Everywhere. Everywhere and everywhere, wherever you look, there are such constructions:
These "constructions" in literate people cause an ambiguous reaction. At least like "is it really so - right?".
In general, personally, I can’t understand where the “fashion” of not closing external quotes came from. The first and only analogy that comes up in this regard is the analogy with brackets. Nobody doubts that two brackets in a row are normal. For example: "Pay for the entire circulation (200 pieces (of which 100 are defective))". But in the normality of setting two quotes in a row, someone doubted (I wonder who was the first?) ... And now everyone without exception began to produce constructions like LLC Firm Pupkov and Co. with a clear conscience.
But even if you have not seen the rule in your life, which will be discussed below, then the only logically justified option (using the brackets as an example) would be the following: Firm Pupkov and Co LLC.
So, the rule itself:
If at the beginning or at the end of a quotation (the same applies to direct speech) there are internal and external quotation marks, then they must differ from each other in a pattern (the so-called "Christmas trees" and "cutes"), and external quotation marks should not be omitted, for example: C The sides of the ship were radioed: "Leningrad has entered the tropics and is continuing on its course." About Zhukovsky, Belinsky writes: “Contemporaries of Zhukovsky’s youth looked at him mainly as an author of ballads, and in one of his messages Batyushkov called him a “ballade player.”
© Rules of Russian spelling and punctuation. - Tula: Autograph, 1995. - 192 p.
Accordingly ... if you do not have the opportunity to type in quotes, "Christmas trees", then what can you do, you will have to use such "" icons. However, the impossibility (or unwillingness) to use Russian quotes is by no means the reason why you can not close the outer quotes.Thus, it seems that they figured out the incorrect design of Firm Pupkov and Co LLC. There are also constructions of the type LLC Firm Pupkov and Co.
From the rule, it is quite clear that such constructions are illiterate ... (Correct: LLC Firm Pupkov and Co.However!
Milchin's Publisher's and Author's Handbook (2004 edition) states that two design options can be used in such cases. The use of "herringbones" and "paws" and (in the absence of technical means) the use of only "herringbones": two opening and one closing.
The directory is “fresh” and personally I immediately have 2 questions here. Firstly, with what joy you can still use one closing quote-herringbone (well, this is illogical, see above), and secondly, the phrase “in the absence of technical means” especially attracts attention. How is that, sorry? Here, open Notepad and type “only Christmas trees: two opening and one closing” there. There are no such characters on the keyboard. Printing a Christmas tree doesn't work... The combination Shift + 2 produces the sign " (which, as you know, is not even a quotation mark). Now open Microsoft Word and press Shift + 2 again. The program will correct " to " (or " ). Well, it turns out that the rule that existed for more than a dozen years was taken and rewritten under Microsoft Word? Like, since the Word from "Firm" Pupkov and Co "does" Firm "Pupkov and Co", then now let it be acceptable and correct ???
It seems so. And if so, then there is every reason to doubt the correctness of such an innovation.Yes, and one more clarification ... about the very "lack of technical means." The fact is that on any Windows computer there are always "technical means" for entering both "Christmas trees" and "paws", so this new "rule" (for me it is in quotes) is wrong from the very beginning!
All special characters in a font can be easily typed by knowing the corresponding number of that character. It is enough to hold down Alt and type on the NumLock keyboard (NumLock is pressed, the indicator light is on) the corresponding symbol number:
„ Alt + 0132 (left foot)
“ Alt + 0147 (right foot)
« Alt + 0171 (left herringbone)
» Alt + 0187 (right herringbone)
Parentheses are used to indicate the order in which actions are performed in numeric and alphabetic expressions, as well as in expressions with variables. It is convenient to pass from an expression with brackets to an identically equal expression without brackets. This technique is called parenthesis opening.
To expand brackets means to rid the expression of these brackets.
Another point deserves special attention, which concerns the peculiarities of writing solutions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as equality. For example, after opening 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 reduce entries, it is customary not to write a plus sign if it is the first in an expression or in brackets. 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 there is a plus in front of the bracket, which is not written, and there is a plus + (+5 + x) in front of the five.
Bracket expansion rule for addition
When opening brackets, if there is a plus before the brackets, then this plus is omitted along with the brackets.
Example. Open the brackets in the expression 2 + (7 + 3) Before the brackets plus, then the characters in front of the numbers in the brackets do not change.
2 + (7 + 3) = 2 + 7 + 3
The rule for expanding brackets 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. Open brackets in expression 2 − (7 + 3)
There is a minus before the brackets, so you need to change the signs before the numbers from the brackets. There is no sign in brackets before the number 7, which means that the seven is positive, it is considered that the + sign is in front of it.
2 − (7 + 3) = 2 − (+ 7 + 3)
When opening the brackets, we remove the minus from the example, which was before 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. At the same time, 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, parentheses in products are expanded in accordance with the distributive property of multiplication.
Example. 2 (9 - 7) = 2 9 - 2 7
When multiplying parenthesis by parenthesis, each term of the first parenthesis is multiplied with every term of the second parenthesis.
(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 one: c(a−b)=ca−cb. Why? Because if we substitute one instead of c, we 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.
Expand parentheses when dividing
If there is a division sign after the brackets, then each number inside the brackets is divisible by the divisor after the brackets, and vice versa.
Example. (9 + 6) : 3=9: 3 + 6: 3
How to expand nested parentheses
If the expression contains nested brackets, then they are expanded in order, starting with external or internal.
