In the English language, words are rarely “inflected” like they are in other languages. The most we do with our regular verbs, for example, is to add an -s in the third person singular, -ed in past tense or past participles, and -ing in gerunds and present participles or present continuous. Occasionally we need to add helping verbs to indicate more nuanced uses, like a form of the “to be” verb for passive voice or a form of “have” for perfective forms of the verb.
In the classical languages like Greek and Latin, such inflections are typically built in to a singular word form. For example, here are the various first-person indicative mood forms of the Latin verb porto with all tense and voice combinations:
| Tense | Active Voice | Passive Voice |
| Present: | porto (I carry) | portor (I am carried) |
| Future: | portabo (I will carry) | portabor (I will be carried) |
| Pluperfect: | portaveram (I had carried) | portatus eram (I had been carried) |
| Imperfect: | portabam (I was carrying) | portabar (I was being carried) |
| Perfect: | portavi (I have carried) | portatus sum (I have been carried) |
| Future Perfect: | portavero (I will have carried) | portatus ero (I will have been carried) |
Source: Copilot request for first-person Latin verb forms.
Notice that in some cases, one word form in Latin requires up to four English words to translate depending on the ending. Greek is the same way, except inflecting verbs involves not just endings, but could also involve prefixes, infixes, and initial reduplications along with the loss of aspiration for aspirated consonants. The words are intended to be one singular form in most cases, and when we translate them, we treat the original word as a single lexemic unit.
We have a similar “inflection” of mathematical values or “terms” that do not have an extant operational sign (+ – × ÷) or that have such signs enclosed within or under a grouping symbol. These values or terms are (or should be) treated as a unit. Just as we don’t separate the root from the ending of a word in Latin and add the root to the end of the preceding word, neither do we (or should we) separate numbers that are “inflected,” so to speak, to represent a certain method of calculation that should be given priority over signed operations. The latter I am calling “implicit constructions,” because they imply a certain way of calculating based on syntax (position and orientation) that reflects a form of grouping and should be treated as a priority element of any expression.
One of the main functions of this kind of inflection is to indicate formulas for common measurements and values. The circumference of a circle is 2πr; the area of a circle is πr2; velocity is distance over time, v = d/t; etc. The juxtaposition of elements of the formula indicates that you’re working with what should be considered a unified value. This juxtaposition carries over into general mathematics as well. Juxtaposition is a form of grouping, but juxtaposition, like the inflection of the verb, is not a monolithic concept of simply being “side by side” with something else. Implicit constructions in mathematics use juxtaposition combined with orientation of the elements (including grouping symbols when necessary) to show the operational relationship of the individual elements.
At the very basic level, any real number, at least in the Indo-Arabic paradigm[1], is a simple juxtaposition of the powers of the base. A whole number (base 10) is, from right to left, a representation of how many 10n (n >= 0) in order right to left with implied addition of the place value (i.e., 10n) multiplied by the number holding the place. If you remember back to basic grade school math, that is how children are taught to understand our number system (ones’ place, tens’ place, hundreds’ place, etc.). In other words, juxtaposition is a foundational element of our number system and should not simply be discarded or replaced when it is used in other ways.
A fraction is a value that is formed by vertical or offset juxtaposition of one number to another with an intervening grouping symbol (vinculum or fraction bar; solidus or slash, respectively) that represents division or ratio but does not necessarily demand that such division be immediately carried out. I give examples of these in this implicit constructions table. Fractions should be considered as a single value first and foremost (because they are grouped) and NOT as a division problem in which the vinculum or solidus is considered equivalent to the obelus (÷) in function and implication. Rational fractions are necessary for precision, especially when the conversion to a decimal involves a nonterminating decimal and you’re working with very large numbers that would not be as precise as needed if we used a two-decimal approximation. For purposes of this paper, I’m working on the assumption that fractions should not be converted to decimal equivalent since this essay is theoretical.
A mixed number is the juxtaposition of a whole number to a fraction, and like a whole number, the side-by-side juxtaposition without any other symbol implies addition. Mixed numbers are considered unique, inseparable values as well when it comes to operations performed on them, but they often must be converted to an improper fraction to work with them more effectively.
