Hi Orion,

Perhaps you will change your mind about using the Order of Operations as PEMDAS, after considering the word problem below, while remembering what a "term" is & that a single term (e.g. "2a") does not need parentheses encasing it to be understood as a single quantity (the product of its factors), and how to divide by a monomial:

In this scenario, I own & run a diner which serves a Breakfast Special from early morning opening until 11 AM. Today, at 10:58 AM, the last of the Breakfast Special customers left the restaurant. But then, at 10:59 AM, two separate groups of a dozen people each come into the diner, wanting to get the Breakfast Special. The 2 separate groups are each seated at their own table. Each member of the 2 groups, consisting of a dozen people apiece, wants eggs. I go into the restaurant kitchen to see how many eggs I have left. When I open the refrigerator, I see that there are 4 full cartons of a dozen eggs each, left on the shelf

Split evenly among the Breakfast Special customers in the diner, how many eggs does each customer get?

The proposition is a single quantity of eggs split evenly among a single quantity of customers:

4 dozen eggs divided by 2 dozen customers

4 dozen ÷ 2 dozen = ?

That same proposition can also be viewed as 4a÷2a when a=12.

In solving this problem, I'd appreciate it if you would please show each mathematical step you take, from beginning to end, so that it can be fully understood how you arrived at the correct answer. If you find more than one mathematical method of solving the monomial division expression in this word problem, please show all of the steps in each alternative.

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I look forward to seeing your methodology to find the correct answer to the word problem.

-- Dee

Thank you for commenting on what I wrote. It's not that I don't know how to solve the word problem -- I was illustrating a point about how to view juxtaposition.

Obviously, the answer to the question of how many eggs each diner customer gets is 2. The word "dozen" equals 12, so 12 is a factor of both of those terms, 4 dozen & 2 dozen, respectively. Thus, dividing 4 dozen by 2 dozen is written mathematically as:

4(12)/2(12)

...or as the top-and-bottom fraction of 4(12) over 2(12).

That is the same division expression as the monomial division expression of 4a/2a when a=12.

In no Basic Algebra textbook I have ever seen, are young math students instructed to ALWAYS encase a monomial inside parentheses in order to be understood to be a single combined quantity, as illustrated by 4 dozen eggs being understood to be a single combined quantity of 48 individual eggs (4 packs of 12 eggs each) & 2 dozen customers is understood to be a single combined quantity of 24 customers (2 groups of 12 customers each).

There is nothing confusing or ambiguous about the expression 6 ÷ 2(1+2), once one remembers the definition of a term (specifically, a monomial) which holds the value of the PRODUCT of its factors, the rules of the Distributive Law (which directly addresses juxtaposition), and how to divide by a monomial. It is also important to remember, as taught in 5th grade math: that Fraction=Division & therefore, Division=Fraction.

4 dozen divided by 2 dozen is precisely the same expression as the internet division meme from 2011:

48 ÷ 2(9+3)

Factoring out 48 makes the expression...

4(9+3) ÷ 2(9+3)

...or you could say it's 4 dozen divided by 2 dozen.

In any case, the quotient is 2, not 288 as the PENDAS crowd keeps insisting.

PEMDAS is taught in elementary school. In 9th grade Basic Algebra, however, a new set of rules is taught which supersede the PEMDAS construct. Included in the superseding rules are defining what a term is (specifically in this case, what a monomial is) & how to divide by a monomial.

The terms "4a" & "2A" are monomials. If you believe that those single terms need to always be encased in parentheses when part of a monomial division expression, please cite specific teaching websites which instruct young algebra students that this is REQUIRED. Here are some teaching sites which say the contrary:

from Mathway's calculator Algebra Examples: https://www.mathway.com/popular-problems/Algebra/1012959

" 2x÷2x

Rewrite the division as a fraction:

shown as 2x over 2x

Cancel the common factor of 2.

shown as x over x

Cancel the common factor of x.

1 "

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Lumen Learning Courses: https://courses.lumenlearning.com/uvu-introductoryalgebra/chapter/9-5-dividing-polynomials-by-a-monomial/

"Dividing Polynomials by a Monomial"

"When there are coefficients attached to the variables, we divide the coefficients and divide the variables.

EXAMPLE Find the quotient: 56x^5 ÷ 7x^2

Solution

Rewrite as a fraction

shown as 56x^5 over 7x^2

...Answer

56x^5 ÷ 7x^2 = 8x^3 "

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from Cue Math teaching website: https://www.cuemath.com/algebra/dividing-monomials/

"Practice Questions on Dividing Monomials"

Q.1. Divide. 15a^2b^3 ÷ 5b

Correct answer is shown as 3a^2b^2.

That can only be true if 15a^2b^3 is the entire numerator & 5b is the entire denominator of the top-and-bottom fraction, even though there are no parentheses anywhere in the original horizontally written division statement (using an obelus).

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from Greene Math teaching website: https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html

"Dividing a Polynomial by a Monomial"

See examples #1, #2 & #3: All are originally written as horizontal expressions using an obelus as the division symbol, which are all immediately rewritten as top-and-bottom fractions with the monomial denominator being the entire term to the right of the division sign -- with no parentheses around it.

In example #1, the denominator is 8x^2

In example #2, the denominator is 6x^2

In example #3, the denominator is also 6x^2

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...and there are plenty more examples all over the web of how young algebra students are taught to divide by a monomial, in which the original expression uses an obelus to represent division, and are then instructed to "Rewrite as a fraction," with the whole monomial term to the right of the obelus as the entire denominator. I can find no examples of young students being told to use only the monomial's numerical coefficient in the division before multiplying that quotient by the other factor or factors of the monomial.

