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 date is today?
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. Than modulus is equal 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.
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.
Have you learned how to perform different actions 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 is like a closed door behind which there is something interesting.) Today we have explored 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:
“Towards mathematics show ability,
Don't be lazy, but develop every day.
Multiply, divide, work, think,
Don’t forget to be friends with mathematics.”
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 identically equal to the expression without parentheses. 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
In this article we will look in detail at the basic rules of such important topic mathematics course, like opening parentheses. You need to know the rules for opening parentheses in order to correctly solve equations in which they are used.
How to open parentheses correctly when adding
Expand the brackets preceded by the “+” sign
This is the simplest case, because if there is an addition sign in front of the brackets, the signs inside them do not change when the brackets are opened. Example:
(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.
How to expand parentheses preceded by a "-" sign
IN in this case you need to rewrite all the terms without brackets, but at the same time change all the signs inside them to the opposite ones. The signs change only for terms from those brackets that were preceded by the sign “-”. Example:
(9 + 3) - (1 - 6 + 9) = 9 + 3 - 1 + 6 - 9 = 8.
How to open parentheses when multiplying
Before the brackets there is a multiplier number
In this case, you need to multiply each term by a factor and open the brackets without changing the signs. If the multiplier has a “-” sign, then during multiplication the signs of the terms are reversed. Example:
3 * (1 - 6 + 9) = 3 * 1 - 3 * 6 + 3 * 9 = 3 - 18 + 27 = 12.
How to open two parentheses with a multiplication sign between them
In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. Example:
(9 + 3) * (1 - 6 + 9) = 9 * 1 + 9 * (- 6) + 9 * 9 + 3 * 1 + 3 * (- 6) + 3 * 9 = 9 - 54 + 81 + 3 - 18 + 27 = 48.
How to open parentheses in a square
If the sum or difference of two terms is squared, the brackets should be opened according to the following formula:
(x + y)^2 = x^2 + 2 * x * y + y^2.
In the case of a minus inside the brackets, the formula does not change. Example:
(9 + 3) ^ 2 = 9 ^ 2 + 2 * 9 * 3 + 3 ^ 2 = 144.
How to expand parentheses to another degree
If the sum or difference of terms is raised, for example, to the 3rd or 4th power, then you just need to break the power of the bracket into “squares”. The powers of identical factors are added, and when dividing, the power of the divisor is subtracted from the power of the dividend. Example:
(9 + 3) ^ 3 = ((9 + 3) ^ 2) * (9 + 3) = (9 ^ 2 + 2 * 9 * 3 + 3 ^ 2) * 12 = 1728.
How to open 3 brackets
There are equations in which 3 brackets are multiplied at once. In this case, you must first multiply the terms of the first two brackets together, and then multiply the sum of this multiplication by the terms of the third bracket. Example:
(1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.
These rules for opening parentheses apply equally to solving both linear and trigonometric equations.
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, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... we 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, mathematical apparatus The use of 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. WITH physical point From a perspective, it 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? Stay in constant units measurements of time and do not go to reciprocal quantities. 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. For the next time interval, equal to 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 in different moments time, but distance cannot be determined from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed 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 “screw 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. Apply mathematical theory sets to the 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 stack and hand it to the mathematician" mathematical set salaries." We explain to the mathematics 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 mud, crystal structure and the arrangement of atoms in 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 the help of 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 numbers 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 measurements. 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 mathematical operation does not depend on the size of the number, the unit of measurement used and who performs the 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 stupid, no knowledgeable in physics. She just has an arch stereotype of perception 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.
Now we'll move on to opening parentheses in expressions in which the expression in parentheses is multiplied by a number or expression. Let us formulate a rule for opening parentheses preceded by a minus sign: the parentheses together with the minus sign are omitted, and the signs of all terms in the parentheses are replaced with the opposite ones.