Exponents also use juxtaposition to imply their operation. A power is superscripted and juxtaposed to the right of the base, and such superscription implies that the base should be multiplied the number of times indicated by the power. Of course, there are special rules when the power is a fraction, but those aren’t relevant to the discussion as the point is the implication of juxtaposition.
Since division is the opposite of multiplication, the question arises as to what is the opposite of a grouped fraction that implies multiplication? I would call this a “collection” because such a construction implies n sets of a quantity A. By way of example, 2(2 + 2) or 6(1 + 2) are collections if A = (2 + 2) or A = (1 + 2). The 2 (number of sets) is juxtaposed to (2 + 2) or (1 + 2) with the parentheses establishing the boundaries of quantity A.
Here is where many make a critical mistake in interpreting the collection: they fail to recognize the otherwise universal application of juxtaposition in real numbers, fractions, mixed numbers, and exponents that demands those forms be treated as single values. By suggesting one can just willy-nilly replace the juxtapositional relationship with a multiplication symbol and “ungroup” the collection, one violates the sacred bond that juxtaposition has in all other basic number forms. Just as fractions should be treated as a single value, so should collections.
By now you should see how this applies to the viral math expressions that some use to troll the Internet and pounce on the unsuspecting with their flawed view of Order of Operations or PEMDAS. An expression like 8 ÷ 2(2 + 2) should NOT require the undoing of a juxtaposed construction. The juxtaposition demands the “collection” 2(2 + 2) be treated as having a single value and should not be undone by an obelus that does not have grouping (or ungrouping) powers. Otherwise, an expression like 8 ÷ ¾ would be seen as 8 ÷ 3 ÷ 4 rather than inverting the fraction and multiplying (the latter being the official way students are taught to work the problem) if that same principle were applied. The expression 8 ÷ 2(2 + 2) = 8 ÷ 8 = 1, pure and simple. Yet some are so locked into a false concept of Order of Operations that fails to recognize the power of juxtaposition as a form of grouping that they can’t see the forest for the trees.
I have written elsewhere about the other issues that arise with the flawed understanding of juxtaposition. I believe this treatise provides a firmer theoretical basis for the concept of grouping and its proper place in the order of operations than the juvenile charts one finds in many textbooks.
Answering an Objection
One of the objections I often hear about juxtapositional grouping is that it violates the place of the exponent in such an order. That objection, however, is based on a flawed understanding of the role of juxtaposition. One example cited is:
8 ÷ 2(2 + 2)2.
Opponents say my theory would interpret the expression as follows:
8 ÷ 2(4)2 becomes 8 ÷ 82 (i.e., multiplication first, then exponent), which becomes 8 ÷ 64 or ⅛.
That is not accurate. If you understand that the exponent is also a form of juxtaposed grouping, then there is no issue with performing the exponent operation first and then performing the implicit multiplication, which complies with a standard view of the Order of Operations. Juxtaposed grouping does NOT supersede regular Order of Operations. The problem correctly interpreted then would be
8 ÷ 2(4)2 becomes 8 ÷ 2(16) (exponent first, then multiplication), which becomes 8 ÷ 32 or ¼.
To date, no one has ever been able to explain why implicit multiplication is treated differently from other implicit constructions in all my discussions and debates.
The bottom line here is that the juxtaposition itself is grouping, not just the symbols used. A fraction is a combination of the vertical juxtaposition of the numerator with a fraction symbol (vinculum, solidus, slash) grouping the denominator below it. A collection is the horizontal position of a cofactor or coefficient with the other cofactor or coefficient grouped within the parentheses. The concept of dealing with what is in the parentheses first without factoring in juxtaposition of other elements ignores the reality of juxtaposition in every other type of implicit construction and represents an imperfect and immature application of the grouping principle.
[1] Languages that use letters to represent numbers often do not have a letter to represent the value “0” and are not based solely on a base with powers. In Hebrew and Greek, for example, the respective alphabets represent 1–9, 10–90 (by 10), and 100–900 (by 100) and then repeat for the next three powers of ten using special symbols above the letters. You need 27 symbols/letters for these values instead of 10, and for numbers >=1,000, you need special symbols over (usually) the letters to indicate multiplication by 1,000 for each of the 27 symbols. If there is no value for one of the powers of 10, they simply do not use a letter to represent that.