If you honestly believe that to be incorrect, please show some examples from algebra teaching websites which specifically instruct young students NOT to consider a monomial term such as "4a" or "2a" (without parentheses completely encasing the entire monomial) as having a single value which is the PRODUCT of its factors.

I just showed what is currently being taught on math teaching websites, as to how to divide by a monomial. See Lumen Learning.com, Cue Math.com, Greene Math.com, BYJUS.com teaching website, etc., instructing students to "Rewrite as a fraction," when they are presented with a linearly written division expression using an obelus as the division symbol. That is because, remembering back to 5th grade math, students are taught that Fraction=Division (and therefore, Division=Fraction). On those teaching websites, there are a number of examples given to current Basic Algebra students, with the solutions, which show this clearly to be the case. Here are a few more examples of this, on math teaching websites to help current Basic Algebra students :

from Teachoo .com:

https://www.teachoo.com/9708/2961/Dividing-monomial-by-monomial/category/Dividing-two-monomials/

"Dividing Monomial by Monomial"

"Let's do some more examples

6x^3 ÷ 3x^2

... = 2x "

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from Cool Math .com: https://www.coolmath.com/algebra/05-division-of-polynomials/01-dividing-by-monomials-01

"Dividing by Monomials"

"Let's do one

(18x^4 -10x^2 + 6x^7) ÷ 2x^2

Let's rewrite it like this:

18x^4 -10x^2 + 6x^7

--------- [over] -------- =

2x^2

...

9x^2 - 5 + 3x^5

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from Mometrix Test Preparation: https://www.mometrix.com/academy/dividing-monomials/

See example #3 on the video:

8x^5y^3 ÷ x^2y

Note that no parentheses are used in that horizontal division expression.

The eventual answer is shown to be 8x^3y^2...which only works if 8x^5y^3 is the entire numerator & x^2y is the entire denominator of the top-and-bottom fraction.

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If you still need to see more of this from other math teaching websites, just say the word & I will provide additional examples of students being instructed to rewrite monomial division expressions as a top-and-bottom fraction, when it was, originally written horizontally with an obelus without parentheses around the divisor (denominator) monomial.

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In teaching your students how to 'process' a division expression such as a/b/c, yes, additional parentheses are definitely necessary, to indicate the order of the multiple divisions in the expression. When there is only one division symbol in an expression, however, it is clear that the dividend/numerator is everything to the left of the division symbol (slash or obelus) & the divisor/denominator is everything to the right of the division symbol. See above examples from teaching websites.

In no Basic Algebra textbook are students taught that a numerical coefficient of a term can be separated from its other factor or factors -- with or without parentheses surrounding the term. Quite the contrary -- students are taught that a monomial term such as "2a" holds a SINGLE VALUE, which is the PRODUCT of "2" & "a." When a=(2+2), the monomial term 2a equals 8.

As demonstrated on a myriad of math teaching websites showing students how to divide by a monomial, the value of a monomial does not change when a monomial such as "2a" is preceded by a division symbol (slash or obelus). The two in "2a" is not a separate, stand-alone quantity any more than the two in "2 dozen" is a stand-alone quantity. Two dozen equals 24, even when there's a division symbol immediately to the left of that term. Once again, see all those Basic Algebra teaching websites for confirmation of how to divide by a monomial.

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As for students using calculators, if they are taught the proper way to execute monomial division in the first place, then if they do not get the same result on a calculator, they know it is due to faulty programming. Some manual & online calculators do get it correctly, however. If they get what they understand is an erroneous answer from a calculator program, then they can go back & add more (unnecessary) parentheses.

With the proper knowledge & understanding of how to divide monomials, there is nothing confusing or ambiguous about an expression such as 6 ÷ 2(1+2). 8 ÷ 2(2+2), or 48 ÷ 2(9+3).

Here's how students reconcile poorly programmed calculators & the correct procedure to do monomial division:

LEARN TO EXECUTE MONOMIAL DIVISION PROPERLY, IN THE FIRST PLACE!

When using calculators, it might be a good idea to use additional parentheses for the calculator to interpret the expression correctly, in the event that a calculator programmed was by someone who missed that day in 5th grade math when it was taught that Division=Fraction (and vice versa) & also missed that day in 9th grade Basic Algebra when the teacher went over what a "term" is (specifically, a monomial) & how to divide by a monomial.

Again -- once it is understood what a monomial is & the correct procedure to divide by a monomial, there is nothing confusing or ambiguous about an expression such as 6 ÷ 2(1+2). 8 ÷ 2(2+2), or 48 ÷ 2(9+3). It's monomial division, with the entire term to the right of the division symbol being the denominator/divisor. And the numerator/dividend (the term to the left of the division symbol) can be factored, too. For example:

6 ÷ 2(1+2) =

6=2(1+2)

...making the expression...

2(1+2) ÷ 2(1+2) =

...which is the same statement as...

2a ÷ 2a

...when a=(1+2)...

...which can also correctly be written as the top-and-bottom fraction of 2a over 2a.

You say, "For what it's worth, I did not miss those days in class (granted, they may have been different years). Yes, division=fraction. But that alone does not tell a parser where to end a subexpression, one needs more rules for that." Actually, it does tell the parser exactly where the division occurs, when it comes to division by a single term!

