An increasing or decreasing number sequence is called monotonous.
Very large and very small numbers are usually written in standard form: a∙10
n, Where 1≤a<10
And n(natural or integer) – is the order of a number written in standard form.
For example, 345.7=3.457∙10 2; 123456=1.23456∙10 5 ; 0.000345=3.45∙10 -4.
Examples.
Write the number in standard form: 1)
40503; 2)
0,0023; 3)
876,1; 4)
0,0000067.
Solution.
1)
40503=4.0503·10 4;
2)
0,0023=2,3∙10 -3 ;
3)
876,1=8,761∙10 2 ;
4)
0,0000067=6,7∙10 -6 .
More examples on the standard form of numbers.
5)
The number of gas molecules in 1 cm 3 at 0°C and a pressure of 760 mm ps.st is equal to
27 000 000 000 000 000 000.
Solution.
27 000 000 000 000 000 000=2,7∙10 19
.
6) 1 parsec(unit of length in astronomy) is equal to 30,800,000,000,000 km. Write this number in standard form.
Solution.
1 parsec=30 800 000 000 000=3.08∙10 13 km.
On topic:
Kilowatt hour is an off-system unit of energy or work, used in electrical engineering, denoted kWh.
1 kWh=3.6∙10 6 J(Joules).
Often you need to find the sum of squares (x 1 2 +x 2 2) or the sum of cubes (x 1 3 +x 2 3) of the roots of a quadratic equation, less often - the sum of the reciprocal values of the squares of the roots or the sum of arithmetic square roots of the roots of a quadratic equation:
Vieta's theorem can help with this:
Sum of roots of the reduced quadratic equation x 2 +px+q=0 is equal to the second coefficient taken with the opposite sign, and the product of the roots is equal to the free term:
x 1 + x 2 = -p; x 1 ∙x 2 =q.
Let's express
through p And q:
1)
sum of squares of the roots of the equation x 2 +px+q=0;
2)
sum of cubes of the roots of the equation x 2 +px+q=0.
Solution.
1)
Expression x 1 2 +x 2 2 obtained by squaring both sides of the equation x 1 + x 2 = -p;
(x 1 +x 2) 2 =(-p) 2 ; open the brackets: x 1 2 +2x 1 x 2 + x 2 2 =p 2 ; we express the required amount: x 1 2 +x 2 2 =p 2 -2x 1 x 2 =p 2 -2q. We got a useful equality: x 1 2 +x 2 2 =p 2 -2q.
2)
Expression x 1 3 +x 2 3 Let us represent the sum of cubes using the formula:
(x 1 3 +x 2 3)=(x 1 +x 2)(x 1 2 -x 1 x 2 +x 2 2)=-p·(p 2 -2q-q)=-p·(p 2 -3q).
Another useful equation: x 1 3 +x 2 3 = -p·(p 2 -3q).
Examples.
3) x 2 -3x-4=0. Without solving the equation, calculate the value of the expression x 1 2 +x 2 2.
Solution.
x 1 +x 2 =-p=3, and the work x 1 ∙x 2 =q=in example 1) equality:
x 1 2 +x 2 2 =p 2 -2q. We have -p=x 1 +x 2 = 3
→ p 2 =3 2 =9; q= x 1 x 2 = -4.
Then x 1 2 +x 2 2 =9-2·(-4)=9+8=17.
Answer: x 1 2 +x 2 2 =17.
4)
x 2 -2x-4=0. Calculate: x 1 3 +x 2 3 .
Solution.
By Vieta's theorem, the sum of the roots of this reduced quadratic equation is x 1 +x 2 =-p=2, and the work x 1 ∙x 2 =q=-4. Let's apply what we have received ( in example 2) equality: x 1 3 +x 2 3 =-p·(p 2 -3q)= 2·(2 2 -3·(-4))=2·(4+12)=2·16=32.
Answer:
x 1 3 +x 2 3 =32.
Question: what if we are given an unreduced quadratic equation? Answer: it can always be “reduced” by dividing term by term by the first coefficient.
5) 2x 2 -5x-7=0. Without deciding, calculate: x 1 2 +x 2 2.
Solution. We are given a complete quadratic equation. Divide both sides of the equality by 2 (the first coefficient) and obtain the following quadratic equation: x 2 -2.5x-3.5=0.
According to Vieta's theorem, the sum of the roots is equal to 2,5
; the product of the roots is equal -3,5
.
We solve it in the same way as the example 3)
using the equality: x 1 2 +x 2 2 =p 2 -2q.
x 1 2 +x 2 2 =p 2 -2q= 2,5 2 -2∙(-3,5)=6,25+7=13,25.
Answer: x 1 2 + x 2 2 = 13,25.