Sunday Morning Greek Blog 
While I concur with the bulk of your analysis, I do take issue with this statement:
“Fractions should be considered as a single value first and foremost (because they are grouped) and NOT as a division problem in which the vinculum or solidus is considered equivalent to the obelus (÷) in function and implication.”
That is contrary to what is being taught worldwide (in 5th grade), which is that Fraction=Division & therefore, Division=Fraction.
from Third Space Learning tutoring website:
https://thirdspacelearning.com/us/math-resources/topic-guides/number-and-quantity/fractions-as-division/#:~:text=Interpreting%20fractions%20as%20division%20is,a%20%C3%B7%20b%20.
“Interpret Fractions as Division
Here you will learn about interpreting fractions as division, including understanding a fraction as a division equation, understanding a division equation as a fraction, and solving word problems involving understanding a fraction as division.
Students will first learn about interpreting fractions as division as part of number and operations–fractions in 5th grade.”
“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.”
~ ~ ~ ~ ~ ~
This is the same thing that the Australian Association of Mathematics Teachers says:
https://topdrawer.aamt.edu.au/Fractions/Big-ideas/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‘.”
~ ~ ~ ~ ~ ~
Here are some solved examples from a math tutoring website which illustrate the point:
from Greene Math .com:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html#:~:text=We%20should%20get%20our%20dividend%20(numerator)%20back:%208%20x,parts%20to%20report%20our%20answer:
“Dividing a polynomial by a monomial”
“Example 1
(4x^4 + 2x^3 + 32x^2) ÷ 8x^2
Step 1) Let’s set up the division problem using a fraction
4x^4 + 2x^3 + 32x^2
————————-
8x^2 ”
Solved examples 2 & 3 show the same procedure, instructing students to “set up the division problem using a fraction,” with the numerator being what is to the left of the obelus as the numerator & the entire monomial term to the right of the obelus as the complete denominator. The subsequent “Skills Check” quiz examples also result in the correct answer, if one converts the division statement originally written with an obelus into a top-and-bottom fraction, using what is to the left of the obelus as the numerator & the whole monomial term (sans parentheses) to the right of the obelus as the entire denominator.
~ ~ ~ ~ ~ ~ ~ ~
As for your statement, “The bottom line here is that the juxtaposition itself is grouping,” there is a plethora of evidence in algebra instruction that that is 100% true. See above solved examples, which show a monomial divisor/denominator with no parentheses, considered as a single quantity. By virtue of the fact that in the above solved example, the term “8x^2” has a single value as the denominator of the rewritten fraction (from the horizontal expression originally written with an obelus), proving once & for all that the numerical coefficient alone is NOT used in the division before multiplying by the variable!
Additionally, the Distributive Law EXPANDS PARENTHESES, so it is part of the “Parentheses” step of the Order of Operations as PEMDAS.
~ ~ ~ ~ ~ ~
I did a Google search for the following:
Is the distributive law part of the parentheses step in the order of operations, as in 2(a+b)÷2(a+b)?
This was the AI answer:
What you’re solving for
Whether the distributive law is part of the parentheses step in the order of operations.
What’s given in the problem
Expression: 2(a+b)÷2(a+b)
Helpful information
Order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Distributive law: x(y+z)=xy+xz
How to solve
Apply the distributive law within the parentheses if necessary, then follow the order of operations.
Apply the distributive law to both terms:
2(a+b)=2a+2b
Rewrite the expression:
(2a+2b)÷(2a+2b)
Divide the expression by itself:
the fraction with numerator 2 a plus 2 b and denominator 2 a plus 2 b
Solution
The expression simplifies to 1.”
Note where it says, “Apply the distributive law within the parentheses if necessary, then follow the order of operations.” Case closed.