I just showed you a boatload of examples from a number of online teaching sites, demonstrating that Division=Fraction. Here's more on that:

from Teaching Better Lesson.com Common Core:

https://teaching.betterlesson.com/browse/common_core/standard/272/ccss-math-content-5-nf-b-3-interpret-a-fraction-as-division-of-the-numerator-by-the-denominator-a-b-a-b-solve-word-problems-invo?from=standard_level1

"Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)."

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Australian Association of Mathematics Teachers: https://topdrawer.aamt.edu.au/Fractions/Big-ideas/Fractions-as-division

"Fractions as Division"

"Anyone who has studied secondary school mathematics would probably be comfortable with the convention of 'a over b' meaning 'a divided by b'."

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from ClubZ Tutoring:

https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-examples-6-7-5/#:~:text=Fraction%20Bar%3A%20The%20fraction%20bar,between%20the%20numerator%20and%20denominator

"FAQ Section

Q1: Can the fraction bar be replaced with the division symbol (/)?

A1: Yes, the fraction bar and the division symbol (/) are interchangeable and convey the same meaning in mathematical notation."

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from MathOnlyMath .com:

https://www.math-only-math.com/Fraction-as-Division.html#:~:text=Fraction%20as%20division%20is%20also%20known%20as%20fraction%20as%20quotient.&text=If%208%20biscuits%20are%20distributed,4%20%C3%B7%202%20%3D%202%20biscuits

"Fraction as Division"

"Fraction as division is also known as fraction as quotient."

"For examples the divisions can be expressed as fractions.

(i) 8 ÷ 2 = 8/2

(ii) 12 ÷ 4 = 12/4

(iii) 5 ÷ 3 = 5/3

(iv) 15 ÷ 5 = 15/5

(v) 11 ÷ 19 = 11/19 "

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from Third Space Learning:

https://thirdspacelearning.com/us/math-resources/topic-guides/number-and-quantity/fractions-as-division/#:~:text=Alternatively%2C%20you%20can%20also%20write,the%20denominator%20of%20the%20fraction

"Interpreting fractions as division is when you understand that a fraction represents a division operation between its numerator and denominator. In other words, when you have a fraction

a

--

b ,

you can interpret it as 'a divided by b,' or a ÷ b."

"Alternatively, you can also write a division equation as a fraction.

To do this, you would write the dividend of the equation as the numerator of a fraction and the divisor of the equation as the denominator of the fraction.

For example:

3 ÷ 4 =

3

4 "

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from Algebra Class .com:

https://www.algebra-class.com/dividing-monomials.html

"Dividing Monomials"

"Remember: A division bar and fraction bar are synonymous!"

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from Siyavula Technology Powered Learning:

https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressions-part-2/08-algebraic-expressions-part-2-02

Grade 8

Algebraic Expressions

"WORKED EXAMPLE 8.2

DIVIDING ALGEBRAIC MONOMIALS

Simplify the following expression:

24t^7 ÷ 4t^5

SOLUTION:

Step 1: Rewrite the division as a fraction

This question is written with a division symbol (÷), but this is the same as writing it as a fraction.

24t^7 ÷ 4t^5 =

24t^7

4t^5

...= 6t^2 "

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from SlideShare a Scribd Company:

https://www.slideshare.net/slideshow/94-16609182/16609182

"Dividing by a polynomial"

"Example 3

Divide a polynomial by a monomial

Divide 4x^3 + 8x^2 + 10x by 2x.

4x^3 + 8x^2 + 10x ÷ 2x =

Write as a fraction

4x^3 + 8x^2 + 10x

------------------------- =

2x

Simplify

2x^2 + 4x + 5 "

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from Toppr teaching website:

https://www.toppr.com/guides/maths-formulas/division-formula/

"Method of Division"

The number which is divided is the dividend. And the number in which the dividend is being divided is the divisor. The answer to a division problem is the quotient. Example signs for “a divided by b”:

a ÷ b &

a

--

b

So the Division Formula is

Dividend ÷ Divisor = Quotient OR

Dividend

------------- = Quotient

Divisor "

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As illustrated by the above examples from teaching websites, the vinculum (fraction bar), the solidus (slash) & the obelus (division sign) are all exactly the same because they all mean "divided by."

In a linearly written division expression using an obelus (division sign), when dividing by a monomial, the entire term to the right of the division sign is the denominator. There is nothing confusing about that at all.

A single term (monomial) cannot be ripped apart & only the coefficient used in a separate operation. because its value is the PRODUCT of its factors. Here's what's wrong with using only the coefficient in the division operation in a linearly written division expression:

Let's look at the simple monomial division expression of x divided by x. Given that x does not equal zero, that quotient is 1, whether the division expression is written with a vinculum (fraction bar), a solidus (slash) or an obelus (division sign).

Every Basic Algebra textbook says that a variable by itself actually has a numerical coefficient of 1, so x=1x. Writing that same division expression of "x divided by x," using its coefficient of 1...

1x

1x

...or it can be written as...

1x/1x

...or as...

1x÷1x

As such, in all cases, the quotient should always be 1.

Using the Order of Operations as PEMDAS, as you & some other prescribe (i.e. breaking apart a monomial & using only its coefficient as the divisor):

1x÷1x =

1*x = 1x

1x÷1 = 1x

1x*x = 1x^2

No, the non-zero monomial "x" (also correctly written as "1x") divided by itself is not "x squared."

