Basic mathematics should not have any ambiguity. It’s time to declare juxtaposition as a property of mathematics (and our Arabic number system in general) and silence the ill-informed ambiguists!
The PEMDAS Convention:
Please
Excuse
Me
Doubting
A
Silly
Convention.
Harvard University math professor Oliver Knill has a wonderful site (Ambiguous PEMDAS) that documents many of the arguments against the PEMDAS convention (at least by the PEMDAS/Order of Operations legalists) of treating juxtaposed implicit multiplication with the same priority as multiplication with an extant operational sign. Of course, he’s referring to hotly debated, viral math expressions like:
8 ÷ 2(2 + 2) and 6 ÷ 2(1 + 2),
both of which equal 1 in my book, but to the underinformed PEMDAS/Order of Operations crowd, they equal 16 and 9 respectively.
Similarly, the Math Doctors site has an excellent article entitled Order of Operations: Historical Caveats – The Math Doctors in which they say that “nobody” made the rules for the order of operations and that the so-called “rules” are only descriptive. Doctor Peterson of the Math Doctors has been kind and gracious in answering many of my pointed questions and objections to the normalization of a PEMDAS convention that does not uniformly acknowledge the power of juxtaposition of numerals to imply certain mathematical operations, and for that I am grateful.
But all this begs the question: Why should there be ANY ambiguity in basic mathematics? Why do we have rules that are “descriptive” and not theoretically supported and sound? Isn’t mathematics a hard science (not Barbie’s “Math is hard!” but lacking or minimizing subjectivity), or at least the foundational backbone of the hard sciences that supports the objectivity needed for the hard sciences? Maybe most of the world doesn’t care about this argument about PEMDAS, but I see the admission or acknowledgment or tolerance (or whatever synonym) of ambiguity as a weakness of the discipline. The refusal to acknowledge the inconsistency of rejecting the binding power of juxtaposition in the 2(2 + 2) or 2(1 + 2) part of the expression flies in the face of what is otherwise inherent, intuitive, and obvious about the way the Arabic number system in its current form is constructed.
I agree with the Math Doctors that “nobody” made the rules of Order of Operations and that the “rules” of Order of Operations are descriptive (as opposed to theoretical, which seems to be the implication), but should we really allow such a laissez-faire, passive convention to be immune to such criticisms as I have raised in my writings and videos?
See, for example:
8 ÷ 2(2 + 2) = 1: A Discussion with ChatGPT on the Implications of Juxtaposition in Mathematics
Stocking’s Order: Implicit Constructions Correct the Misapplication of Order of Operations
My original critique, ever expanded and updated, 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough
My Rumble videos:
Stocking’s Order: Juxtapositional Grouping Is Foundational
Stocking’s Order; or Why 8 ÷ 2(2 + 2) = 1).
If a mathematical convention has no clear origins or has no theoretical background, then either someone should propose a theoretical background to support it or promulgate a rule based on demonstrated, objective properties and principles (as I have done above) that rightly and fairly critique the convention to bring it into compliance with the rest of the discipline. In other words, “SOMEONE MAKE A LEGITIMATE RULE AND ENFORCE IT!” Lay down the law instead of allowing this abuse of Order of Operations to tarnish the otherwise marvelous discipline of mathematics!
I understand that part of this comes from linear computer algorithms that take operations as they come. But that is an archaic algorithm now when compared the complexity of language models we have with artificial intelligence (AI). If a language model can figure out the parts of speech of a sentence, then certainly the math algorithms can be overhauled to recognize juxtaposition. The fact that this hasn’t happened yet some 50 years after the advent of the hand-held calculator is astounding to me. We’re basically programming our computers with a primitive, elementary model of mathematics and not accounting for the nuances that arise in Algebra. The order of operations rules should not be different for algebra than they are for arithmetic. Algebra is more or less a theoretical description of arithmetic, so the rules of Algebra should prevail. They should be uniform throughout the field of mathematics.
In my articles above, I go into detail on the influence of juxtaposition in the Arabic number system. For those who haven’t read them yet, here’s a quick summary of the prevalence of juxtaposition and its function in creating a singular value that takes priority in processing.
A fraction is the product of one operand multiplied by the reciprocal of another nonzero operand of the same or different value.[1] The operand below the vinculum (and by definition grouped by the vinculum[2],[3]) should be rationalized, when possible. When looked at this way, then, you have a quantity vertically juxtaposed to a grouped quantity, just like in 8 ÷ 2(2 + 2) you have a quantity (2) horizontally juxtaposed to a grouped quantity. The only difference is the orientation of the juxtaposition.
Yet when dividing by a fraction, we do not pull the numerator/dividend away from the fraction and divide that first, then divide by the denominator/divisor. We treat the fraction as a single quantity, invert the fraction first, then multiply. In other words, the element with a value juxtaposed to another grouped value is addressed first, then the expression is worked (and it’s not insignificant that such a transformation puts multiplication before division, just like it is in PEMDAS). In the same way then, the implied multiplication in 2(2 + 2) MUST be addressed first to be consistent with its cognate function in the former example. The obelus is NOT a grouping symbol, so it is WRONG to treat it as such and give it some nonexistent power to ungroup a juxtaposed quantity.
It is the same way when multiplying or dividing by a mixed number. We do not undo the implied addition of the juxtaposition of a whole number and a fraction but must first convert the mixed number to an improper fraction, then apply the rules for fractions in the previous paragraph.
This is the undeniable reality of how our number system works. I fail to see how anyone can disprove or ignore this reality simply by citing an untested and subjective convention that isolates one of several similar forms and says, “We’re treating it differently because it’s easier than trying to teach it the correct way.” To that I say “Poppycock!”
It’s time to stop capitulating to ambiguity! Mathematics is not worthy of such capitulation. Have some courage and take a stand for what is demonstrably true! Mathematics has hard and fast rules and properties, and the property of juxtaposition should be elevated to the same level as the associative, distributive, and commutative properties. Need a definition? Here it is:
Juxtaposition Property: When two or more values are juxtaposed without any intervening free-standing operators (i.e., operators not included in a grouped value), the juxtaposed values are considered inseparable and must be given priority over extant signed operators in the Order of Operations, regardless of what the implied operators are.
Scott Stocking
My highly informed opinions are my own and are the product of my own research.
[1] To put it another way, a basic division problem using the obelus can be converted to a multiplication problem by inverting (i.e., taking the reciprocal of) the divisor and multiplying by the dividend.
[2] “2 : a straight horizontal mark placed over two or more members of a compound mathematical expression and equivalent to parentheses or brackets about them,” VINCULUM Definition & Meaning – Merriam-Webster, accessed 05/25/25.
[3] “A horizontal line placed above multiple quantities to indicate that they form a unit. vinculum – Wolfram|Alpha, accessed 05/25/25.
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