Comment by Dee R. — February 24, 2025 @ 1:44 pm |
Hi, Dee. Thank you for responding. I understand that a fraction by itself is essentially a “potential” division problem, but as I’ve said all along, rational fractions that do not convert into a terminating decimal equivalent are preferable for the sake of precision. The precision bit isn’t necessarily a 5th grade concept. My concern is not about stand-alone fractions, though. My concern is about how fractions are treated in more complex expressions. I think the article you posted might confuse the matter for someone. If you say a fraction is just a division problem, then what do you do about an expression like 6 ÷ ¾? If all a student had to go on was the article you posted, they might think such an expression could simply be converted to 6 ÷ 3 ÷ 4 if all we’re doing is replacing the fraction bar with the obelus and ignoring the grouping, which we know would be incorrect. My point is that the vertical juxtaposition of the 3 to the 4 with the intervening fraction symbol REQUIRES the fraction to be treated as a monomial, and thus it has an imaginary set of parentheses around it. Thus it would properly be interpreted as 6 ÷ (3 ÷ 4). The juxtapositional grouping is what is key; otherwise, the ONLY way to group a division expression using an obelus is to place the expression in parentheses.
You would also need to look at https://thirdspacelearning.com/us/math-resources/topic-guides/number-and-quantity/dividing-fractions/ to understand that dividing by a fraction means you invert and multiply the fraction in the divisor, so the expression would be 6 x 4 ÷ 3. That’s a different looking expression from, but equal to 6 ÷ (3 ÷ 4).
Keep in mind that I’m developing a new paradigm here, so the standard or conventional way of looking at an expression may ignore the broader distinctions I am making about implicit constructions.
Scott
Comment by Scott Stocking — February 24, 2025 @ 9:10 pm |
Hi Scott —
You say:
“If all a student had to go on was the article you posted, they might think such an expression could simply be converted to 6 ÷ 3 ÷ 4”
That’s correct!
from Medium .com:
“Seriously, It’s Just Division”
https://medium.com/cw-math/seriously-its-just-division-58bac796f1a1
“I teach Algebra II. Fractions don’t exist.
I’m not saying, of course, that 1/2 and 5/31 aren’t things that might occur. I mean that I encourage students to stop obsessing on “fractions” as an isolated concept.”
“In “Elements of Arithmetic” (1893), William J. Milne writes, ‘(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.’ ”
“Most importantly, from a mathematician’s perspective, the “fraction bar” notation simply represents division.
3
___ = 3/4 = 3÷4
4
There are certainly cases where we want to think in terms of fractions, but from the perspective of notation, there’s no difference between “fractions” and “division”. It’s just a notation.”
“I suspect (without explicit confirmation) that both the obelus and the colon were chosen specifically because they suggest the fractional notation: The two dots represent the numerator and the denominator. In the case of the obelus, we also write the fraction bar itself. Cajori even reports at least one occurrence of ·/· (compare to 3/4).”
~ ~ ~ ~ ~ ~ ~
The expression 6 ÷ ¾ can be written as you have shown it with two obeluses [6 ÷ 3 ÷ 4], or with two slashes, thusly:
6/3/4
As a result, as in any expression with more than one division symbol, the author needs to put in at least one set of parentheses to clarify the order of the divisions. In the case of 6 ÷ ¾, if the author wants to indicate that 6 should be divided by three-quarters, it should be written thusly, to avoid confusion:
6 ÷ (3/4) or 6/(3/4)
That way, no one will perform the division as 6÷3 & then divide the quotient of 2 by 4 & arrive at the erroneous answer.
~ ~ ~ ~ ~ ~ ~
from Club Z Tutoring:
https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-examples-6-7-5/#:~:text=A1%3A%20Yes%2C%20the%20fraction%20bar,same%20meaning%20in%20mathematical%20notation.
“The Role of the Fraction Bar in Fraction Notation
The fraction bar serves as a visual representation of the division operation between the numerator and the denominator. It shows that the numerator is divided by the denominator, indicating the fraction’s value. For instance, the fraction 3/4 is interpreted as ‘3 divided by 4’ ”
“FAQ
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 notatiion.”
. * . * . * . * . * . *
You say:
“…the ONLY way to group a division expression using an obelus is to place the expression in parentheses.”