"1x" is a single term (a monomial) with a single value which is the PRODUCT of its factors. It never needs parentheses around it to be understood as one term which holds a single value, any more than the number 46 does, even though 46 has implied multiplication (4 tens & 6 ones) and implied addition (4 tens + 6 ones). Therefore, a monomial cannot be broken apart & used in separate operations.

That also applies to monomials such as "4a" & "2a." Given that the variable "a" does not equal zero, 4a divided by 2a has a quotient of 2, regardless of whether the expression is written with a vinculum (fraction bar), a solidus (slash) or an obelus (division sign).

Here's an example of a calculator getting it right:

from Mathway: https://www.mathway.com/popular-problems/Algebra/1012959

"Enter a Problem"

" 2x ÷ 2x

Rewrite the division as a fraction

2x

2x

... 1 "

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And from Texas Instruments:

“Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators”

https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/11773#:~:text=Implied%20multiplication%20has%20a%20higher,as%20they%20would%20be%20written

"Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2X), while other products may evaluate the same expression as 1/2X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper."

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There are plenty more examples of calculators getting the proposition correct. But because some calculator programmers missed those days in school when it was taught that Division=Fraction & how to divide monomials, students can be instructed to use additional parentheses when entering monomial terms into a calculator. Students must be helped to understand that a calculator can get things wrong because the programmer did not have a strong understanding of the underlying concepts. -- "Garbage in, Garbage out."

No, not "most if not all" computer calculators treat 1x/1x the same as you keep insisting -- that multiplication & division have the same precedence from left to right & that the numerical coefficient can be detached from the other factor(s) of a monomial & used in a separate operation. I showed you the Texas Instruments write-up on the issue. Here's an example of how messed up calculators are & the reason that students need to be properly educated not to just accept a result from a calculator as if it came from God:

Wolfram Alpha gives two different answers to the same division expression:

MATH INPUT:

https://www.wolframalpha.com/input?i2d=true&i=6%C3%B72%5C%2840%291%2B2%5C%2841%29

6 ÷ 2(1+2) =

9

But...

NATURAL LANGUAGE INPUT:

https://www.wolframalpha.com/input?i2d=true&i=Divide+6+by+2%5C%2840%291%2B2%5C%2841%29

Divide 6 by 2(1+2) =

6

2(1+2)

one ~ ~ ~ ~ ~ ~ ~

That demonstrates that the programmer's understanding was poor, when it came to the underlying principles of Fraction=Division (so Division=Fraction) & how to properly execute monomial division. The programmer failed to account for the fact that, as taught in 5th grade math, since Fraction=Division (and vice versa) & the vinculum (fraction bar), the solidus (slash) & the obelus (division sign) all mean "divided by," and are thus interchangeable.

This is why I've just disciplined myself to use the Oxford comma rather than rely upon someone's particular justification for why sentences can be written unambiguously without it.

Was this reply intended for me?

No worries, Daniel. FWIW, I agree with your points. Personally, I think this thread ran its course as soon as it started (apparently 9 years ago!). Why would anyone get het up over something purely conventional like order of operations?

Look, the fundamental mathematics for the typical number systems you're talking about (the stuff we encounter in k-12 curriculum) defines multiplication and addition as binary operations. It does not define order of operations. It assumes that operations will be specified unambiguously. The semantics don't dictate a presentation, the burden is on the presentation to indicate the semantics. In today's world, parentheses are the presentation mechanism for indicating the grouping of these binary operations. Any particular convention for order of operations is simply an agreement within a community of people about which parentheses can be omitted. Different communities can come up with different conventions and communicate their intentions perfectly clearly. Your argument simply boils down to you justifying your preferred convention. And that's fine, absolutely and totally fine. And if you are working in a community that shares that preference, then all is good. But there is no deep significance to it (it has nothing to do with the math itself), and you cannot demand universal acceptance. Different communities will find different conventions useful. I'm sure that accountants, engineers, scientists, school teachers, professional mathematicians, software developers, etc, not to mention all of those communities at different times in history, all have different perspectives that could easily lead to different conventions, and the underlying mathematical semantics survive intact in every case. The sequential constraints of a mechanical adding machine would probably lead to different conventions than those arising in the presence of modern GUIs. Given that conventions are not rigorous mathematical concepts, it's not surprising that ambiguous cases arise, especially when communication crosses community boundaries. I think the practical thing to do in such cases is simply to include more parentheses. You seem to think the practical thing to do is force everyone to adopt a universal convention. Try telling all English speakers to settle on one spelling convention, one pronunciation convention, or one dictionary. It's just not a practical exercise. It's also completely unnecessary, as we all know how to disambiguate the expressions when needed. (Well, I'd like it to be completely unnecessary, but just these kinds of assumptions have led to serious software defects. I'd argue that's actually the result of being unwilling to accept the arbitrariness of conventions--being so self-referential that you can't even conceive that your assumptions aren't universal.)

I'm not disparaging your research. I'm not arguing with your findings. In fact, I'm willing to stipulate all of that. I guess I shouldn't have buried the lede, so here it is: the original question was stupid.

The original question was how should we evaluate

6 ÷ 2(1 + 2)

All of your research on order of operations is of no value to us if we don't know the provenance of the question. It may be a statistical fact that 99.99% of situations expect to the result to be 1. But it's pretty easy to conceive of variations. Mathematica evaluates that expression to 9. It would be perverse to barge into the Mathematica community and complain that Mathematica deviates from the other 99.99% of usage. The users of Mathematica can get along fine with their own convention.

