|
Question 50110This question is from textbook Intermediate Algebra
: What I really need is an explaination, I've posted the ISBN number of the text I'm using...although it does explain the order of operations on page 26 it does not explain it to the point I understand. For instance, one problem asked was to evaluate: -9-(-11)-4-(-6) by our instructor. To be quit frank I'm lost and need to understand the basic concepts starting at the bottom floor and not in the middle. However, since the college I am attending will not put me in a pre-algebra class to learn the basic's needed as I scored too high on the pre-math exam I stuck with intermediate algebra which starts me in the middle. I've also checked out other web sites for help and they don't explain it any better either, at least not simplified in steps as to why I'm suppost to do it that way, so I can understand it. I would greatly appreciate any help...thank you...littledarling55.
This question is from textbook Intermediate Algebra
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! OK. Not sure i can help you 100% with an explanation of "why" but i can start.
+ and - are used in maths for 2 operations each, which is a source of confusion for the beginner and most teachers seem to forget it too and resort to the age-old explanation of "because!"
+ indicates either a positive number like +5 or +23
OR
+ indicates the "ADDING" operation such as 3+4 --> 7.
Now, mathematicians are very quick to drop symbols that are not required, since "we know they are really there". But of course, the beginner does not know they are there and it all looks like magic!
3+4=7 really should be written as (+3)+(+4)=(+7)... the brackets are not needed. I have just used them to collect terms together. So what we have is "positive 3" ADDED to "positive 4" equals "positive 7"
It gets tedious writing +4++5 all the time for 4+5... we accept that the missing POSITIVE SYMBOLS are there. But more importantly, it looks complicated and is prone to misreading
Now, how about adding negative numbers? Here we do need to write the negatives in: (-3)+5 really means (-3)+(+5) which is +2. Or written as you would write it:
-3+5=2
Now the issue behind all this (apart from mathematicians not writing some + symbols) is how do we manage/work with ++ or -- or +- or -+. Eg
(+4)-(+3) is what? Well it means "positive 4" take away "positive 3". The answer is, i hope you can see, "positive 1".
How i tend to visualise some of this is a number line or a "ladder" where each rung is a number... somewhere is zero and above that is +1 then +2 etc and below the zero rung is -1 then -2 etc. The number +4 is in reality "start at zero and move UP 4" and the number -6 is "start at zero and move DOWN 6"
Now for the + and - meaning of ADD and SUBTRACT.
ADD means "keep the UP/DOWN sense" so by this i mean:
(+2)+(+3) is "start at zero and go UP 2" then ADD which means keep the sense unaltered and "go UP a further 3". You end up at +5.
(+5)+(-3) is "start at zero and go UP 5" then ADD which means keep the sense unaltered and "go DOWN 3". You end up at +2.
Now for SUBTRACT and what i mean by "keep the sense unaltered" :-)
SUBTRACT means "reverse the sense of UP/DOWN so by this i mean:
(+6)-(+4) is "start at zero and go UP 6" then SUBTRACT +4 which means instead of going UP by 4, we go DOWN by 4... we end up at +2
Now for the classic example:
(+2)-(-5). The answer is +7 and the explanation is "start at zero and go UP 2" then we need to SUBTRACT (-5). Well the number (-5) is "GO DOWN 5 rungs" but since we have SUBTRACT, that means reverse the sense, so we go up 5 instead and we end up at +7.
You can obviously use this in your head every time you do a calculation but it slows you down. BUT DEFINITELY HAVE IT IN YOUR HEAD, SINCE THIS IS WHAT IS GOING ON. Wht mathematicians tend to do is look for patterns to simplify all this and then they learn and apply that simpler model.
What you will hear is "like terms add and unlike terms subtract" or some such which is correct (applied correctly).
So in 3+4... visualise all the symbols: (+3)+(+4) or... +3++4 and you can see ++ together. ++ is thought of as + (since ++ are the same symbols) and we "simplify" +3++4 to +3+4 --> +7. Now from what I have explained so far, you can appreciate that +3+4 is in fact WRONG maths since what does the second + now signify? ADD or POSITIVE?
So... -3+-5. Looking at the +-... these are different so we write then as - and we have -3-5 is -8. ALWAYS APPRECIATE THAT THIS METHOD IS A FUDGE ON THE REAL MATHS...THE LADDER and ADD(keep the direction) or SUBTRACT(reverse).
So, (-4)-(-1) is (without the brackets) -4--1 and -- becomes + --> -4+1 is -3.
The CORRECT explanation of this is "start at zero and go DOWN 4 rungs" then we subtract -1. Well, -1 means go down a further one rung but because we have SUBTRACT, we reverse that and we go UP 1 instead...We end up at -3.
Hope this helps? :-)
In terms of multiplication and division, the same rules apply:
(-3)x(+4) is:
- and a + are different --> go to -
3x4 is 12
Answer is -12
Always split down terms...think about the symbols then do the numbers.
As for you question:
-9-(-11)-4-(-6)... even here you have mixed messages since some symbols have been missed (either you copying incorrectly) or the questioner being lazy. Your maths skill is in understanding this as -9-(-11)-(+4)-(-6) and this is what i shall calculate...
We should have the real explanation as follows:
start at zero and go DOWN 9 then we have a further 11 to go down but we go up instead because of the SUBTRACT... We now are at rung +2.
From here we subtract +4 so this means "go 4 rungs UP" but actually go 4 down because we are subtracting... so we were at +2 and now at -2.
Finally from here we subtract -6. So this means go DOWN 6 rungs but actually go UP since we are SUBTRACTING...we end up at +4 as the final answer. Hopefully this all makes sense but the last thing you want to be doing is visualising the bloody ladder every time :-)
So, the quicker way (i am doing every step):
-9-(-11)-(+4)-(-6)
-9+11-4+6
+2-4+6
-2+6
+4
Or my method (where i collect terms)
-9-(-11)-(+4)-(-6)
-9+11-4+6
-9-4+11+6
-13+17
+4
You find the method that suits you... just get the corret answer :-)
Hope this helps
jon
|
|
|
| |