One type of expression transformation is the expansion of parentheses. Numeric, literal expressions and expressions with variables can be composed using parentheses, which can indicate the order in which actions are performed, contain a negative number, etc. Let us assume that in the expressions described above, instead of numbers and variables, there can be any expressions.
And let us pay attention to one more point regarding the peculiarities of writing a solution when opening brackets. In the previous paragraph, we dealt with what is called opening parentheses. To do this, there are rules for opening brackets, which we will now review. This rule is dictated by the fact that positive numbers It is customary to write without parentheses; in this case, parentheses are unnecessary. The expression (−3.7)−(−2)+4+(−9) can be written without parentheses as −3.7+2+4−9.
Finally, the third part of the rule is simply due to the peculiarities of writing negative numbers on the left in the expression (which we mentioned in the section on brackets for writing negative numbers). You may encounter expressions made up of a number, minus signs, and several pairs of parentheses. If you open the brackets, moving from internal to external, then the solution will be as follows: −(−((−(5))))=−(−((−5)))=−(−(−5))=−( 5)=−5.
How to open parentheses?
Here's an explanation: −(−2 x) is +2 x, and since this expression comes first, +2 x can be written as 2 x, −(x2)=−x2, +(−1/ x)=−1/x and −(2 x y2:z)=−2 x y2:z. The first part of the written rule for opening parentheses follows directly from the rule for multiplying negative numbers. Its second part is a consequence of the rule for multiplying numbers with different signs. Let's move on to examples of opening parentheses in products and quotients of two numbers with different signs.
Opening brackets: rules, examples, solutions.
The above rule takes into account the entire chain of these actions and significantly speeds up the process of opening brackets. The same rule allows you to open parentheses in expressions that are products and partial expressions with a minus sign that are not sums and differences.
Let's look at examples of the application of this rule. Let us give the corresponding rule. Above we have already encountered expressions of the form −(a) and −(−a), which without parentheses are written as −a and a, respectively. For example, −(3)=3, and. These are special cases of the stated rule. Now let's look at examples of opening parentheses when they contain sums or differences. Let's show examples of using this rule. Let us denote the expression (b1+b2) as b, after which we use the rule of multiplying the bracket by the expression from the previous paragraph, we have (a1+a2)·(b1+b2)=(a1+a2)·b=(a1·b+a2· b)=a1·b+a2·b.
By induction, this statement can be extended to an arbitrary number of terms in each bracket. It remains to open the brackets in the resulting expression using the rules from previous paragraphs, as a result we get 1·3·x·y−1·2·x·y3−x·3·x·y+x·2·x·y3.
The rule in mathematics is opening parentheses if there are (+) and (-) in front of the brackets.
This expression is the product of three factors (2+4), 3 and (5+7·8). You will have to open the brackets sequentially. Now we use the rule for multiplying a bracket by a number, we have ((2+4) 3) (5+7 8)=(2 3+4 3) (5+7 8). Degrees whose bases are some expressions written in brackets, with in kind can be thought of as the product of several brackets.
For example, let's transform the expression (a+b+c)2. First, we write it as a product of two brackets (a+b+c)·(a+b+c), now we multiply a bracket by a bracket, we get a·a+a·b+a·c+b·a+b· b+b·c+c·a+c·b+c·c.
Let's also say that to raise the sums and differences of two numbers in natural degree It is advisable to use Newton's binomial formula. For example, (5+7−3):2=5:2+7:2−3:2. It is no less convenient to first replace division with multiplication, and then use the corresponding rule for opening parentheses in a product.
It remains to understand the order of opening brackets using examples. Let's take the expression (−5)+3·(−2):(−4)−6·(−7). We substitute these results into the original expression: (−5)+3·(−2):(−4)−6·(−7)=(−5)+(3·2:4)−(−6·7) . All that remains is to finish opening the brackets, as a result we have −5+3·2:4+6·7. This means that when moving from the left side of the equality to the right, the opening of the parentheses occurred.
