Is There a Universal Standard for Order of Operations in Mathematics?

[chwala]

Homework Statement

solve 6/2(2+1)

Relevant Equations

BODMAS

attempt 1.
6 /2×3 = 3×3=3×3=9
attempt 2
6/2(3)= 6 divide 6 = 1.
sorry i cannot see the divide symbol.

 

[sysprog]

You can use / to represent the division operation --
that makes the problem statement as follows: 6/2(2+1) --
the addition operation indicated by the + sign is inside parentheses --
that means it should be done before the division and multiplication operations --
so the problem can be interpreted in English as:
what number is: six divided by the quantity, two times the quantity 'two plus one' --
after you determine what '2+1' adds up to, then you multiply that number by 2,
then divide 6 by the resulting number, and you'll have the answer.

 

[Ray Vickson]

attempt 1.
6 divide 2×3 = 3×3=3×3=9
attempt 2
6divide 2(3)= 6 divide 6 = 11.
sorry i cannot see the divide symbol.

NO! 6 divide 6 is not 11; it is 1!

 

[symbolipoint]

attempt 1.
6 divide 2×3 = 3×3=3×3=9
attempt 2
6divide 2(3)= 6 divide 6 = 11.
sorry i cannot see the divide symbol.

The quoting is not working right.

Want show this:

solve 6 divide 2(2+1)

SIX divided by 2(2+1)

SIX divided by 2*3

SIX divided by 6

6 divided by 6

1chwala, did you mean something else?

 

[Mark44]

You can use / to represent the division operation --
that makes the problem statement as follows: 6/2(2+1)

This notation is ambiguous.
It could mean
$$1)~~ \frac 6 {2(2 + 1)} = \frac 6 6 = 1$$
or it could mean
$$2)~~ \frac 6 2 (2 + 1) = 9$$

that means it should be done before the division and multiplication operations

And further, reading from left to right, the division operation comes before the multiplication operation, most would interpret 6/2(2 + 1) as #2 above. Also, every programming language that I know about would interpret 6/2*(2 + 1) as 9, for the reason I gave.

@chwala, this is not an algebra problem -- it's plain old arithmetic.

 

[symbolipoint]

@chwala, this is not an algebra problem -- it's plain old arithmetic.

That's okay. Often the first time students are forced to pay attention to Order Of Operations is in Pre-Algebra(the course) or Algebra 1. Before Pre-Algebra, this stuff is not given as much attention as it should.

 

[sysprog]

This notation is ambiguous.

I agree.

It could mean
$$1)~~ \frac 6 {2(2 + 1)} = \frac 6 6 = 1$$
or it could mean
$$2)~~ \frac 6 2 (2 + 1) = 9$$

Correct, and nicely presented, both as usual for you.

And further, reading from left to right, the division operation comes before the multiplication operation, most would interpret 6/2(2 + 1) as #2 above. Also, every programming language that I know about would interpret 6/2*(2 + 1) as 9, for the reason I gave.

I partially agree, but with a proviso: in parsing of expressions that mix mathematical notation with English, such as the OP's "solve 6 divide 2(2+1)", the mathematical notation is generally evaluated separately from, and prior to, any mathematical operation specified in the English, so in this instance, I would interpret the expression as meaning 'divide 6 by the result of multiplying 2 by the result of 2 being added to 1'.

I did not point that out to the OP, and my explanation included only the idea of doing the parenthetical addition before doing the "division and multiplication", which could be interpreted as suggesting that the division should be done before the multiplication.

To make the expression 6/2(2+1) unambiguous, without resorting to $\frac 6 {2 (2 + 1)}$ or $\frac 6 2 (2 + 1)$ notation that here on PF requires $\TeX$, one could use another pair of parentheses, to make it either 6/(2(2+1)) or (6/2)(2+1).

I think you're right to have pointed out the ambiguity.

 

[chwala]

NO! 6 divide 6 is not 11; it is 1!

amended...

 

[Mark44]

solve 6/2(2+1)

As minor point, you're not "solving" anything here. You can solve an equation or inequality, but in this case the goal is to simplify the given expression, by writing it in a simpler form.