So, for the original question, we're stuck with ignorance of its provenance. My answer to the question is "I don't know". If I had come up with that expression for some reason and wanted to communicate it to an audience, I personally would add parentheses first to disambiguate it. I wouldn't just assume that my audience shared my preference for order of operations or, perhaps of more relevance, that they would remember it correctly in the moment.

You could say that, given your research, the statistically most likely answer is 1. You could make a contingent statement: "assuming convention so-and-so, the answer is 1". What you can't say is that there are no other valid interpretations whatsoever (it seems that you are indeed saying that, so forgive me if that's not what you're saying).

There is no context-free method for determining a correct answer to the original question. You can't insist on retro-fitting your interpretation, no matter how statistically valid, onto the utterance without a method of validation.

I submit for your consideration this Dijkstra memo: https://www.cs.utexas.edu/~EWD/transcriptions/EWD13xx/EWD1300.html

Things to notice: Even though he's documenting his convention he recognizes the potential for ambiguity; he explains his personal conventions as a favor to his readers rather than arguing for some universal purity; he acknowledges limitations; he allows for deviations--clarity trumps consistency; he argues that optimization at the system ("macro") level is more important than optimization at the local ("micro") level; he completely recognizes that each piece of notation depends upon context (e.g. juxtaposition). Personally, I think this is the kind of teaching we should be doing rather than trying to enforce adherence to any one particular convention any type of notation.

Regarding syntax, by “provenance” I would take as meaning “intent of problem poser”. For example, 2a+b divided by 2a+b will equal 1 (extending slightly a recurring example). But that does not determine the result of evaluating the expression ‘2a+b/2a+b’. For that one requires an agreed-upon precedence and (possibly) association laws. More below.

Most of this is not relevant to later pedagogy, by the way. The obelus kind of disappears by the time the students hit calculus.

Back to the example above. I entered it on a bunch of online calculators, using slash or obelus, whichever was supported. From these I got four different results, which I will show with unambiguous grouping.

(2x+y)/(2x+y)

2x+(y/2)+y

2x+(y/(2x+y))

2x+(y/(2x))+y

I only saw that last on one calculator. Also I will note that most that give the third and some that give the first reformat the input in a way that makes obvious the interpretation.

I do not claim any to be wrong. It’s all in the precedence convention being used, not the underlying math.

Since I have a moment, I'll just belabor the point. If I were an engineer, and any of these arithmetic expressions under discussion showed up in the documents that I must work from, and if the consequence of getting it wrong was loss of life or millions of dollars, should I just push forward with certainty knowing that there is only one valid interpretation of the expression? Personally, I would not. I'd ask some clarifying questions. I would furthermore expect that in life-critical domains the convention would be to not rely on conventions for order of operations. The goal is not to be able to defend one's computation; the goal is to communicate effectively.

You said:

"I entered it on a bunch of online calculators, using slash or obelus, whichever was supported. From these I got four different results, which I will show with unambiguous grouping.

(2x+y)/(2x+y)

2x+(y/2)+y

2x+(y/(2x+y))

2x+(y/(2x))+y

I only saw that last on one calculator. Also I will note that most that give the third and some that give the first reformat the input in a way that makes obvious the interpretation."

You have just proven the point I was making all along -- that a number of calculator programs were not written in accordance with what has been taught in 5th grade & reiterated again in Basic Algebra class, which is that Fraction=Division & therefore Division=Fraction, along with the definition of what a "term" is (specifically, a monomial) & how to divide by a monomial.

The programming of a given calculator is only as good as the programmer's grasp of the underlying concepts. Some programmers obviously have a lack of understanding of the aforementioned concepts that Fraction=Division (and vice versa) & what a "term" is -- specifically a monomial, which, by definition, has a numerical coefficient which is inseparable from its factor(s).

Because some programmers had a poor understanding of those concepts, users of a given calculator may be instructed to place parentheses around each term in an expression, so that the calculator program registers the operations correctly, but that does NOT mean that it is strictly required in all cases when performing the division by a monomial, manually! See Texas Instruments:

“Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators”

https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/11773#:~:text=Implied%20multiplication%20has%20a%20higher,as%20they%20would%20be%20written

“Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written [manually]. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2X), while other products may evaluate the same expression as 1/2X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper."

Here is an online calculator which gives different answers to the exact same division proposition:

from Mathway online calculator:

https://www.mathway.com/Algebra

Whether the input is 2a/2a or 2a÷2a, the answer comes out as 1, shown as the top-and-bottom fraction 2a over 2a in both cases, even though the original input was in linear form using a slash or obelus.

However, when the input is 2(1+2)÷2(1+2), the answer shown is 9.

In the expression 2a÷2a when a=(1+2) or a=3, would you please explain how is that possible?

~ ~ ~ ~ ~ ~ ~

I previously supplied you with a myriad of solved examples from a variety of math teaching websites, instructional videos & textbooks from all over the world, which instruct Basic Algebra students to "Rewrite as a fraction," when presented with a division expression using an obelus, showing the monomial divisor/denominator as the entire term -- not just dividing by its coefficient.

What is taught in high school Algebra is the foundation for more complex math later on. Without understanding basic concepts such as what a "term" is & how to divide by a monomial, people will not know how to solve more complex expressions which they encounter down the road, as they advance.

Can you please link to some math textbooks or teaching websites which specifically instruct students to detach the numerical coefficient from a monomial such as "2a" or "4xyz" & use it in a separate division operation from the rest of the single term, showing several solved examples demonstrating that specific method of calculating "Division by a Monomial." to support that what you are saying is a common practice among educators & the global math community?