That is not correct. Only if there is more than one division symbol in the expression, must at least one set of parentheses be used to indicate the order in which the multiple divisions are to be done.
The real grouping is juxtaposition, because the Distributive Law MANDATES that the parentheses be EXPANDED. Thus, juxtaposition (implied multiplication) is PART OF the “Parentheses” step in The Order of Operations as PEMDAS.
— Dee
Comment by Dee R. — February 25, 2025 @ 9:38 am
Hi Scott —
You say:
“If all a student had to go on was the article you posted, they might think such an expression could simply be converted to
6 ÷ 3 ÷ 4″
That’s correct!
from Medium .com:
“Seriously, It’s Just Division“
https://medium.com/cw-math/seriously-its-just-division-58bac796f1a1
“I teach Algebra II. Fractions don’t exist.
I’m not saying, of course, that 1/2 and 5/31 aren’t things that might occur. I mean that I encourage students to stop obsessing on “fractions” as an isolated concept.”
“In “Elements of Arithmetic” (1893), William J. Milne writes, ‘(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.’ “
“Most importantly, from a mathematician’s perspective, the “fraction bar” notation simply represents division.
3
___ = 3/4 = 3÷4
4
There are certainly cases where we want to think in terms of fractions, but from the perspective of notation, there’s no difference between “fractions” and “division”. It’s just a notation.”
“I suspect (without explicit confirmation) that both the obelus and the colon were chosen specifically because they suggest the fractional notation: The two dots represent the numerator and the denominator. In the case of the obelus, we also write the fraction bar itself. Cajori even reports at least one occurrence of ·/· (compare to 3/4).”
~ ~ ~ ~ ~ ~ ~
The expression 6 ÷ ¾ can be written as you have shown it with two obeluses [6÷3÷4], or with two slashes, thusly:
6/3/4
As a result, as in any expression with more than one division symbol, the author needs to put in at least one set of parentheses to clarify the order of the divisions. In the case of 6 ÷ ¾, if the author wants to indicate that 6 should be divided by three-quarters, it should be written thusly, to avoid any confusion:
6 ÷ (3/4) or 6/(3/4)
That way, no one will perform the division as 6÷3 & then divide that quotient of 2 by 4 & arrive at an erroneous answer.
~ ~ ~ ~ ~ ~ ~
from Club Z Tutoring:
https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-examples-6-7-5/#:~:text=A1%3A%20Yes%2C%20the%20fraction%20bar,same%20meaning%20in%20mathematical%20notation.
“The Role of the Fraction Bar in Fraction Notation
The fraction bar serves as a visual representation of the division operation between the numerator and the denominator. It shows that the numerator is divided by the denominator, indicating the fraction’s value. For instance, the fraction 3/4 is interpreted as ‘3 divided by 4’ “
“FAQ
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.”
. * . * . * . * . * . *
You say:
“…the ONLY way to group a division expression using an obelus is to place the expression in parentheses.”
That is not correct. Only if there is more than one division symbol in the expression, must at least one set of parentheses be used to indicate the order in which the multiple divisions are to be done.
With only one division symbol, however, the expression can be rewritten as a top-and-bottom fraction (using the position of the division sign or slash as the fraction bar, with the numerator as everything to the left & the denominator as everything to the right) & divided accordingly.
The real grouping is juxtaposition, because the Distributive Law MANDATES that the parentheses be EXPANDED. Thus, juxtaposition (implied multiplication) is PART OF the “Parentheses” step in The Order of Operations as PEMDAS.
Additionally, because Fraction=Division (and therefore, Division=Fraction), The Order of Operations needs to be revised to reflect the correct procedure to calculate the value of a fraction: First, do all of the operations in the numerator, then do all of the operations in the denominator, and then finally, divide the numerator by the denominator.
DIVISION MUST GO LAST!
Therefore, The Order of Operations should be…
Parentheses/Juxtaposition
Exponents
Multiplication (explicit)
Addition/Subtraction
Division
…which can be remembered as:
“Please Just Excuse My Aunt Sally Dee.”
