Question 27473: what is the rule for adding and subtracting multipying and diving intergers .
for example:-2+2
-3*27
-12/4
Found 2 solutions by longjonsilver, sdmmadam@yahoo.com:
Answer by longjonsilver(2297)   (Show Source):
Answer by sdmmadam@yahoo.com(530)  (Show Source):
You can put this solution on YOUR website! what is the rule for adding and subtracting multipying and diving intergers .
for example:-2+2
-3*27
-12/4
You
what is the rule for adding and subtracting multipying and diving intergers .
for example:-2+2
-3*27
-12/4
The answers for the given problems:
1) -2+2 = 0
A loan of 2 dollars and an income of 2 dollars is quits and the final amount at hand is 0 dollar
2) -3*27 = (-3) X (27) = -(3X27) = -81
3) -12/4 = (-12)/4 = -(12/4) = (3X4)/4 = 3 (cancelling 4 in the numerator and in the denominator)
Given any two integers how to apply addition, subtraction, multiplication and division?
Say the given integers are a and b
Let a and b be both of the same sign
The Rule is if two integers(or numbers for that matter) of the same sign are added,
then give the same sign and add the quantities.
Case I) a and b are both positive (which we express as a>0 and b>0)
Then according to the rule (a+b)>0
Examples:
1) 2 and 5 Answer: 2 + 5 = +7
2) 115 and 20 Answer: 115 + 20 = +135
Case II) a and b are both negative (which we express as a<0 and b<0)
Then according to the rule (a+b)<0
Examples:
1) -2 and -5 Answer: (-2) + (-5) = -(2+5)= -7
2) -115 and -20 Answer: (-115) + (-20) = -(115+20) = -135
I can give you a real life example for this.
Just consider quantities positive as things that you possess or get
and quantities negative as things that you borrow or give
Then the whole thing is like this:
You have or you get 2 Cadburys chocolates and you get 5 more cadburys chocolates,
then you have in the box, now a total of 2 and 5 equal to 7 chocolates.
The answer is you HAVE and hence the sign of the sum is +
Suppose you owe 20 dollars to Tom (which means a minus of 20)
and you again borrow 115 dollars from Jack (which is another minus of 115)
Suppose you had had money at home. Then you pay back a loan of $20 +$115 = $135
which means you GIVE $135 which amounts to minus
or say you didn't have any money. Then you have a loan of $135 which is minus
because according to our rule having or getting is positive and giving or borrowing is negative.
Two integers of opposite signs when added
The Rule is if two integers(or numbers for that matter) of opposite signs are added,
then give the sign of the larger integer and take away the smaller from the larger
Case III) Let a be numerically larger and b be numerically smaller (when we say one larger and another smaller what we mean is consider only the numerical aspect without paying attention to the sign)
Then [(+a) + (-b)] = +(a-b)
Examples: 1)a = +100, b = -19
Between 100 and 19 which is larger? of course 100
So give +, the sign of the larger number 100 and take away the smaller 19 from the larger 100
and the answer is (100-19) = +81
Note: Here it means you have $100 with you and you borrow or give $19 and so after giving away the loan $19, what is left with you of your $100 is $81 and hence the answer is +81
2) a = -150, b = +100
Between 150 and 100,which is larger? of course 150
So give -, the sign of the larger number 150 and take away the smaller number 100
from the bigger number 150 and the answer is (-150+100) = -(150-100) = -50
Note: Here it means you owe or you have to give $150 and as you have with you $100, you pay it off and still you have a loan of $50 with you and hence the answer is -50
A Big NOTE: YOU DO NOT HAVE TO STUDY SUBTRACTION SEPARATELY!
Addition of two integers of opposite signs is actually Subtraction or a better way of putting it is
Subtraction regarding two integers, what we mean difference between two integers is actually
Addition of two integers of oppostie signs.
So it is enough if we learn the rules for addition of two given integers. Subtraction takes care of itself .
Examples of the first integer a numerically smaller and the second integer b numerically larger.
1) a = +12, b = -26
(a+b) = [(12) + (-26)] = -(26-12) = -14
You have 12 dollars and you owe 26 dollars and so the net business is
you still owe 14 dollars after paying off 12 dollars.
