
Opening the parentheses in the expression 3 x 2 1 - x + 1 x + 2, we get 3 x 2 1 - 3 x 2 x + 3 x 2 1 x + 2.
Multiplying parenthesis by parenthesis
Consider the product of two brackets of the form (a 1 + a 2) · (b 1 + b 2) . This will help us obtain a rule for opening parentheses when performing bracket-by-bracket multiplication.
In order to solve the given example, we denote the expression (b 1 + b 2) like b. This will allow us to use the rule for multiplying a parenthesis by an expression. We get (a 1 + a 2) · (b 1 + b 2) = (a 1 + a 2) · b = (a 1 · b + a 2 · b) = a 1 · b + a 2 · b. By performing a reverse replacement b by (b 1 + b 2), again apply the rule of multiplying an expression by a bracket: a 1 b + a 2 b = = a 1 (b 1 + b 2) + a 2 (b 1 + b 2) = = (a 1 b 1 + a 1 b 2) + (a 2 b 1 + a 2 b 2) = = a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 b 2
Thanks to a number of simple techniques, we can arrive at the sum of the products of each of the terms from the first bracket by each of the terms from the second bracket. The rule can be extended to any number of terms inside the brackets.
Let us formulate the rules for multiplying brackets by brackets: to multiply two sums together, you need to multiply each of the terms of the first sum by each of the terms of the second sum and add the results.
The formula will look like:
(a 1 + a 2 + . . . + a m) · (b 1 + b 2 + . . . + b n) = = a 1 b 1 + a 1 b 2 + . . . + a 1 b n + + a 2 b 1 + a 2 b 2 + . . . + a 2 b n + + . . . + + a m b 1 + a m b 1 + . . . a m b n
Let's expand the brackets in the expression (1 + x) · (x 2 + x + 6) It is the product of two sums. Let's write the solution: (1 + x) · (x 2 + x + 6) = = (1 · x 2 + 1 · x + 1 · 6 + x · x 2 + x · x + x · 6) = = 1 · x 2 + 1 x + 1 6 + x x 2 + x x + x 6
It is worth mentioning separately those cases where there is a minus sign in parentheses along with plus signs. For example, take the expression (1 − x) · (3 · x · y − 2 · x · y 3) .
First, let's present the expressions in brackets as sums: (1 + (− x)) · (3 · x · y + (− 2 · x · y 3)). Now we can apply the rule: (1 + (− x)) · (3 · x · y + (− 2 · x · y 3)) = = (1 · 3 · x · y + 1 · (− 2 · x · y 3) + (− x) · 3 · x · y + (− x) · (− 2 · x · y 3))
Let's open the brackets: 1 · 3 · x · y − 1 · 2 · x · y 3 − x · 3 · x · y + x · 2 · x · y 3 .
Expanding parentheses in products of multiple parentheses and expressions
If there are three or more expressions in parentheses in an expression, the parentheses must be opened sequentially. You need to start the transformation by putting the first two factors in brackets. Within these brackets we can carry out transformations according to the rules discussed above. For example, the parentheses in the expression (2 + 4) · 3 · (5 + 7 · 8) .
The expression contains three factors at once (2 + 4) , 3 and (5 + 7 8) . We will open the brackets sequentially. Let's enclose the first two factors in another bracket, which we'll make red for clarity: (2 + 4) 3 (5 + 7 8) = ((2 + 4) 3) (5 + 7 8).
In accordance with the rule for multiplying a bracket by a number, we can carry out the following actions: ((2 + 4) · 3) · (5 + 7 · 8) = (2 · 3 + 4 · 3) · (5 + 7 · 8) .
Multiply bracket by bracket: (2 3 + 4 3) (5 + 7 8) = 2 3 5 + 2 3 7 8 + 4 3 5 + 4 3 7 8 .
Bracket in kind
Degrees whose bases are some expressions written in brackets, with in kind can be thought of as the product of several brackets. Moreover, according to the rules from the two previous paragraphs, they can be written without these brackets.
Consider the process of transforming the expression (a + b + c) 2 . It can be written as the product of two brackets (a + b + c) · (a + b + c). Let's multiply bracket by bracket and get a · a + a · b + a · c + b · a + b · b + b · c + c · a + c · b + c · c.
Let's look at another example:
Example 8
1 x + 2 3 = 1 x + 2 1 x + 2 1 x + 2 = = 1 x 1 x + 1 x 2 + 2 1 x + 2 2 1 x + 2 = = 1 x · 1 x · 1 x + 1 x · 2 · 1 x + 2 · 1 x · 1 x + 2 · 2 · 1 x + 1 x · 1 x · 2 + + 1 x 2 · 2 + 2 · 1 x · 2 + 2 2 2
Dividing parenthesis by number and parentheses by parenthesis
Dividing a bracket by a number requires that all terms enclosed in brackets be divided by the number. For example, (x 2 - x) : 4 = x 2: 4 - x: 4 .