Note that in all three examples we simply removed the parentheses. First, add 445 to 889. This action can be performed mentally, but it is not very easy. Let's open the brackets and see that the changed procedure will significantly simplify the calculations.
How to expand parentheses to another degree
Illustrating example and rule. Let's look at an example: . You can find the value of an expression by adding 2 and 5, and then taking the resulting number with the opposite sign. The rule does not change if there are not two, but three or more terms in brackets. Comment. The signs are reversed only in front of the terms. In order to open the brackets, in this case we need to remember the distributive property.
For single numbers in brackets
Your mistake is not in the signs, but in incorrect handling of fractions? In 6th grade we met positive and negative numbers. How will we solve examples and equations?
How much is in brackets? What can you say about these expressions? Of course, the result of the first and second examples is the same, which means we can put an equal sign between them: -7 + (3 + 4) = -7 + 3 + 4. What did we do with the parentheses?
Demonstration of slide 6 with rules for opening brackets. Thus, the rules for opening parentheses will help us solve examples and simplify expressions. Next, students are asked to work in pairs: they need to use arrows to connect the expression containing brackets with the corresponding expression without brackets.
Slide 11 Once in Sunny City, Znayka and Dunno argued about which of them solved the equation correctly. Next, students solve the equation on their own using the rules for opening brackets. Solving equations” Lesson objectives: educational (reinforcement of knowledge on the topic: “Opening brackets.
Lesson topic: “Opening parentheses. In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. First, the first two factors are taken, enclosed in one more bracket, and inside these brackets the parentheses are opened according to one of the already known rules.
rawalan.freezeet.ru
Opening brackets: rules and examples (grade 7)
The main function of parentheses is to change the order of actions when calculating values numerical expressions . 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: \(5·(3+7)=5·10=50\).
However, if we are dealing with algebraic expression containing variable- for example, like this: \(2(x-3)\) - then it’s impossible to calculate the value in the bracket, the variable is in the way. Therefore, in this case, the brackets are “opened” using the appropriate rules.
Rules for opening parentheses
If there is a plus sign in front of the bracket, then the bracket is simply removed, the expression in it remains unchanged. In other words:
Here it is necessary to clarify that in mathematics, to shorten notations, it is customary not to write the plus sign if it appears first in the expression. 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.
Example
. Open the bracket and bring similar terms: \((x-11)+(2+3x)\).
Solution
: \((x-11)+(2+3x)=x-11+2+3x=4x-9\).
If there is a minus sign in front of the bracket, then when the bracket is removed, each term of the expression inside it changes sign to the opposite:
Here it is necessary to clarify that while a was in the bracket, there was a plus sign (they just didn’t write it), and after removing the bracket, this plus changed to a minus.
Example
: Simplify the expression \(2x-(-7+x)\).
Solution
: inside the bracket there are two terms: \(-7\) and \(x\), and before the bracket there is a minus. This means that the signs will change - and the seven will now be a plus, and the x will now be a minus. Open the bracket and we present similar terms .
Example.
Open the bracket and give similar terms \(5-(3x+2)+(2+3x)\).
Solution
: \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).
If there is a factor in front of the bracket, then each member of the bracket is multiplied by it, that is:
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 \(-2\).
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:
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 and multiply each member by the second bracket:
Step 2. Expand the products of the brackets and the factor as described above:
- First things first...
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 solve successfully similar tasks, need to:
- 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:
Let's begin the task by opening the inner bracket (the one inside). Expanding it, we are dealing only with what directly relates to it - this is the bracket itself and the minus in front of it (highlighted in green). We rewrite everything else (not highlighted) the same way it was.
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While the program is running:
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A little theory.
Product of a monomial and a polynomial. The concept of a polynomial
Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
The sum of monomials is called a polynomial. The terms in a polynomial are called terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.
Let's represent all terms in the form of monomials standard view:
Let us present similar terms in the resulting polynomial:
The result is a polynomial, all terms of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
Behind degree of polynomial of a standard form take the highest of the powers of its members. Thus, a binomial has the third degree, and a trinomial has the second.