Daniel, do you suppose we're talking to an AI?

The only thing being "ignored" by you & others is:

Vinculum (fraction bar)=Slash (solidus)=Obelus (division sign)

That's because they all represent "divided by."

According to every math book ever written, the vinculum (fraction bar) is a grouping symbol. And since the vinculum, solidus & obelus are all synonymous with one another & thus interchangeable, all division symbols are grouping symbols.

If you believe that that's wrong, prove it with links to reputable teaching sites which expressly state that the obelus is somehow different to the solidus & vinculum, even though they all mean "divided by."

When it comes to conducting division manually (not entering the expression into a calculator), everyone with more than a 5th grade education would write the fraction twenty-four twelfths as the top-and-bottom fraction of 24 over 12 (using a fraction bar to separate the two terms).

And if someone was asked to write that same fraction of twenty-four twelfths on a single line, the person would write that as 24/12.

Everyone with more than a 5th grade education understands that the quotient in both cases is 2.

Isn't is true that 24/12 can also be written as 24÷12 because both the slash & the division sign mean "divided by"?

You say:

" 'Isn't is true that 24/12 can also be written as 24÷12 because both the slash & the division sign mean "divided by"?'

We've strayed way out of scope for this forum. The answer is yes but that has nothing to do with parsing arithmetic expressions which in turn only tangentially intersects Wolfram Community."

The title of this discussion thread here on the Wolfram Alpha forum is:

"Math breaking Facebook: 6 ÷ 2(1+2) = ?"

This subject is all about how the expression should be parsed.

6÷2(1+2) is 6 divided by 2(1+2), which is the precisely the same as 2a÷ 2a when a=(1+2).

You just acknowledged that twenty-four twelfths can be correctly written as the top-and-bottom fraction of 24 over 12 (with a fraction bar separating the two terms), or as 24/12, and that 24/12 is the same as 24÷12 -- all with a quotient of 2, since they all mean 24 divided by 12.

Using the same logic, then, it follows that the monomial division of 2a÷2a can be correctly written as 2a over 2a (with a fraction bar separating the two terms). When the variable "a" does not equal zero, 2a divided by itself has a quotient of 1, regardless of which division notation was used to write it.

The monomial division expression 2a÷2a is also correctly written as the fraction...

2a

2a

When a=(1+2), the expression becomes...

2a=2(1+2)=6

6/2(1=2)

6/6=1

There is nothing confusing or ambiguous there, once one understands that the division sign, the slash & the fraction bar are all one and the same, and that a monomial is one term with a singular value which is the product of its factors (i.e. the coefficient cannot be lopped off & used in a separate division operation, without its factors).

Education forms the foundation for more complex mathematics down the road. What gets taught in 5th grade & reiterated in Basic Algebra is used later on, as calculations become more complex. Failure by a programmer to understand that Fraction=Division (and therefore, Division=Fraction), what a monomial is (i.e. what it consists of & its singular value) & how to conduct division of a monomial means that the programming (and/or its language) will be faulty.

"Garbage In, Garbage Out!"

Pointing to computer programs as some kind of "proof" of how to conduct division by a monomial proves only that the creator of the programming language and/or the writer of the calculator program did not have a good grasp of the basic concepts necessary to correctly carry out those calculations.

Some online calculators do get it right, because input of the monomial division is automatically converted to a top-and-bottom fraction.

BYJUS teaching website:

https://byjus.com/dividing-monomials-calculator/

"In Algebra, a polynomial with a single term is known as a monomial. When a monomial is divided by a monomial, first divide the coefficients of the variable and then divide the variable when the variables are present in both the numerator and denominator. For example, assume two monomials, 50 xy and 5y. Now the monomial 50xy is divided by 5y, we will get

= 50xy/5y

= 10x

Thus, the quotient value obtained is 10x, which is the result of the division process."

~ ~ ~ ~ ~ ~

from Snapxam .com:

https://www.snapxam.com/calculators/polynomial-long-division-calculator

Input:

2(1+2)÷2(1+2)

Answer:

1

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Here's an online calculator that gives two different answers, depending on which division symbol is used:

https://www.symbolab.com/solver/polynomial-long-division-calculator

When input with an obelus, this calculator shows a result of 2a÷2a=1.

In that expression, a=(1+2)

When input with a slash as 2(1+2)/2(1+2), the expression is solved as the top-and-bottom fraction of 2(1+2) over 2(1+2), thus rendering a quotient of 1.

...but when inputting the same division expression with an obelus as 2(1+2)÷2(1+2), the answer comes out to 9.

~ ~ ~ ~ ~ ~ ~

Since every math textbook on the planet affirmatively specifies that the slash & the obelus are synonymous & are thus interchangeable (as well as every algebra textbook showing how to plug in the value of a variable), I'd appreciate it if you would please explain the difference in the method of calculation between using the slash vs. using the obelus in the expression 2(1+2) divided by 2(1+2), or 2a divided by 2a when a=(1+2).

I look forward to that complete contradiction being explained as something other than the programmer's lack of understanding of the basic concepts behind the acknowledged methodology, used worldwide, to execute division by a monomial.

You say:

"The language committees know what they're doing; that has nothing to do with any particular programmer's understanding. The programmers who write the language compilers know how to parse to meet the design specs (I can say this with assurance, having been one myself prior to math grad school)."

It appears that it's not a "collective decision," because some calculator programs do get it right!

However, some language writers & some programmers did not absorb the concept in every math textbook on the planet, that...