–Dee
Comment by Dee R. — February 25, 2025 @ 4:32 pm
Hi Scott —
You say:
“If all a student had to go on was the article you posted, they might think such an expression could simply be converted to
6 ÷ 3 ÷ 4″
That’s correct!
from Medium .com:
“Seriously, It’s Just Division“
https://medium.com/cw-math/seriously-its-just-division-58bac796f1a1
“I teach Algebra II. Fractions don’t exist.
I’m not saying, of course, that 1/2 and 5/31 aren’t things that might occur. I mean that I encourage students to stop obsessing on “fractions” as an isolated concept.”
“In “Elements of Arithmetic” (1893), William J. Milne writes, ‘(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.’ “
“Most importantly, from a mathematician’s perspective, the “fraction bar” notation simply represents division.
3
___ = 3/4 = 3÷4
4
There are certainly cases where we want to think in terms of fractions, but from the perspective of notation, there’s no difference between “fractions” and “division”. It’s just a notation.”
“I suspect (without explicit confirmation) that both the obelus and the colon were chosen specifically because they suggest the fractional notation: The two dots represent the numerator and the denominator. In the case of the obelus, we also write the fraction bar itself. Cajori even reports at least one occurrence of ·/· (compare to 3/4).”
~ ~ ~ ~ ~ ~ ~
The expression 6 ÷ ¾ can be written as you have shown it with two obeluses [6÷3÷4], or with two slashes, thusly:
6/3/4
As a result, as in any expression with more than one division symbol, the author needs to put in at least one set of parentheses to clarify the order of the divisions. In the case of 6 ÷ ¾, if the author wants to indicate that 6 should be divided by three-quarters, it should be written thusly, to avoid any confusion:
6 ÷ (3/4) or 6/(3/4)
That way, no one will perform the division as 6÷3 & then divide that quotient of 2 by 4 & arrive at an erroneous answer.
~ ~ ~ ~ ~ ~ ~
from Club Z Tutoring:
https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-examples-6-7-5/#:~:text=A1%3A%20Yes%2C%20the%20fraction%20bar,same%20meaning%20in%20mathematical%20notation.
“The Role of the Fraction Bar in Fraction Notation
The fraction bar serves as a visual representation of the division operation between the numerator and the denominator. It shows that the numerator is divided by the denominator, indicating the fraction’s value. For instance, the fraction 3/4 is interpreted as ‘3 divided by 4’ “
“FAQ
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.”
. * . * . * . * . * . *
You say:
“…the ONLY way to group a division expression using an obelus is to place the expression in parentheses.”
That is not correct. Only if there is more than one division symbol in the expression, must at least one set of parentheses be used to indicate the order in which the multiple divisions are to be done.
With only one division symbol, however, the expression can be rewritten as a top-and-bottom fraction (using the position of the division sign or slash as the fraction bar, with the numerator as everything to the left & the denominator as everything to the right) & divided accordingly.
The real grouping is juxtaposition, because the Distributive Law MANDATES that the parentheses be EXPANDED. Thus, juxtaposition (implied multiplication) is PART OF the “Parentheses” step in The Order of Operations as PEMDAS.
Additionally, because Fraction=Division (and therefore, Division=Fraction), The Order of Operations needs to be revised to reflect the correct procedure to calculate the value of a fraction: First, do all of the operations in the numerator, then do all of the operations in the denominator, and then finally, divide the numerator by the denominator.
DIVISION MUST GO LAST!
Therefore, The Order of Operations should be…
Parentheses/Juxtaposition
Exponents
Multiplication (explicit)
Addition/Subtraction
Division
…which can be remembered as:
“Please Just Excuse My Aunt Sally Dee.”