6) x 2 -5x-2=0. Find:
Let us transform this equality and, using Vieta’s theorem, replace the sum of roots through -p, and the product of the roots through q, we get another useful formula. When deriving the formula, we used equality 1): x 1 2 +x 2 2 =p 2 -2q.
In our example x 1 +x 2 =-p=5; x 1 ∙x 2 =q=-2.
We substitute these values into the resulting formula:
7) x 2 -13x+36=0. Find:
Let's transform this sum and get a formula that can be used to find the sum of arithmetic square roots from the roots of a quadratic equation.
We have x 1 +x 2 =-p=13; x 1 ∙x 2 =q=36.
We substitute these values into the resulting formula:
Advice
: Always check the possibility of finding the roots of a quadratic equation using a suitable method, because 4
reviewed useful formulas allow you to quickly complete a task, especially in cases where the discriminant is an “inconvenient” number. In all simple cases, find the roots and operate on them. For example, in the last example we select the roots using Vieta’s theorem: the sum of the roots should be equal to 13
, and the product of the roots 36
. What are these numbers? Certainly, 4 and 9. Now calculate the sum of the square roots of these numbers: 2+3=5.
That's it!
Only because for integers you need to calculate the sign of the quotient. How to calculate the sign of the quotient of integers? Let's look at it in detail in the topic.
Terms and concepts of quotient of integers.
To perform division of integers, you need to remember terms and concepts. In division there is: the dividend, the divisor and the quotient of integers.
Dividend is the integer that is being divided. Divider is the integer that is being divided by. Private is the result of dividing integers.
You can say “Division of integers” or “Quotient of integers”; the meaning of these phrases is the same, that is, you need to divide one integer by another and get the answer.
Division originates from multiplication. Let's look at an example:
We have two factors 3 and 4. But let’s say we know that there is one factor 3 and the result of multiplying the factors is their product 12. How to find the second factor? Division comes to the rescue.
Rule for dividing integers.
Definition:
Quotient of two integers is equal to the quotient of their modules, with a plus sign as a result if the numbers have the same signs, and with a minus sign if they have different signs.
It is important to consider the sign of the quotient of integers. Brief rules for dividing integers:
Plus on plus gives plus.
“+ : + = +”
Two negatives make an affirmative.
“– : – =+”
Minus plus plus gives minus.
“– : + = –”
Plus plus minus gives minus.
“+ : – = –”
Now let's look in detail at each point of the rule for dividing integers.
Dividing positive integers.
Recall that positive integers are the same as natural numbers. We use the same rules as when dividing natural numbers. The quotient sign for dividing positive integers is always plus. In other words, when dividing two integers “ plus on plus gives plus”.
Example:
Divide 306 by 3.
Solution:
Both numbers have a “+” sign, so the answer will be a “+” sign.
306:3=102
Answer: 102.
Example:
Divide the dividend 220286 by the divisor 589.
Solution:
The dividend of 220286 and the divisor of 589 have a plus sign, so the quotient will also have a plus sign.
220286:589=374
Answer: 374
Dividing negative integers.
The rule for dividing two negative numbers.
Let us have two negative integers a and b. We need to find their modules and perform division.
The result of division or the quotient of two negative integers will have a “+” sign. or "two negatives make an affirmative".
Let's look at an example:
Find the quotient -900:(-12).
Solution:
-900:(-12)=|-900|:|-12|=900:12=75
Answer: -900:(-12)=75
Example:
Divide one negative integer -504 by a second negative integer -14.
Solution:
-504:(-14)=|-504|:|-14|=504:14=34
The expression can be written more briefly:
-504:(-14)=34
Dividing integers with different signs. Rules and examples.
When executing dividing integers with different signs, the quotient will be equal to a negative number.
Whether a positive integer is divided by a negative integer or a negative integer is divided by a positive integer, the result of the division will always be equal to a negative number.
Minus plus plus gives minus.
Plus plus minus gives minus.
Example:
Find the quotient of two integers with different signs -2436:42.
Solution:
-2436:42=-58
Example:
Calculate division 4716:(-524).
Solution:
4716:(-524)=-9
Zero divided by an integer. Rule.
When zero is divided by an integer, the answer is zero.
Example:
Perform division 0:558.
Solution:
0:558=0
Example:
Divide zero by the negative integer -4009.
Solution:
0:(-4009)=0
You cannot divide by zero.
You cannot divide 0 by 0.
Checking partial division of integers.
As stated earlier, division and multiplication are closely related. Therefore, to check the result of dividing two integers, you need to multiply the divisor and the quotient, resulting in the dividend.
Checking the division result is a short formula:
Divisor ∙ Quotient = Dividend
Let's look at an example:
Perform division and check 1888:(-32).
Solution:
Pay attention to the signs of integers. The number 1888 is positive and has a “+” sign. The number (-32) is negative and has a “–” sign. Therefore, when dividing two integers with different signs, the answer will be a negative number.
