Handling signs
Signs can be confusing. You handle them differently depending on whether it is a addition/subtraction operation or a multiplication/division operation. The rules are different, but both rules are rather simple.
For addition/subtraction:
There are only two possibilities in addition/subtraction. We will call them Case I and Case II.
Case I is when the sign of both numbers is the same and
Case II is when they are not the same (they are different).
Case I (same sign), with simple rules.
1- Add the numbers together.
2- Use the sign.
A few examples:
45+32 Add the numbers together and you get 77
Use the sign, which is positive, so answer is +77
and normally we just write this as 77.
-55-32 Add the numbers together and you get 87
Use the sign, which is negative, so answer is -87.
A few more examples:
-50-35 = -85 signs same, add numbers, use sign
34+23 = 77
-4-8-15-3= four numbers, do them two at a time, as such
-12-15-3=
-27-3=-30
-101-5-50-35-43-3= Even harder one, two numbers at a time
-106-50-35-43-3=
-156-35-43-3=
-191-43-3=
-234-3=-237
Case II (different signs) is when the signs of the numbers are different and the rule is different.
After all, how could it not be.
Simple rules for Case II (different signs).
1- Ignore the sign.
2- Subtract small number from large number.
3- Use the sign of the larger number.
So, lets look at -76+44
Ignoring the sign, 76 is the larger number and
44 is the smaller number
Subtract the smaller from the larger
76-44=32
Then use the sign of the larger which was 76
and which has a negative sign.
So, the answer is
-32
Now lets do several more of these. Just apply the rules.
1- Ignore the sign.
2- Subtract small number from large number.
3- Use the sign of the larger number.
Do these problems and make sure you get these answers.
-65+30=-35
56-80=-24
90-60=30
120-125=-5
47-50=-3
47-150=-103
-400+500=100
-893+894=1
Lets do a few more problems.
45-50=-5
and also
-50+45=-5
The point here is that the order in which positive/negative numbers
are combined is unimportant, the answer is the same.
So,
-300+325-500 is the same thing as
-500-300+325 = -475
and also the same as 325-500-300 = -475 .
If you do not see that with the above rules, then you have not
gotten a clear picture of the rules and you should do the
following:
-3+6 and 6-3 both equal 3
-6-3+2 and 2-3-6 both equal -7
8+24-32 and -32+24+8 both equal 0
But one of the real problems (challenges) is to distinguish
between a sign for addition/subtraction and one for multiplication
and division. In the following problems, all the signs are
addition/subtraction.
-7+8-7+8
-4-5-3+18-4-5-3+18
But the next problem has some signs associated with multiplication/division (specifically multiplication).
1- Add the numbers together.
2- Use the sign.
A few examples:
45+32 Add the numbers together and you get 77
Use the sign, which is positive, so answer is +77
and normally we just write this as 77.
-55-32 Add the numbers together and you get 87
Use the sign, which is negative, so answer is -87.
A few more examples:
-50-35 = -85 signs same, add numbers, use sign
34+23 = 77
-4-8-15-3= four numbers, do them two at a time, as such
-12-15-3=
-27-3=-30
-101-5-50-35-43-3= Even harder one, two numbers at a time
-106-50-35-43-3=
-156-35-43-3=
-191-43-3=
-234-3=-237
Simple rules for Case II (different signs).
1- Ignore the sign.
2- Subtract small number from large number.
3- Use the sign of the larger number.
So, lets look at -76+44
Ignoring the sign, 76 is the larger number and
44 is the smaller number
Subtract the smaller from the larger
76-44=32
Then use the sign of the larger which was 76
and which has a negative sign.
So, the answer is
-32
1- Ignore the sign.
2- Subtract small number from large number.
3- Use the sign of the larger number.
Do these problems and make sure you get these answers.
-65+30=-35
56-80=-24
90-60=30
120-125=-5
47-50=-3
47-150=-103
-400+500=100
-893+894=1
-7+8-7+8
-4-5-3+18-4-5-3+18
But the next problem has some signs associated with multiplication/division (specifically multiplication).
-8-(-7)-5+2(-15)
In the "order of operations" rules of math there is a
popular saying
"Please Excuse My Dear Aunt Sally"
The first letter of each word is a key in the
order of operations.
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
This is the required order of operations - the hierarchy which must be followed.
And actually division and multiplication are of "equal importance" and addition and subtraction are equal.
Other than that, you must do the operations in this order:
Parenthesis Exponents Mult Div Add SubSo, looking at -8-(-7)
There is a parenthesis, so we must do all the operations inside the parenthesis. But in this case there are no operations only a negative 7. So, we have taken that step, and nothing further is required for the parenthesis. Now we must do multiplication/division. The -(-7) is multiplication. If you choose you could replace this with -1(-7) which means negative 1 times negative 7. So it is multiplication.
So, -8-(-7)-5+2(-15) becomes -8+7-5+2(-15) . Now, one step at a time
-8+7-5+2(-15) do the multiplication and 2 times -15 is -30
-8+7-5-30 now, one step at a time add/subtract
-1-5-30
-6-30
-36
Multiplication/division "sign rules" are simple.
If the two signs are the same the result is positive. If the two signs are different, the result is negative. One should not worry about the "logic" of this, just memorize it.Now back to -1(-7) which means negative 1 times negative 7. The signs are the same and the result is positive. One times seven is seven. The result is positive and -1(-7) therefore equals +7.So, working the problem from the beginning, one step at a time
-8-(-7) Do the multiplication first
-8+7 Using our addition/subtraction rules
-1
It is that easy. Now consider -(-6)+8+(-3-5)
-(-6)+8+(-3-5) Do parenthesis first
-(-6)+8+(-8)
Now do multiplication
+6+8-8
Now apply add/sub rules, one at a time
+14-8
+6 <- Answer
One more
-(-(8-12))+14 All this foolishness -(-(8-12))
must be handled one parenthesis at a time,
with the "inner" parenthesis first.
-(-(-4))+14 Now handle the multiplication, remember -(-4)
is the same as -1(-4) or negative one times
negative 4.
A negative times a negative is a plus and
one times 4 is 4.
-(+4)+14 Now handle -(+4) because that is multiplication
-1(+4) which equals -4
-4+14 Now apply add/sub rules
+10 <- The answer
| -(-4/2+3)-(-6-3)+(12-15/-3) | One step at a time, work on parenthesis first. |
| -(-2+3) -(-6-3)+(12-15/-3) | Negative 4 div by positive 2 is negative 2, now finish the first parenthesis. |
| -(+1) -(-6-3)+(12-15/-3) | Now work on that second parenthesis |
| -(+1) -(-9)+(12-15/-3) | Now the last parenthesis, remember divide before subtracting |
| -(+1) -(-9)+(12+5) | Negative 15 div by negative 3 is positive 5, keep working until only one number is inside the parenthesis |
| -(+1) -(-9)+(+17) | Now there is only one number
in each parenthesis, so we
have handled the "Please"
portion of "please excuse
my dear aunt sally".
And we have no exponents, so
move on to multiplication.
One step at a time. |
| -1 -(-9)+(+17) | |
| -1 +9 +(+17) | Remember negative 1 times negative 9 is positive 9 and +1 times +17 is +17 |
| -1 +9 +17 | |
| +8 +17 | -1+9 is +8 |
| +25 | < - the answer |