At the same time, when opening one of the brackets, it is important not to touch the other brackets, just rewriting them as they are.
Example. 12 - (a + (6 - b) - 3) = 12 - a - (6 - b) + 3 = 12 - a - 6 + b + 3 = 9 - a + b
In this article, we will consider in detail the basic rules for such an important topic in a mathematics course as opening brackets. You need to know the rules for opening brackets in order to correctly solve equations in which they are used.
How to properly open parentheses 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, when the brackets are opened, the signs inside them do not change. Example:
(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.
How to open brackets preceded by a "-" sign
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 the 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 brackets when multiplying
The parentheses are preceded by a multiplier
In this case, you need to multiply each term by a factor and open the brackets without changing signs. If the multiplier has the sign "-", then when multiplying, 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 brackets 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 brackets in a square
If the sum or difference of two terms is squared, the brackets should be expanded 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 open parentheses in a different degree
If the sum or difference of the terms is raised, for example, to the 3rd or 4th power, then you just need to break the degree of the bracket into “squares”. The powers of the same factors are added, and when dividing, the degree of the divisor is subtracted from the degree 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 among themselves, 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 bracket opening rules apply equally to both linear and trigonometric equations.
A + (b + c) can be written without brackets: a + (b + c) \u003d a + b + c. This operation is called parenthesis expansion.
Example 1 Let's open the brackets in the expression a + (- b + c).
Decision. a + (-b + c) = a + ((-b) + c) = a + (-b) + c = a-b + c.
If there is a “+” sign before the brackets, then you can omit the brackets and this “+” sign, retaining the signs of the terms in brackets. If the first term in brackets is written without a sign, then it must be written with a “+” sign.
Example 2 Let's find the value of the expression -2.87+ (2.87-7.639).
Decision. Opening the brackets, we get - 2.87 + (2.87 - 7.639) \u003d - - 2.87 + 2.87 - 7.639 \u003d 0 - 7.639 \u003d - 7.639.
To find the value of the expression - (- 9 + 5), you need to add numbers-9 and 5 and find the number opposite to the amount received: -(- 9 + 5)= -(- 4) = 4.
The same value can be obtained in a different way: first write down the numbers opposite to these terms (i.e. change their signs), and then add: 9 + (- 5) = 4. Thus, - (- 9 + 5) = 9 - 5 = 4.
To write the sum opposite to the sum of several terms, it is necessary to change the signs of these terms.
So - (a + b) \u003d - a - b.
Example 3 Find the value of the expression 16 - (10 -18 + 12).
Decision. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.
To open the brackets preceded by the “-” sign, you need to replace this sign with “+”, changing the signs of all the terms in the brackets to the opposite ones, and then open the brackets.
Example 4 Let's find the value of the expression 9.36-(9.36 - 5.48).
Decision. 9.36 - (9.36 - 5.48) = 9.36 + (- 9.36 + 5.48) == 9.36 - 9.36 + 5.48 = 0 -f 5.48 = 5 .48.
Bracket opening and the use of commutative and associative properties additions make calculations easier.
Example 5 Find the value of the expression (-4-20)+(6+13)-(7-8)-5.
Decision. First, we open the brackets, and then we find separately the sum of all positive and separately the sum of all negative numbers, and, finally, add the results:
(- 4 - 20)+(6+ 13)-(7 - 8) - 5 = -4-20 + 6 + 13-7 + 8-5 = = (6 + 13 + 8)+(- 4 - 20 - 7 - 5)= 27-36=-9.
Example 6 Find the value of the expression
Decision. First, we represent each term as the sum of their integer and fractional parts, then open the brackets, then add the whole and separately fractional parts and finally sum up the results:
How do you open parentheses that are preceded by a "+" sign? How can you find the value of an expression that is the opposite of the sum of several numbers? How to open brackets preceded by a "-" sign?
1218. Expand the brackets:
a) 3.4+(2.6+ 8.3); c) m+(n-k);
b) 4.57+(2.6 - 4.57); d) c+(-a + b).
1219. Find the value of the expression:
1220. Expand the brackets:
a) 85+(7.8+ 98); d) -(80-16) + 84; g) a-(b-k-n);
b) (4.7 -17) + 7.5; e) -a + (m-2.6); h) - (a-b + c);
c) 64-(90 + 100); e) c+(-a-b); i) (m-n)-(p-k).
1221. Expand the brackets and find the value of the expression:
1222. Simplify the expression:
1223. Write amount two expressions and simplify it:
a) - 4 - m and m + 6.4; d) a + b and p - b
b) 1.1+a and -26-a; e) - m + n and -k - n;
c) a + 13 and -13 + b; e)m - n and n - m.
1224. Write the difference of two expressions and simplify it:
1226. Use the equation to solve the problem:
a) There are 42 books on one shelf, and 34 on the other. Several books were removed from the second shelf, and as many as were left on the second from the first. After that, 12 books remained on the first shelf. How many books were taken off the second shelf?
b) There are 42 students in the first class, 3 students less in the second than in the third. How many students are in the third grade if there are 125 students in these three grades?
1227. Find the value of the expression:
1228. Calculate orally:
1229. Find the largest value of the expression:
1230. Enter 4 consecutive integers if:
a) the smaller of them is equal to -12; c) the smaller of them is equal to n;
b) the greater of them is equal to -18; d) the larger of them is equal to k.