1888:(-32)=-59
Now let’s check the found answer:
1888 – divisible,
-32 – divisor,
-59 – private,
We multiply the divisor by the quotient.
-32∙(-59)=1888
The numbers in division are arranged as follows: the dividend is in the first place, the divisor is in the second, and the quotient is after the equal sign.
Dividend: divisor = quotient.
Let's denote all unknown numbers by letters
Let the dividend be equal to a, the divisor equal to b, and the quotient equal to c.
According to the condition, the product (that is, multiplication) of the dividend, the divisor and the quotient is equal to 3136. Let's create an equation.
- a * b * c = 3136.
- Since c is equal to a/b, we replace the letter c with the fraction a/b.
- a * b * a/b = 3136.
- The variable in is reduced, leaving a * a = 3136 or a 2 = 3136.
- Using the table of squares, we find the value of a, a is equal to 56.
The dividend is 56. The following equation is obtained: 56: b = c
Let's express the known dividend in terms of unknown variables
To find the dividend, you need to multiply the divisor and the quotient, that is, 56 = in * s.
By condition, all participating numbers are natural numbers, that is, positive integers. As we know, 56 is equal to the product of only two integers - 7 and 8.
This results in two expressions:
This means that the quotient (the number after the equal sign) can only be equal to either 7 or 8.
Answer: The quotient can be 7 or 8.
Let's denote the dividend by x and the divisor by y.
Then the quotient of dividing these two numbers will be equal to x/y.
According to the conditions of the problem, the product of the dividend, divisor and quotient is equal to 3136, therefore, we can write the following relation:
x * y * (x/y) = 3136.
Simplifying the resulting relationship, we get:
According to the conditions of the problem, the dividend, divisor and quotient are natural numbers, therefore, the value x = -56 is not suitable.
Let's decompose the number 56 into a product of prime factors:
56 = 2 * 28 = 2 * 2 * 14 = 2 * 2 * 2 * 7.
Let us list all possible divisors of the number 56 for which the quotient is a natural number.
Divisor 1, quotient 56;
divisor 2, quotient 28;
divisor 4, quotient 14;
divisor 8, quotient 7;
divisor 7, quotient 8;
divisor 14, quotient 4;
divisor 28, quotient 2.
divisor 56, quotient 1.
Answer: the quotient can take the values 1, 2, 4, 8, 7, 14, 28, 56.
Division is defined as the inverse of multiplication.
To divide one number by another means to find a third number that, when multiplied by the divisor, will give the dividend in the product:
Based on this definition, we derive the division rule for rational numbers.
First of all, let us point out once and for all that the divisor cannot be zero. Division by zero is excluded for the same reason it was excluded in arithmetic.
The absolute value a is equal to the product of the absolute values and c. This means that the absolute value of b is equal to the absolute value of a divided by the absolute value
Let us define the sign of the quotient s.
If the dividend and divisor have the same signs, then the quotient is a positive number. Indeed, if a and are positive, then the quotient o will also be a positive number.
Example. because
If a and are negative, then the quotient of c must also be positive in this case, since by multiplying by its negative number we must obtain a negative number a.
Example. because
If the dividend and divisor have different signs, then the quotient is a negative number. Indeed, if a is positive and a is negative, then c must be negative, since by multiplying a negative number by it we must obtain a positive number a.
Example. because
If a is negative and a is positive, then in this case c must be a negative number, since by multiplying a positive number by it we must obtain a negative number a.
Example. because
So, we came to the following division rule:
To divide one thing by another, you need to divide the absolute value of the dividend by the absolute value of the divisor and put a plus sign in front of the quotient, if the dividend and the divisor have the same signs, and a minus sign,
if the dividend and divisor have opposite signs.
As we have already said, division by zero is impossible, let us explain this in more detail. Suppose you need to divide some non-zero number, for example -3, by 0.
If the number a is the desired quotient, then by multiplying it by the divisor, that is, by 0, we must obtain the dividend, that is, - 3. But the product is equal to 0, and the dividend - 3 cannot be obtained. From this we conclude that the number
You can't divide 3 by zero.
Let the number 0 be divided by 0. Let a be the required quotient; multiplying a by the divisor 0, we obtain 0 in the product for any value of a:
Thus, we did not get any specific number: multiplying any number by 0, we get 0. Therefore, dividing zero by zero is also considered impossible.
For rational numbers, the following basic property of the quotient remains in force:
The quotient of two numbers will not change if the dividend and divisor are multiplied by the same number (not equal to zero).
Let us explain this with the following examples.
1. Consider the quotient, multiply the dividend and the divisor by - 4; then we get a new quotient
So, in the new quotient we got the same number 2.
2. Consider the quotient, multiply the dividend and the divisor by - then we get the following quotient:
The quotient has not changed since the result is the same number