Typically, the terms of standard form polynomials containing one variable are arranged in descending order of exponents. For example:
The sum of several polynomials can be transformed (simplified) into a polynomial of standard form.
Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since enclosing parentheses is the inverse transformation of opening parentheses, it is easy to formulate rules for opening brackets:
If a “+” sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.
If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.
Transformation (simplification) of the product of a monomial and a polynomial
By using distributive property multiplications can be converted (simplified) into a polynomial, the product of a monomial and a polynomial. For example:
The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.
This result is usually formulated as a rule.
To multiply a monomial by a polynomial, you must multiply that monomial by each of the terms of the polynomial.
We have already used this rule several times to multiply by a sum.
Product of polynomials. Transformation (simplification) of the product of two polynomials
In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.
Usually the following rule is used.
To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.
Abbreviated multiplication formulas. Sum squares, differences and difference of squares
With some expressions in algebraic transformations have to deal with more often than others. Perhaps the most common expressions are u, i.e. the square of the sum, the square of the difference and the difference of squares. You noticed that the names of these expressions seem to be incomplete, for example, this is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.
Expressions can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered such a task when multiplying polynomials:
It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.
- square of the sum equal to the sum squares and double the product.
- the square of the difference is equal to the sum of the squares without the double product.
- the difference of squares is equal to the product of the difference and the sum.
These three identities allow one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. The most difficult thing is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.
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Expanding parentheses
We continue to study the basics of algebra. In this lesson we will learn how to expand parentheses in expressions. Expanding parentheses means removing the parentheses from an expression.
To open parentheses, you need to memorize only two rules. With regular practice, you can open the brackets with eyes closed, and those rules that were required to be memorized can be safely forgotten.
The first rule for opening parentheses
Consider the following expression:
The value of this expression is 2 . Let's open the parentheses in this expression. Expanding parentheses means getting rid of them without affecting the meaning of the expression. That is, after getting rid of the parentheses, the value of the expression 8+(−9+3) should still be equal to two.
The first rule for opening parentheses is as follows:
When opening brackets, if there is a plus in front of the brackets, then this plus is omitted along with the brackets.
So, we see that in the expression 8+(−9+3) There is a plus sign before the parentheses. This plus must be omitted along with the parentheses. In other words, the brackets will disappear along with the plus that stood in front of them. And what was in brackets will be written without changes:
8−9+3 . This expression equals 2 , like the previous expression with brackets, was equal to 2 .
8+(−9+3) And 8−9+3
8 + (−9 + 3) = 8 − 9 + 3
Example 2. Expand parentheses in expression 3 + (−1 − 4)
There is a plus in front of the brackets, which means this plus is omitted along with the brackets. What was in brackets will remain unchanged:
3 + (−1 − 4) = 3 − 1 − 4
Example 3. Expand parentheses in expression 2 + (−1)
IN in this example opening the parentheses became a kind of reverse operation of replacing subtraction with addition. What does it mean?
In expression 2−1 subtraction occurs, but it can be replaced by addition. Then we get the expression 2+(−1) . But if in the expression 2+(−1) open the brackets, you get the original 2−1 .
Therefore, the first rule for opening parentheses can be used to simplify expressions after some transformations. That is, rid it of brackets and make it simpler.
For example, let's simplify the expression 2a+a−5b+b .
To simplify this expression, similar terms can be given. Let us recall that to reduce similar terms, you need to add the coefficients of similar terms and multiply the result by the common letter part:
Got an expression 3a+(−4b). Let's remove the parentheses in this expression. There is a plus in front of the brackets, so we use the first rule for opening brackets, that is, we omit the brackets along with the plus that comes before these brackets:
So the expression 2a+a−5b+b simplifies to 3a−4b .