 

[chwala]

As minor point, you're not "solving" anything here. You can solve an equation or inequality, but in this case the goal is to simplify the given expression, by writing it in a simpler form.

absolutely...that is what i was looking for, i have read your comments on this and taken note...

 

[sysprog]

As minor point, you're not "solving" anything here. You can solve an equation or inequality, but in this case the goal is to simplify the given expression, by writing it in a simpler form.

Also a minor point, the problem could be posed algebraically as:

Solve for $x$:

$x=6/2(2+1)$

or with the expression formulated non-ambiguously as:

$x=6/(2(2+1))$ or $x=(6/2)(2+1)$

or also non-ambiguously as:

$x= \frac 6 {2(2+1)}$ or $x=\frac 6 2(2+1)$

Please note that, as a computer language, $\TeX$ parses as @Mark44 explained, resulting in, if not otherwise specified by braces, the latter interpretation as the default.

 

[Mark44]

Also a minor point, the problem could be posed algebraically as:
Solve for $x$:
$x=6/2(2+1)$

That's kind of a stretch, as x is already isolated (solved for). The only thing left to do is to simplify the right side.

 

[sysprog]

That's kind of a stretch, as x is already isolated (solved for). The only thing left to do is to simplify the right side.

It's nice of you to call it "kind of a stretch", instead of denouncing it altogether.

 

[symbolipoint]

The original posted description made the expression 2(2+1) look like a separate expression to be evaluated separately from the "6". The DIVISION operation was placed between "6" and the "2(2+1)".

 

[sysprog]

The original posted description made the expression 2(2+1) look like a separate expression to be evaluated separately from the "6". The DIVISION operation was placed between "6" and the "2(2+1)".

I agree with that, but I still think @Mark44 was right to point out the ambiguity.

 

[Mark44]

The original posted description made the expression 2(2+1) look like a separate expression to be evaluated separately from the "6". The DIVISION operation was placed between "6" and the "2(2+1)".

Yes, and there's an implied multiplication between the 2 and (2+ 1). The result hinges on whether the division is perfomed first (in which we get a value of 9) or the multiplication is done first (in which the result is 1).
Back when I was learning algebra, we had the acronym MDAS (My Dearl Aunt Sally). More recently we have BODMAS or PEMDAS to indicate the relative priority of operations.

What is lacking in all these acronyms are the notions of precedence and associativity, which are elaborated on in modern programming languages. In C, C++, C#, Java, Python, and many others, multiplication and division have the same precedence, and they associate left-to-right. Same for addition and multiplication. This means that 6/2*(2+1) evaluates unambiguously to 9.

Here's from the opening post:

solve 6/2(2+1)

Python evaluates 6/2*(2+1) as 9, as does C and all the others I listed.

 

[symbolipoint]

The original, post#1, was later adjusted. I remember the word "division" appearing. As it is now stated, it does contain an ambiguity.

 

[sysprog]

Here's from the opening post:

solve 6/2(2+1)

That's from an edited version. The original said "solve 6 divide 2(2+1)" , and the OP said that he didn't know what to type to indicate the division operation. I introduced the ambiguity inadvertently in my post in which I told the OP that he could use / to represent division.

 

[symbolipoint]

That's from an edited version. The original said "solve 6 divide 2(2+1)" , and the OP said that he didn't know what to type to indicate the division operation.

Certainly! This has now been mostly clarified.

 

[sysprog]

Certainly! This has now been mostly clarified.

The OP also specified 'Relevant Equations: BODMAS' -- I think the application of the associated operational priority was probably the main point of the exercise -- and I went and effed that part of it up --

 

[symbolipoint]

The OP also specified 'Relevant Equations: BODMAS' -- I think the application of the associated operational priority was probably the main point of the exercise -- and I went and effed that part of it up --

You should not feel too bad. For many years, I thought I knew Order Of Operations perfectly, until I looked at a few mixed multiplications & divisions without parentheses being present. Then, I had to review very carefully how that precedence was supposed to be. Today, I sometimes forget again, and when I am writing the expression, I must use parentheses to make clear what I want.