2) a = +1000, b = -8456
(a+b) = [(1000) + (-8456)] = -(8456-1000) = -7456
You have 1000dollars and you owe 8456 dollars and so the net business is
you still owe 7456dollars after paying off 1000dollars.
3) a = -500, b = +812
(a+b) = [(-500)+(812)] = (-500+812)= +(812-500) = +312
You have 812 dollars and you owe 500 dollars and so the net business is
you have 312 with you after paying off the loan 500 dollars.
4) a = -1500, b = +2100
(a+b) = [(-1500)+(2100)] = (-1500+2100)= +(2100-1500) = +600
You have 2100 dollars and you owe 1500 dollars and so the net business is
you have 600 dollars with you after paying off the loan 1500 dolalrs.
In all the above four examples if you carefully study the steps
(especially examples 3) and 4)) you see additive commutativity in operation.
Multiplication of two integers of the same sign
The Rule is if two integers(or numbers for that matter) of the same sign are multiplied,
then give the positive sign always and multiply the quantities in the usual way (using arithmetic tables)
Case I) a and b are both positive (which we express as a>0 and b>0)
Then according to the rule (ab)>0
Examples:
1) 2 and 5 Answer: 2 X 5 = +10
2) 20 and 115 Answer: 20 X 115 = + 2300
Case II) a and b are both negative (which we express as a<0 and b<0)
Then according to the rule (ab)>0
Examples:
1) -2 and -5 Answer: (-2) X (-5) = +10
2) 20 and 115 Answer: (-20) X (-115) = + 2300
Multiplication of two integers of opposite sign
The Rule is if two integers(or numbers for that matter) of opposite signs are multiplied,
then give the negative sign always and multiply the quantities in the usual way.
That is EITHER a>0, b<0 giving (ab)<0 OR a<0, b>0 giving (ab)<0
Whether it is (+a) X (-b) or it is (-a) X (+b) the answer is always -(ab)
Examples:
1) (-2) X (5) = -(2X5) = -10 (2) (2) X (-5) = -(2X5) = -10
3) (-1000) X (12) = -(1000X12) = - 12000
(4)(1000) X (-12) = -(1000X12) = -12,000
Rreal life example for this.
Just consider quantities positive as things that you possess or get
and quantities negative as things that you borrow or give
Then the whole thing is like this:
2 X 5 is explained as Your having or getting 5 Cadburys chocolates
and your getting 2 times such (5 chocolates)
or by multiplicative commutativity it is 5 X 2 which is your having 2 Cadburys chocolates and your getting
5 times such (2 chocolates)
2 X 5 is actually 5 considered two times which amounts to (5+5) =10
And 5 X 2 = 2 considered 5 times = (2+2+2+2+2) = 10
Note: Real examples are given to augment your status of understanding things.
And when they tend to call for a fresh level of understanding to understand something, then it is wise enough to chuck the real life example. So we better not dive into real life examples for multiplying two integers both negative or for multiplying two integers of opposite signs.
A Big NOTE: YOU DO NOT HAVE TO STUDY DIVISION SEPARATELY!
Division of one integer a by another b (when b is not zero) is a/b which is nothing but a X (1/b)
So simply apply the rules for multiplication and be done with it.
Only we must remember to present the final answer in the most simplified form
after cancelling out common factors in the numerator a and in the denominator b.
Most simplified form means the final nr and dr having only 1 as common factor.
Examples:
1) a = +12, b = +26
ab = 12 X 26 = +(12X26) = +312
(a/b) = 12/26 = +(6/13) = 6/13
2) a = -1000, b = -8456
(ab) = (-1000) X (-8456) = +(1000X8456) = + 84,56,000
(a/b) = 1000/ (-8456) = -(1000/8456) = -(500/4228) = -(250/2114) = -(125/1057)
3) a = +12, b = -26
ab = 12 X (-26) = -(12X26) = -312
(a/b) = 12/(-26) = -(6/13)
4) a = -1000, b = +8456
(ab) = (-1000) X (8456) = - 84,56,000
(a/b) = (-1000)/ 8456 = -(1000/8456) = -(500/4228) = -(250/2114) = -(125/1057)
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