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 times 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 for division natural numbers. The quotient sign for dividing integers positive numbers always a 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 the second a negative number -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.
By doing 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 division will always be equal to a negative number.
Minus plus plus gives minus.
Plus times 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 function a n =f (n) of the natural argument n (n=1; 2; 3; 4;...) is called a number sequence.
Numbers a 1; a 2 ; a 3 ; a 4 ;…, forming a sequence, are called members of a numerical sequence. So a 1 =f (1); a 2 =f (2); a 3 =f (3); a 4 =f (4);…
So, the members of the sequence are designated by letters indicating indices - serial numbers their members: a 1 ; a 2 ; a 3 ; a 4 ;…, therefore, a 1 is the first member of the sequence;
a 2 is the second term of the sequence;
a 3 is the third member of the sequence;
a 4 is the fourth term of the sequence, etc.
Briefly the numerical sequence is written as follows: a n =f (n) or (a n).
There are the following ways to specify a number sequence:
1) Verbal method. Represents a pattern or rule for the arrangement of members of a sequence, described in words.
Example 1. Write a sequence of all non-negative numbers, multiples of 5.
Solution. Since all numbers ending in 0 or 5 are divisible by 5, the sequence will be written like this:
0; 5; 10; 15; 20; 25; ...
Example 2. Given the sequence: 1; 4; 9; 16; 25; 36; ... . Ask it verbally.
Solution. We notice that 1=1 2 ; 4=2 2 ; 9=3 2 ; 16=4 2 ; 25=5 2 ; 36=6 2 ; ... We conclude: given a sequence consisting of squares of natural numbers.
2) Analytical method. The sequence is given by the formula of the nth term: a n =f (n). Using this formula, you can find any member of the sequence.
Example 3. The expression for the kth term of a number sequence is known: a k = 3+2·(k+1). Compute the first four terms of this sequence.
a 1 =3+2∙(1+1)=3+4=7;
a 2 =3+2∙(2+1)=3+6=9;
a 3 =3+2∙(3+1)=3+8=11;
a 4 =3+2∙(4+1)=3+10=13.
Example 4. Determine the rule for composing a numerical sequence using its first few members and express the general term of the sequence using a simpler formula: 1; 3; 5; 7; 9; ... .
Solution. We notice that we are given a sequence of odd numbers. Any odd number can be written in the form: 2k-1, where k is a natural number, i.e. k=1; 2; 3; 4; ... . Answer: a k =2k-1.
3) Recurrent method. The sequence is also given by a formula, but not by a general term formula, which depends only on the number of the term. A formula is specified by which each next term is found through the previous terms. In the case of the recurrent method of specifying a function, one or several first members of the sequence are always additionally specified.
Example 5. Write out the first four terms of the sequence (a n ),
if a 1 =7; a n+1 = 5+a n .
a 2 =5+a 1 =5+7=12;
a 3 =5+a 2 =5+12=17;
a 4 =5+a 3 =5+17=22. Answer: 7; 12; 17; 22; ... .
Example 6. Write out the first five terms of the sequence (b n),
if b 1 = -2, b 2 = 3; b n+2 = 2b n +b n+1 .
b 3 = 2∙b 1 + b 2 = 2∙(-2) + 3 = -4+3=-1;
b 4 = 2∙b 2 + b 3 = 2∙3 +(-1) = 6 -1 = 5;
b 5 = 2∙b 3 + b 4 = 2∙(-1) + 5 = -2 +5 = 3. Answer: -2; 3; -1; 5; 3; ... .
4) Graphic method. The numerical sequence is given by a graph, which represents isolated points. The abscissas of these points are natural numbers: n=1; 2; 3; 4; ... . Ordinates are the values of the sequence members: a 1 ; a 2 ; a 3 ; a 4 ;… .
Example 7. Write down all five terms of the numerical sequence given graphically.
Every point in this coordinate plane has coordinates (n; a n). Let's write down the coordinates of the marked points in ascending order of the abscissa n.
We get: (1 ; -3), (2 ; 1), (3 ; 4), (4 ; 6), (5 ; 7).
Therefore, a 1 = -3; a 2 =1; a 3 =4; a 4 =6; a 5 =7.
Answer: -3; 1; 4; 6; 7.
Reviewed number sequence as a function (in example 7) is given on the set of the first five natural numbers (n=1; 2; 3; 4; 5), therefore, is finite number sequence(consists of five members).
If a number sequence as a function is given on the entire set of natural numbers, then such a sequence will be an infinite number sequence.