 

[sysprog]

You should not feel too bad. For many years, I thought I knew Order Of Operations perfectly, until I looked at a few mixed multiplications & divisions without parentheses being present. Then, I had to review very carefully how that precedence was supposed to be. Today, I sometimes forget again, and when I am writing the expression, I must use parentheses to make clear what I want.

I've been a programmer ever since childhood (early '70s -- I'm 60 now -- my Dad taught at the university, and 'they' let us faculty brats play with the IBM 370 computer), and I know the conventional order of operations well; however, when evaluating "6 divide 2(2+1)", I allowed that one could substitute / for 'divide', without giving due account for that substitution making the expression no longer separable into English and mathematical notational subunits, so that the idea of automatically evaluating 2(2+1) prior to doing the division was no longer valid. So, yeah, I feel like I dropped the ball, but it's not as if a championship was at stake, so I'm not too terribly crestfallen about it.

 

[Mark44]

Cheer up guys, we get this kind of question regularly, and it always seems to stump some people. IMO, the people whose focus is primarily mathematics aren't as much up to speed as those whose focus is programming, particularly in how operations at the same precedence level associate.

This is all very well described in programming language definitions, because computer programs are expected to be consistent in the the results they produce in arithmetic operations. The acronyms PEMDAS (or is it PEDMAS) and BODMAS do a reasonably good job at reinforcing operator precedence, the idea that a parenthesized expression (Brackets) should be evaluated before, say, division, but they don't do a good job at specifying how same-precedence operations such as mult/division or addition/subtraction should be evaluated. That's where the concept of associativity comes into play.

For C and its derivatives, how 2 + 3 + 5 is evaluated is governed by the associativity rules, as are 2 - 3 + 5 and 6/6*2. I.e., all of the operations here proceed from left to right, as if these three expressions were (2 + 3) + 5, (2 - 3) + 5, and (6/6)*2.

 

[FactChecker]

The rule is PEMDAS:
Parentheses. Exponents. Multiplication and Division (from left to right) Addition and Subtraction (from left to right)
Notice that "Multiplication and Division (from left to right)" implies that any order of multiplication and division operations are performed as they are encountered from left to right. So the division 6/2 should be done before the multiplication by 3. But that is obscure.
It is far better to use parentheses to make the calculation clear:
(6/2)(2+1)
or
6/(2*(2+1))

 

[Mark44]

Emphasis added...

The rule is PEMDAS:
Parentheses. Exponents. Multiplication and Division (from left to right) Addition and Subtraction (from left to right)

I agree that this is how it should be, but I'm not sure that it's presented this way; i.e., that arithmetic operations at the same precedence are evaluated left to right.

This PEMDAS (or BEDMAS/BODMAS/BIDMAS as are used in Canada/New Zealand and a bunch of other countries) came about after my time in grade school and high school, so I don't have any idea how it's normally presented. The wiki article here (https://en.wikipedia.org/wiki/Order_of_operations#Mnemonics) makes no mention of associativity of operations at the same precedence, but they do give some exceptions, further clouding the water.

Mixed division and multiplication
Similarly, there can be ambiguity in the use of the slash symbol / in expressions such as 1/2x

The article interprets this as (1/2)x.
Later in the same section, is this:

However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x.

Certainly $\frac 1 {2x}$ should be interpreted as division by 2x, and I can make a case for 1 ÷ 2x, but the symbol ÷ isn't on computer keyboards, so the symbol / gets recruited for this purpose, and we're back to ambiguity again.

It's just as bad with calculators.

An expression like 1/2x is interpreted as 1/(2x) by TI-82, but as (1/2)x by TI-83 and every other TI calculator released since 1996

The Windows calculator evaluates $1 + 2 \times 3$ in two different ways, depending on whether you're using it in Standard mode vs. Scientific mode. For the record, Standard mode evaluates this expression as 9, and Scientific mode evaluates it as 7.

My whole point is that the programming world has its act together as regards expression evaluation, but the math community, not so much.