Having opened some brackets, you may encounter others along the way. We apply the same rules to them as to the first ones. For example, let's expand the parentheses in the following expression:
There are two places where you need to open the parentheses. In this case, the first rule of opening parentheses applies, namely, omitting the parentheses along with the plus sign that precedes these parentheses:
2 + (−3 + 1) + 3 + (−6) = 2 − 3 + 1 + 3 − 6
Example 3. Expand parentheses in expression 6+(−3)+(−2)
In both places where there are parentheses, they are preceded by a plus. Here again the first rule of opening parentheses applies:
Sometimes the first term in parentheses is written without a sign. For example, in the expression 1+(2+3−4) first term in brackets 2 written without a sign. The question arises, what sign will appear in front of the two after the brackets and the plus sign in front of the brackets are omitted? The answer suggests itself - there will be a plus in front of the two.
In fact, even being in parentheses there is a plus in front of the two, but we don’t see it because it’s not written down. We have already said that the complete notation of positive numbers looks like +1, +2, +3. But according to tradition, pluses are not written down, which is why we see the positive numbers that are familiar to us 1, 2, 3 .
Therefore, to expand the parentheses in the expression 1+(2+3−4) , you need to omit the brackets as usual, along with the plus sign in front of these brackets, but write the first term that was in the brackets with a plus sign:
1 + (2 + 3 − 4) = 1 + 2 + 3 − 4
Example 4. Expand parentheses in expression −5 + (2 − 3)
There is a plus in front of the brackets, so we apply the first rule for opening brackets, namely, we omit the brackets along with the plus that comes before these brackets. But the first term, which we write in parentheses with a plus sign:
−5 + (2 − 3) = −5 + 2 − 3
Example 5. Expand parentheses in expression (−5)
There is a plus in front of the parentheses, but it is not written down because there were no other numbers or expressions before it. Our task is to remove the parentheses by applying the first rule of opening parentheses, namely, omit the parentheses along with this plus (even if it is invisible)
Example 6. Expand parentheses in expression 2a + (−6a + b)
There is a plus in front of the brackets, which means this plus is omitted along with the brackets. What was in brackets will be written unchanged:
2a + (−6a + b) = 2a −6a + b
Example 7. Expand parentheses in expression 5a + (−7b + 6c) + 3a + (−2d)
There are two places in this expression where you need to expand the parentheses. In both sections there is a plus before the brackets, which means this plus is omitted along with the brackets. What was in brackets will be written unchanged:
5a + (−7b + 6c) + 3a + (−2d) = 5a −7b + 6c + 3a − 2d
The second rule for opening parentheses
Now let's look at the second rule for opening parentheses. It is used when there is a minus before the parentheses.
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.
For example, let's expand the parentheses in the following expression
We see that there is a minus before the brackets. This means that you need to apply the second expansion rule, namely, omit the brackets along with the minus sign in front of these brackets. In this case, the terms that were in brackets will change their sign to the opposite:
We got an expression without parentheses 5+2+3 . This expression is equal to 10, just like the previous expression with brackets was equal to 10.