–Dee
Comment by Dee R. — February 25, 2025 @ 4:35 pm
Dee, I’m sorry, but you misrepresented what I was saying. The expression 6 ÷ ¾ should NEVER be interpreted as 6 ÷ 3 ÷ 4 (without parentheses) because the ¾ is an offset monomial grouped by the solidus (or whatever fraction symbol). The answer to 6 ÷ 3 ÷ 4, if one works strictly left to right, is 1/2. The answer to 6 ÷ ¾ is 8. It must ALWAYS be interpreted as 6 ÷ (3 ÷ 4), or more commonly applying the “invert and multiply” principle, 6 x (4/3) or 6 x (4 ÷ 3) (parentheses are irrelevant in the latter because of standard OOO). In other words, the juxtapositional grouping implies parentheses around the fraction. The stuff you posted from the Internet sites is just plain wrong and misleading in my opinion because it fails to recognize juxtapositional grouping. That’s the whole premise of my argument.
Scott
Comment by Scott Stocking — February 25, 2025 @ 9:41 pm
The way my class was taught (50+ years ago) to convert an inline division expression written with an obelus (or slash) to a top-and-bottom fraction was first to write the dividend (i.e. everything that is to the left of the obelus or slash) across a line. Then, take the obelus (or slash) and write that underneath the numerator. Then, underneath the division symbol, write everything that was to the right of the obelus or slash. For example…
4x + 4 ÷ 2x +2 =
4x + 4
÷
2x + 2
Now replace the obelus (or slash) with a fraction bar:
4x + 4
————-
2x + 2
The division symbol in the inline expression (obelus or slash) is in the same position as the fraction bar — it’s just written from left-to-right instead of top-to-bottom. We were taught that, as with the expression written with a fraction bar, no parentheses were necessary because it was clear what the “pieces” of the division expression were (i.e. what was the numerator & what was the denominator). We were taught that only if there was some departure from that standard setup, parentheses need to be used to indicate a different order of operations from the fraction it would otherwise appear to be.
Think about it — the improper fraction of “Forty-eight twenty-fourths” is the same, whether written as…
48
—–
24
…or written as…
48/24 .
If “Forty-eight twenty-fourths” is ‘factorized’ as…
4(12)
———
2(12)
…and the “12” in parentheses is replaced by the variable “a,” then the fraction becomes…
4a
——
2a
…which, when converted to an inline fraction, becomes…
4a/2a
When a=12, that’s exactly the same division proposition as “Forty-eight twenty-fourths” which is the numerator of 48 divided by the denominator of 24. And that, of course, has a quotient of 2.
There is nothing in this methodology which “conflicts” with anything taught in elementary school vs. algebra. In fact, every 5th grade math textbook affirmatively states that Fraction=Division, and therefore, Division=Fraction. That concept is referred back to in teaching algebra, as evidenced by students being specifically instructed to “Rewrite as a fraction,” when presented with an algebraic division expression written with an obelus, in textbooks & on math tutoring websites. And the way that it is demonstrated for students to do that is exactly what I just outlined — even when there are no parentheses used in the originally written expression using an obelus.
Comment by Dee R. — July 12, 2025 @ 10:25 am
The way my class was taught (50+ years ago) to convert an inline division expression written with an obelus (or slash) to a top-and-bottom fraction was first to write the dividend (i.e. everything that is to the left of the obelus or slash) across a line. Then, take the obelus (or slash) and write that underneath the numerator. Then, underneath the division symbol, write everything that was to the right of the obelus or slash. For example…
4x + 4 ÷ 2x +2 =
4x + 4
÷
2x + 2
Now replace the obelus (or slash) with a fraction bar:
4x + 4
————-
2x + 2
The division symbol in the inline expression (obelus or slash) is in the same position as the fraction bar — it’s just written from left-to-right instead of top-to-bottom. We were taught that, as with the expression written with a fraction bar, no parentheses were necessary because it was clear what the “pieces” of the division expression were (i.e. what was the numerator & what was the denominator). We were taught that only if there was some departure from that standard setup, parentheses need to be used to indicate a different order of operations from the fraction it would otherwise appear to be.
Think about it — the improper fraction of “Forty-eight twenty-fourths” is the same, whether written as…
48
—–
24
…or written as…
48/24 .
If “Forty-eight twenty-fourths” is ‘factorized’ as…
4(12)
———
2(12)
…and the “12” in parentheses is replaced by the variable “a,” then the fraction becomes…
4a
——
2a
…which, when converted to an inline fraction, becomes…
4a/2a
When a=12, that’s exactly the same division proposition as “Forty-eight twenty-fourths” which is the numerator of 48 divided by the denominator of 24. And that, of course, has a quotient of 2.