Vinculum (fraction bar)=Solidus (slash)=Obelus (division sign)

...and also failed to fully comprehend how to plug in the value of a variable into a monomial such as "2a."

Given that "a" does not equal zero, the monomial "2a" divided by itself will always equal 1, regardless of the division notation used.

When a=(1+2), then 2a=6.

So 2(1+2) divided by 2(1+2) has a quotient of 1 -- because 6 divided by 6 equals 1, no matter whether it's written as a top-and-bottom fraction, with a slash, or with a division sign. All division symbols are synonymous with one another & are thus interchangeable -- they all mean "divided by" & they all function as a grouping symbol.

In answer to your statement...

" Expressions encountered might be the same but the parsing/grouping claimed for algebra class is different from what's done by language compilers and interpreters. If that's a fundamental obstacle for you, avoid the programming languages."

Because some computer language writers & calculator programmers apparently only have a limited grasp of the underlying concepts of monomial division (and that all division symbols are synonymous with one another & thus interchangeable), when entering an expression into a calculator, it is best to encase a monomial divisor (denominator) in parentheses, so the machine will not misconstrue the expression's meaning.

With that said, however, that does not change the meaning & value of the original monomial expression, regardless of which division symbol was used.

You say:

"Okay, so you're not only saying that there is only one acceptable convention for order of operations for use in general conversation, but you're adding that no community is free to choose for its own use within its own domain a different convention that they believe works better for them in their domain."

What I'm saying is that this particular expression of 6 ÷ 2(1+2) is a simple monomial division expression.(i.e. division by a monomial). A monomial has a specific definition, which I have linked to here, from various websites. I encourage you to go look it up yourself if you don't believe me.

I have also provided links to numerous websites demonstrating the way that division by a monomial is being taught today -- which is the same way I learned it decades ago. A monomial such as "2a" is one entity with a single value which is the PRODUCT of its factors. By all means, go look that up too, if you have any doubts about my interpretation of what a monomial consists of & how to divide by a monomial such as "2a."

You seem stuck on the idea that computer languages & some calculators force users to encase a monomial within parentheses, to insure that the program interprets the monomial correctly (i.e. as a single combined entity, which holds the value of the product of a coefficient & its factor or factors). A computer program is only as good as its writer's understanding of the propositions involved.

A programming language and/or calculator program has flaws in it, if it gives different answers to the same division proposition, depending on what division symbol is used.

Example from Wolfram Alpha's calculator program:

https://www.wolframalpha.com/input?i=divide+2a+by+2a

Natural language input:

2a divided by 2a = a^2

Natural language input:

Divide 2a by 2a

2a

_____ = 1

2a

Same division proposition -- different answers. Please explain the discrepancy..

I have sent feedback about the discrepancy in the Natural Language input that gives two different answers. What I got was, "We're looking into it." It was obviously never addressed.

The "shortcoming" is that there is a flaw in the program, since there should be no difference in understanding the identical division proposition of dividing 2a by itself, no matter what phraseology is used.

On Wolfram Alpha's calculator's "Math Input," the slash & obelus are not included in their symbols array (on a purple background underneath the entry bar). The only division symbol provided is a "blank" fraction set-up, on the far left of those operations choices. When I used that fraction symbol to conduct the division, I typed in 6 as the numerator & 2(1+2) as the denominator & got "one" as the result. The only time Wolfram Alpha gives a different result is if you input a division symbol that is not provided by their own calculator system. That's another flaw in the program, since all division symbols are synonymous with one another & are therefore interchangeable.

Here's teaching site BYJUS's "Dividing Monomials Calculator," which provides input as numerator & denominator:

https://byjus.com/dividing-monomials-calculator/

...and below that, an explanation & example...

"What is Meant by Dividing Monomials?"

"In Algebra, a polynomial with a single term is known as a monomial. When a monomial is divided by a monomial, first divide the coefficients of the variable and then divide the variable when the variables are present in both the numerator and denominator. For example, assume two monomials, 50 xy and 5y. Now the monomial 50xy is divided by 5y, we will get

= 50xy/5y

= 10x

Thus, the quotient value obtained is 10x, which is the result of the division process."

Note that the monomial division example is written with a slash (the same as the obelus -- meaning "divided by") & no parentheses around "5y," which is regarded as the full denominator in their explanatory example. The solution was not 50xy divided by the coefficient 5 & then multiply that quotient by "xy." The correct solution was shown as the numerator 50xy divided by the denominator 5y, which equals 10x.

Using that method demonstrated for students to learn how to divide monomials, given that "a" does not equal zero...

2a/2a=

2a

2a

...which equals 1.

When a=(1+2), 2a/2a is...

2(1+2) / 2(1+2)

2(1+2)= 6

6/2(1+2)

6/6 = 1

Every math textbook & teaching website supports that parsing of the expression 6/2(1+2) because that is the acknowledged method to divide by a monomial -- in this case, "2a" when a=(1+2) & "6" can also be factored out as 2(1+2) or "2a" when a=(1+2).

For my edification, can you help me identify the monomials in the following expression?

x^2 + 7 + 3x^2 + x + 16 + 5x

And just for clarity, in the original

6 ÷ 2(1+3)

what are the monomials?

For all of these monomials you've found, I'd also like to know their degree.