Thus, between the expressions 5−(−2−3) And 5+2+3 you can put an equal sign, since they are equal to the same value:
5 − (−2 − 3) = 5 + 2 + 3
Example 2. Expand parentheses in expression 6 − (−2 − 5)
There is a minus before the brackets, so we apply the second rule for opening brackets, namely, we omit the brackets along with the minus that comes before these brackets. In this case, we write the terms that were in brackets with opposite signs:
6 − (−2 − 5) = 6 + 2 + 5
Example 3. Expand parentheses in expression 2 − (7 + 3)
There is a minus before the brackets, so we apply the second rule for opening brackets:
Example 4. Expand parentheses in expression −(−3 + 4)
Example 5. Expand parentheses in expression −(−8 − 2) + 16 + (−9 − 2)
There are two places where you need to open the parentheses. In the first case, you need to apply the second rule for opening parentheses, and when it comes to the expression +(−9−2) you need to apply the first rule:
−(−8 − 2) + 16 + (−9 − 2) = 8 + 2 + 16 − 9 − 2
Example 6. Expand parentheses in expression −(−a − 1)
Example 7. Expand parentheses in expression −(4a + 3)
Example 8. Expand parentheses in expression a − (4b + 3) + 15
Example 9. Expand parentheses in expression 2a + (3b − b) − (3c + 5)
There are two places where you need to open the parentheses. In the first case, you need to apply the first rule for opening parentheses, and when it comes to the expression −(3c+5) you need to apply the second rule:
2a + (3b − b) − (3c + 5) = 2a + 3b − b − 3c − 5
Example 10. Expand parentheses in expression −a − (−4a) + (−6b) − (−8c + 15)
There are three places where you need to open the brackets. First you need to apply the second rule for opening parentheses, then the first, and then the second again:
−a − (−4a) + (−6b) − (−8c + 15) = −a + 4a − 6b + 8c − 15
Bracket opening mechanism
The rules for opening parentheses that we have now examined are based on the distributive law of multiplication:
In fact opening parentheses call the procedure when common multiplier multiplied by each term in parentheses. As a result of this multiplication, the brackets disappear. For example, let's expand the parentheses in the expression 3×(4+5)
3 × (4 + 5) = 3 × 4 + 3 × 5
Therefore, if you need to multiply a number by an expression in brackets (or multiply an expression in brackets by a number), you need to say let's open the brackets.
But how is the distributive law of multiplication related to the rules for opening parentheses that we examined earlier?
The fact is that before any parentheses there is a common factor. In the example 3×(4+5) the common factor is 3 . And in the example a(b+c) the common factor is a variable a.
If there are no numbers or variables before the parentheses, then the common factor is 1 or −1 , depending on what sign is in front of the brackets. If there is a plus in front of the parentheses, then the common factor is 1 . If there is a minus before the parentheses, then the common factor is −1 .
For example, let’s expand the parentheses in the expression −(3b−1). There is a minus sign in front of the brackets, so you need to use the second rule for opening brackets, that is, omit the brackets along with the minus sign in front of the brackets. And write the expression that was in brackets with opposite signs:
We expanded the brackets using the rule for expanding brackets. But these same brackets can be opened using the distributive law of multiplication. To do this, first write before the brackets the common factor 1, which was not written down:
The minus sign that previously stood before the brackets referred to this unit. Now you can open the brackets using the distributive law of multiplication. For this purpose the common factor −1 you need to multiply by each term in brackets and add the results.
For convenience, we replace the difference in parentheses with the amount:
−1 (3b −1) = −1 (3b + (−1)) = −1 × 3b + (−1) × (−1) = −3b + 1
As in last time we got the expression −3b+1. Everyone will agree that this time more time was spent solving such a simple example. Therefore, it is wiser to use ready-made rules for opening brackets, which we discussed in this lesson:
But it doesn't hurt to know how these rules work.
In this lesson we learned one more thing identical transformation. Together with opening the brackets, putting the general out of brackets and bringing similar terms, you can slightly expand the range of problems to be solved. For example:
Here you need to perform two actions - first open the brackets, and then bring similar terms. So, in order:
1) Open the brackets:
2) We present similar terms:
In the resulting expression −10b+(−1) you can expand the brackets:
Example 2. Open the parentheses and add similar terms in the following expression:
1) Let's open the brackets:
2) Let us present similar terms. This time, to save time and space, we will not write down how the coefficients are multiplied by the common letter part
Example 3. Simplify an expression 8m+3m and find its value at m=−4
1) First, let's simplify the expression. To simplify the expression 8m+3m, you can take out the common factor in it m outside of brackets:
2) Find the value of the expression m(8+3) at m=−4. To do this, in the expression m(8+3) instead of a variable m substitute the number −4
m (8 + 3) = −4 (8 + 3) = −4 × 8 + (−4) × 3 = −32 + (−12) = −44