There is nothing in this methodology which “conflicts” with anything taught in elementary school vs. algebra. In fact, every 5th grade math textbook affirmatively states that Fraction=Division, and therefore, Division=Fraction. That concept is referred back to in teaching algebra, as evidenced by students being specifically instructed to “Rewrite as a fraction,” when presented with an algebraic division expression written with an obelus, in textbooks & on math tutoring websites. And the way that it is demonstrated for students to do that is exactly what I just outlined — even when there are no parentheses used in the originally written expression using an obelus.
Comment by Dee R. — July 12, 2025 @ 10:25 am
“Implicit construction” is a good way to express the concept of a single quantity, in the case of any whole number which is two digits or more long. For example, the number 48 is 4 tens plus 8 ones. That can be expressed numerically as:
48 = 4(10) + 8(1)
…which is implied multiplication and implied addition, within the “grouping” of the number 48. The same goes for the number 24:
24 = 2(10) + 4(1)
Now let’s divide 48 by 24:
48 ÷ 24 = 2
…which can also be written as the top-and-bottom fraction…
48
_____
24
…which still equals 2.
Both “48′ & “24” are each understood to be a single quantity, even though they both use implied multiplication & implied addition in the ‘term’ — without being encased in parentheses.
No math teacher has ever instructed students to ‘deconstruct’ 48 divided by 24 to be written as…
4(10) + 8(1) ÷ 2(10) +4(1)
…which some argue would entitle them to insert multiplication signs where there actually aren’t any, making the expression…
4* 10 + 8 *1 ÷ 2 * 10 + 4 * 1
Therefore, students are NOT taught to first do the multiplications & divisions as the come up from left-to-right & then go back to the left & take care of the additions, as prescribed by The Order of Operations as PEMDAS. If they did, it would go like this:
4* 10 = 40
8 * 1 = 8
8 ÷ 2 = 4
4 * 10 = 40
4 * 1 = 4
Now, going back to the left to do the additions as they come up:
40 + 4 + 4 = 48
…which of course is demonstrably incorrect to divide 48 by 24, even though that’s how the monomial terms “48” & “24” would be presented, if all of the implied operations were written out & then carried out according to PEMDAS after that.
Here’s another example of the flaw in using PEMDAS without taking care of juxtaposition (i.e. implied multiplication) first:
Given that x does not equal zero:
x ÷ x = 1
In algebra, it is taught that a variable by itself has a coefficient of 1. In other words, x=1x.
No algebra textbook insists that putting parentheses around a monomial is mandatory. A teacher might suggest it for clarity (especially if inputting an expression into a computer calculator program), but it is not taught to students that they MUST put parentheses around a monomial in order for it to be understood as one term with a single value. Therefore, 1x divided by 1x should still equal 1. But according to those applying PEMDAS after inserting a multiplication sign where there actually isn’t one, going left to right, doing the multiplications & division as they come up…
1x÷1x
1*x=1x
1x÷1=1x
1x*x=1x^2
In no algebra class, anywhere in the world, has that that ever be shown to be the case.
~ ~ ~ ~ ~ ~ ~
Have you covered “factoring out” a term, before dividing one term by another?
For example:
48÷24 can be factored out as…
4(12)÷2(12)
Cancelling out the like-factor of 12, makes the division…
4÷2
…which of course equals 2 — just as 48÷24=2.
The division expression of 48÷24 can also be written as follows, when a=12:
4a÷2a
…so after cancelling out the like-factor of “a,” it’s back to…
4÷2
…which of course equals 2 — just as 48÷24=2.
Comment by Dee R. — February 24, 2025 @ 5:32 pm |
https://rumble.com/v6q98es-stockings-order-juxtapositional-grouping-is-foundational.html
Comment by Scott Stocking — March 6, 2025 @ 10:00 pm |
[…] Stocking’s Order: Implicit Constructions Correct the Misapplication of Order of Operations | Sunda… February 23, 2025 […]
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