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.

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## 2 Answers By Expert Tutors

Matthew S. answered • 11/13/17

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The most important thing to note here: When talking about the order of operations, "parentheses" just means that any calculations

*completely**inside*parentheses should be done first. e.g for 10 * (6 + 3), we add 6 and 3 first. In the order of operations, parentheses tell you "combine whatever's inside to a single number before doing anything with it". Of course we can have multiple layers of parenthesis, in which case the innermost parentheses are taken care of first, e.g. 2 * (6 * (5 + 3) + 2) = 2 * (6 * 8 + 2) = 2 * (48 + 2) = 2 * 50 = 100.Once the values inside parentheses have been turned into a single value, they can usually go away (though a couple exceptions are below). So, in your example, putting the 3 into parentheses after flipping the sign isn't necessary. A single value in parentheses doesn't get any special treatment, it's just treated like number.

Here we also see two other ways parentheses are used:

1) to show subtraction of negative numbers. Changing it to a plus doesn't have to be the first step. It actually doesn't matter when it happens (as long as it's before adding/subtracting it, of course). In fact, the "parentheses" instruction of order of operations doesn't apply here - remember that only applies to what's

*inside*.2) to show multiplication. They're common when multiplying with negative numbers so we don't get confused with subtraction. Again, if it's just a single value inside, there's nothing to be done and the "parentheses" instruction doesn't apply. Parentheses could be replaced with the times symbol or any of the other ways we denote multiplication.

So, the main point again: When talking about the order of operations, "parentheses" just means that any calculations completely inside parentheses should be done first.

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.)

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

You have made a couple astute observations, I'll mention them and add my own notes.

First, note that writing the full expression on one line (for a calculator or a computer) has introduced "/" in place of the older division sign (÷) , so computer-based computation has changed some rules. Also, the long lines of fractions are gone in this scenario {note: although they still appear in textbooks and scientific papers).

Second, PEMDAS is a convention, not a rule. It attempts to describe the "order of operations" so everyone will perform them the same way. There's more work to do on this.

You wrote: "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.)"

Great observation. Why doesn't PEMDAS introduce consistency?

You also recognize the difference between unary minus (negative numbers, implied binary subtraction from 0) and binary subtraction. Good for you! That's hard for some people, especially when it's implied multiplication like the current Facebook viral post: 6 ÷ 2 ( 1 + 2 ) = ?

So, the parentheses in "6-(-3)" merely make the expression easier to read than "6--3," which might get changed to "6—3" [em dash vs. en dash] when using autocorrect in a word processor. You are correct, the value "(-3)" should be treated as a number, not as an expression inside parentheses. So, "-10+6-(-3)" becomes "-4-(-3)" because the value inside the parentheses has been completely evaluated;

**it is just a matter of when to remove the parentheses that's in question.**Now, regarding "steps." PEMDAS is performed left-to-right with M&D at an equal precedence level and A&S at an equal precedence level.

**When any replacement is made, the scan re-starts**, moving left-to-right (this in recursive inside parentheses).The steps listed in this problem statement group too many actions.

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Mark M.

11/13/17