[And just to save some time... Why can't I say that in the first case the monomials are 4x^2, 6x, and 23? In the second, why can't I say that there is only one monomial of degree 0? The only reason I ask for degree, is that in the second case what justifies extracting out sub-monomials from a single monomial for no apparent computational reason? Why can't I say that 4x^2 is 8x^2 ÷ 2x * (2x - x)? And having said that, why can't I say that "" and juxtaposition are synonyms for "multiply" and write it as 8x^2 ÷ 2x(2x - x)? Thus, 4x^2 can be rewritten as 4. Or, since "" and juxtaposition are synonyms for "multiply", maybe it's 8x^2 ÷ 2 * x * (2x - x) and therefore 4x^2 is equivalent to 4x^4.]

Ah, that's the explanation! LOL.

Note where it says: "Implied multiplication has a higher priority than explicit multiplication"

Note also where it says "...on TI graphing calculators"

Also, it's curious that you are willing to admit that the synonymous syntax of implicit and explicit multiplication can have different precedence, but that the synonymous syntax of ÷ and / must have the same precedence.

Not that it matters, because I'm still pretty sure you're a bot.

"but the fact remains that this discussion thread's subject is about how to parse the expression 6 ÷ 2(1+2)."

The thread opened a discussion on the various ways that expression might be parsed. We probably should have locked the thread years ago. I may recommend we do that now, after removing all the posts from the past couple of weeks-- they only serve as a noisy distraction.

"Here's something pertinent to that subject from the Texas Instruments site:"

Note as ever that TI produces calculators, not computer languages. They have their own requirements and preferred parsing, and for good reason. Indeed, they made a clear choice to allow two different precedences, so as to meet expectations of two different user communities (education and scientific uses). The languages all went in a different direction in this regard. Other calculators make various choices and I will guess they each base their decision on what their user base is likely to expect and/or prefer. As has been pointed out, these choices are neither right nor wrong, they are simply different conventions for handling arithmetic expressions. Not all users share the wants of algebra teachers.

"Probably back in the 1970's or 1980's, a small handful of computer programmers wrote the first calculator programs."

The language standards for Algol, Cobol and Fortran all date to the late 50's-early 60's. By the 80's these conventions were the norm if not universal (certainly they were in every language I used or implemented during the span 81-85, when I worked on compiler construction). I am not aware of anyone back then thinking "we need to adhere to this rule because it's what every other language has" although by now this might well be the case. I suspect certain consistencies in expression handling were more the desired goal; see next paragraph.

I do not know offhand why the particular set in current use was preferred. Taking a guess, it was so that (i) addition and subtraction could have equal precedence, (ii) likewise multiplication and division could have equal precedence, and (iii) both could associate from the left. To give an idea of what I mean, transporting your insistent (and frankly incessant) "never break up a monomial" rule to addition operators, a-b+c would be parsed as a-(b+c) just as you have a/b*c parsed as a/(b*c). That is to say, when the first infix operator is subtract respectively divide and the second is plus respectively times, then associate from the right instead of the left. This makes for troublesome parsing when one is confronted with a chain of such operators. Now it could have been resolved in the way you seem to think absolutely necessary, by simply giving divide operators lower precedence than multiplication. Again, for whatever reasons, this was not the path taken.

You have spent a lot of time lecturing about parsing absolute requirements to people who have done that for a living, and lecturing on math to people who do that for a living (and take note: none have lectured you on teaching algebra). And you've displayed essentially no understanding of the former topic, and none of the latter beyond algebra 1. And no clear indication that you can distinguish between mathematical conventions and mathematical concepts. Some of this is possibly fine for a math education forum, but that's not what this is. As I noted at the top, this just makes for noise. You can make more of it if you really like, but be aware that, absent some new and sensible direction. the most likely effect of repetition will be to have these notes go away.

Some claim the equation is ambiguous, but the notation is common. It is understood that we apply the Distributive Law in the Parentheses step of PEMDAS. 2(1+2) = (2x1) + (2x2) = 6.

The only number you can divide 6 by to arrive at 9 is 2/3.

2(1+2) != 2/3

2(1+2) == 6

The left to right solvers, by replacing implied multiplication with implicit multiplication, are flipping the parenthetical expression to the inverse.

2÷(1+2) == 2/3

Therefore:

6÷2(1+2) != 6÷2*(1+2)

The Desmos online calculator will return the correct answer when using the keypad.

The Casio fx9860GII SD returns the correct answer to the equation as given 6÷2(1+2) and provides the alternate notation if you were expecting a result of 9.

Yes, without further hints it is not good notation, you need to know the rules of the game. It is like using parenthesis like

 [3*(2+3]*5)

we don't know how to deal with that without some extra rules...

and similarly things like a^b^c could be (a^b)^c or a^(b^c), so always add parenthesis when there is no clear answer. You can avoid part of it by always doing multiplication first and then division:

a/b c (to be interpreted as (a/b)*c)

should be written as : c*a/b, so it doesn't matter what the order is...

Yes it comes down to whether a space in front of parenthesis in an equation has any significance when parsing a formula. Mathematica doesn't care and even inserts a space to make the formula appear nice. Of course in Mathematice as in many other notations, the multiplication sign can be replaced by a space like 4 5=45 obviously leaving out the space there would be the number 45. with parenthesis the space might not be necessary 2(1+2) = 2 (1+2)=2(1+2), I have not seen the interpretation 2(1+2)=(2*(1+2)), but it is all a matter of agreed convention.

In[1]:= 6 / 2 (1 + 2)

Out[1]= 9

When in doubt, the best way to insure clarity one can write out the formula like this:

In[2]:= 6 / 2*(1 + 2)

Out[2]= 9

In[3]:= 6 / (2*(1 + 2))

Out[3]= 1