Division can first be replaced by multiplication, after which you can use the appropriate rule for opening parentheses in a product. The same rule applies when dividing a parenthesis by a parenthesis.
For example, we need to open the parentheses in the expression (x + 2) : 2 3 . To do this, first replace division by multiplying by the reciprocal number (x + 2): 2 3 = (x + 2) · 2 3. Multiply the bracket by the number (x + 2) · 2 3 = x · 2 3 + 2 · 2 3 .
Here's another example of division by parenthesis:
Example 9
1 x + x + 1: (x + 2) .
Let's replace division with multiplication: 1 x + x + 1 · 1 x + 2.
Let's do the multiplication: 1 x + x + 1 · 1 x + 2 = 1 x · 1 x + 2 + x · 1 x + 2 + 1 · 1 x + 2 .
Order of opening brackets
Now consider the order of application of the rules discussed above in the expressions general view, i.e. in expressions that contain sums with differences, products with quotients, parentheses to the natural degree.
Procedure:
- the first step is to raise the brackets to a natural power;
- at the second stage, the opening of brackets in works and quotients is carried out;
- The final step is to open the parentheses in the sums and differences.
Let's consider the order of actions using the example of the expression (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) . Let us transform from the expressions 3 · (− 2) : (− 4) and 6 · (− 7) , which should take the form (3 2:4) and (− 6 · 7) . When substituting the obtained results into the original expression, we obtain: (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) = (− 5) + (3 · 2: 4) − (− 6 · 7) . Open the brackets: − 5 + 3 · 2: 4 + 6 · 7.
When dealing with expressions that contain parentheses within parentheses, it is convenient to carry out transformations by working from the inside out.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Parentheses are used to indicate the order in which actions are performed in numeric, literal, and variable expressions. It is convenient to move from an expression with brackets to an identically equal expression without brackets. This technique is called opening brackets.
Expanding parentheses means removing the parentheses from an expression.
One more point deserves special attention, which concerns the peculiarities of recording decisions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as an equality. For example, after expanding the parentheses instead of the expression
3−(5−7) we get the expression 3−5+7. We can write both of these expressions as the equality 3−(5−7)=3−5+7.
And one more important point. In mathematics, to shorten notations, it is customary not to write the plus sign if it appears first in an expression or in parentheses. For example, if we add two positive numbers, for example, seven and three, then we write not +7+3, but simply 7+3, despite the fact that seven is also a positive number. Similarly, if you see, for example, the expression (5+x) - know that before the bracket there is a plus, which is not written, and before the five there is a plus +(+5+x).
The rule for opening parentheses during addition
When opening brackets, if there is a plus in front of the brackets, then this plus is omitted along with the brackets.
Example. Open the brackets in the expression 2 + (7 + 3) There is a plus in front of the brackets, which means we do not change the signs in front of the numbers in brackets.
2 + (7 + 3) = 2 + 7 + 3
Rule for opening parentheses when subtracting
If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite. The absence of a sign before the first term in parentheses implies a + sign.
Example. Expand the parentheses in the expression 2 − (7 + 3)
There is a minus before the brackets, which means you need to change the signs in front of the numbers in the brackets. In parentheses there is no sign before the number 7, this means that seven is positive, it is considered that there is a + sign in front of it.
2 − (7 + 3) = 2 − (+ 7 + 3)
When opening the brackets, we remove from the example the minus that was in front of the brackets, and the brackets themselves 2 − (+ 7 + 3), and change the signs that were in the brackets to the opposite ones.
2 − (+ 7 + 3) = 2 − 7 − 3
Expanding parentheses when multiplying
If there is a multiplication sign in front of the brackets, then each number inside the brackets is multiplied by the factor in front of the brackets. In this case, multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.
Thus, the parentheses in the products are expanded in accordance with the distributive property of multiplication.
Example. 2 (9 - 7) = 2 9 - 2 7
When you multiply a bracket by a bracket, each term in the first bracket is multiplied with each term in the second bracket.
(2 + 3) (4 + 5) = 2 4 + 2 5 + 3 4 + 3 5
In fact, there is no need to remember all the rules, it is enough to remember only one, this: c(a−b)=ca−cb. Why? Because if you substitute one instead of c, you get the rule (a−b)=a−b. And if we substitute minus one, we get the rule −(a−b)=−a+b. Well, if you substitute another bracket instead of c, you can get the last rule.
