Dl P.

asked • 11/13/17

What is included under ‘Parentheses’ in the Order of Operations?

I don’t mean the types of grouping symbols. 
 
What I don’t understand is how far are parentheses the most important? I know that they go first, but is that only for actual problems inside parentheses, or does that rule also include plain single numbers inside parentheses?
 
there are some particular problems I’ve seen that confuse me:
 
 
 
Problem:               Step 1:
-6(-2)–8              12–8
_________ =       ________ =
-10+6–(-3)          -10+6+(3)
 
Step 2:          Step 3:       Answer:  
12+(-8)         4
______ =      __ =               -4
-4+3             -1
 
 
(note: the long lines are fraction bars, I couldn’t get them closer to the numbers; and the shortest lines mean negative while the longer- short lines mean minus)
 
so my question is, why for the denominator of the fraction bar, is the first step the “change subtraction to adding the opposite” step? I understand the step I just don’t know why it’s first. The problem “6–(-3)” or “six minus negative three” is clearly a subtraction problem, a problem partially outside of the parentheses, and the rule is if there are only adding and subtracting problems to go from left to right. 
 
and then in the second step it doesn’t do the ”6+(3)” problem, it does the “-10+6” part. if parentheses are priority why aren’t they taken care of in the next step seeing as they’re still there? (thats exactly how they were written.)
 
i also put this problem into cymath and it gave the same process, except after it solved that part it didn’t put the 3 into parentheses. this makes it seem as though parentheses are sometimes optional/preferential depending on the context, which states that parentheses aren’t always priority.  
 

Mark M.

You have answered your own question.
The parenthesis rule is for something like (x + 2x) become 3x
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11/13/17

2 Answers By Expert Tutors

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Matthew S. answered • 11/13/17

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Math and Science Tutor - MIT Grad, Google - Available on Short Notice!

Dl P.

First off, thank you very much for explaining it to me. 
 
Originally, before i came across the problem in my post, I knew and understood exactly what you said, that the parentheses rule only applies to an actual problem inside them, a single number with parentheses around it is just the same as any other.
 
But the way that problem was done in the book and then the cymath website also doing it that way threw me off; both were done out of order, because with division and multiplication and then addition and subtraction, you are supposed to start from the left and go right (Ex. 4-3+1 equals 3; left to right.
but if I just decided to do the opposite then I would get 1.)
I didn’t realize at the time that they did it without rhyme or reason and that it actually doesn’t matter because its a different type of operation. 
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11/13/17

David W.

tutor
Well, yes it does matter.
 
Because operations at the same precedence level (e.g., M&D or A&S) may produce different results due to truncation (limited precision of intermediate results).  In computers, two's-complement notation was introduced because +0 and -0 were different numbers (depends on the order of calculations to get here).  Also, I knew an experienced analyst converting a program from a 36-bit computer to a 32-bit computer who spent days to find the place where simply changing the order of operations (done automatically by the compiler to improve performance, not by the programmer) produced a different result.
 
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11/13/17

Dl P.

Sorry, you’re right. thanks for the explanation 
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11/14/17

David W. answered • 11/13/17

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4.6 (63)

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