Opening parentheses when dividing
If there is a division sign after the brackets, then each number inside the brackets is divided by the divisor after the brackets, and vice versa.
Example. (9 + 6) : 3=9: 3 + 6: 3
How to expand nested parentheses
If an expression contains nested parentheses, they are expanded in order, starting with the outer or inner ones.
In this case, it is important that when opening one of the brackets, do not touch the remaining brackets, simply rewriting them as is.
Example. 12 - (a + (6 - b) - 3) = 12 - a - (6 - b) + 3 = 12 - a - 6 + b + 3 = 9 - a + b
The main function of parentheses is to change the order of actions when calculating values. For example, V numerically\(5·3+7\) the multiplication will be calculated first, and then the addition: \(5·3+7 =15+7=22\). But in the expression \(5·(3+7)\) the addition in brackets will be calculated first, and only then the multiplication: \(5·(3+7)=5·10=50\).
Example.
Expand the bracket: \(-(4m+3)\).
Solution
: \(-(4m+3)=-4m-3\).
Example.
Open the bracket and give similar terms \(5-(3x+2)+(2+3x)\).
Solution
: \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).
Example.
Expand the brackets \(5(3-x)\).
Solution
: In the bracket we have \(3\) and \(-x\), and before the bracket there is a five. This means that each member of the bracket is multiplied by \(5\) - I remind you that The multiplication sign between a number and a parenthesis is not written in mathematics to reduce the size of entries.
Example.
Expand the brackets \(-2(-3x+5)\).
Solution
: As in the previous example, the \(-3x\) and \(5\) in the parenthesis are multiplied by \(-2\).
Example.
Simplify the expression: \(5(x+y)-2(x-y)\).
Solution
: \(5(x+y)-2(x-y)=5x+5y-2x+2y=3x+7y\).
It remains to consider the last situation.
When multiplying a bracket by a bracket, each term of the first bracket is multiplied with each term of the second:
\((c+d)(a-b)=c·(a-b)+d·(a-b)=ca-cb+da-db\)
Example.
Expand the brackets \((2-x)(3x-1)\).
Solution
: We have a product of brackets and it can be expanded immediately using the formula above. But in order not to get confused, let's do everything step by step.
Step 1. Remove the first bracket - multiply each of its terms by the second bracket:
Step 2. Expand the products of the brackets and the factor as described above:
- First things first...
Then the second.
Step 3. Now we multiply and present similar terms:
It is not necessary to describe all the transformations in such detail; you can multiply them right away. But if you are just learning how to open parentheses, write in detail, there will be less chance of making mistakes.
Note to the entire section. In fact, you don't need to remember all four rules, you only need to remember one, this one: \(c(a-b)=ca-cb\) . Why? Because if you substitute one instead of c, you get the rule \((a-b)=a-b\) . And if we substitute minus one, we get the rule \(-(a-b)=-a+b\) . Well, if you substitute another bracket instead of c, you can get the last rule.
Parenthesis within a parenthesis
Sometimes in practice there are problems with brackets nested inside other brackets. Here is an example of such a task: simplify the expression \(7x+2(5-(3x+y))\).
To successfully solve such tasks, you need:
- carefully understand the nesting of brackets - which one is in which;
- open the brackets sequentially, starting, for example, with the innermost one.
It is important when opening one of the brackets don't touch the rest of the expression, just rewriting it as is.
Let's look at the task written above as an example.
Example.
Open the brackets and give similar terms \(7x+2(5-(3x+y))\).
Solution:
Example.
Open the brackets and give similar terms \(-(x+3(2x-1+(x-5)))\).
Solution
:
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\(-(x+3(2x-1\)\(+(x-5)\) \())\) |
There is triple nesting of parentheses here. Let's start with the innermost one (highlighted in green). There is a plus in front of the bracket, so it just comes off. |
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\(-(x+3(2x-1\)\(+x-5\) \())\) |
Now you need to open the second bracket, the intermediate one. But before that, we will simplify the expression of the ghost-like terms in this second bracket. |
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\(=-(x\)\(+3(3x-6)\) \()=\) |
Now we open the second bracket (highlighted in blue). Before the bracket is a factor - so each term in the bracket is multiplied by it. |
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\(=-(x\)\(+9x-18\) \()=\) |
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And open the last bracket. There is a minus sign in front of the bracket, so all signs are reversed. |
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Expanding parentheses is a basic skill in mathematics. Without this skill, it is impossible to have a grade above a C in 8th and 9th grade. Therefore, I recommend that you understand this topic well.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
