Stocking’s Reciprocation :: “Inverting the dividend of a division problem and multiplying by the divisor yields the reciprocal of inverting the divisor of a division problem and multiplying by the dividend.”
Oh no, here he goes again, off on one of his wild math/PEMDAS/Order of Operations tangents. But you know, I’m really not doing anything different in this area than I would do in exegeting a biblical passage. I’m looking for patterns in language and logic, grammar and syntax, that reveals clues to the meaning and intent of the words or figures on a page. I’m applying a scientific method to all this as well, forming new or related hypotheses and testing them out to see whether they can be more than just a harebrained rambling.
That is what’s been happening to me with this whole PEMDAS/Order of Operations obsession I’ve found myself pursuing. Every time I think I’ve got the definitive solution, proof, or argument, I get whacked upside the head by another even more convincing argument. Initially those were coming fast and furious, and I couldn’t keep up with myself, as you might discern if read my regularly updated original article, 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough | Sunday Morning Greek Blog or any of the other several articles I’ve written on the subject. But as I feel like I’ve addressed most of the obvious weaknesses of the ignorance of the priority of juxtapositional binding over extant operational signs, those moments of inspiration (dare I say “genius”?) are getting farther apart.
But this new concept I’ve described (I’m not sure how “new” it is, but I know I was never taught anything like this in math) as it turns out is really another way of double checking your division homework. I had gotten close to this when I demonstrated that the PEMDAS crowd’s way of interpreting 8 ÷ 2(2 + 2) was equivalent to the expression 8 ÷ [2/(2 + 2)], the latter of which is subject to “invert and multiply” to solve and therefore more naturally solved without needing to remember or manipulate PEMDAS. [Note: Copilot called this concept “dual symmetry” in division.]
What Stocking’s Reciprocation does is move the emphasis from the divisor to the dividend, thus eliminating the argument about juxtaposition all together, because the answer you get when you double-check with Stocking’s Reciprocation agrees with the answer you get when you consider the juxtaposed divisor to be a single, unbreakable unit or monomial. You can’t get to the PEMDAS crowd’s answer of 16 using Stocking’s Reciprocation, so that proves the answer 16 is not correct and that the PEMDAS crowd’s method of working that expression is greatly flawed.
Here’s how Stocking’s Reciprocation works. Instead of inverting and multiplying the divisor (especially in this case where the divisor is disputed), we invert the dividend and multiply. This will yield the reciprocal of what the correct answer should be.
8 ÷ 2(2 + 2) becomes (1/8) x 2(2 +2).
You can already see how this is going to go then. Now, you have eliminated the obelus and are left with a fraction (which is NOT an explicit division problem) multiplied by a number multiplied by a value in parentheses. With all three factors of the expression multiplied now, it doesn’t matter what order you multiply them (Commutative Property)!
At this point, we can still apply undisputed PEMDAS principles by calculating the value in parentheses first and then multiplying straight across.
(1/8) x 2(4) becomes (1/8) x 8 (remember, order doesn’t matter at this point because it’s all multiplication), so the product is 1. Then you take the reciprocal of the product to compensate for the initial reciprocal conversion, and of course you get 1.
Just to demonstrate this isn’t an anomaly because the answer was 1, let’s do a couple more complicated expressions, as I did on my Facebook response today with one of these expressions:
8 ÷ 4(6 + 8) would be incorrectly calculated by the PEMDAS crowd as 28. But using Stocking’s Reciprocation, you get:
(1/8) x 4(6 + 8) = (1/2)(14) = 7. The reciprocal of that (and the answer to the original expression using Stocking’s Order, i.e., juxtaposed multiplication takes priority) is 1/7. If the 4(6 + 8) is inseparable in that expression, as I have claimed all along, you solve the 4(6 + 8) first, yielding 56, so in the original expression, you would get 8/56 = 1/7. Voila!
So let’s apply it to this expression.
48 ÷ 2(9 + 3) = (1/48) x 2(12) = 24/48 = 1/2, and the reciprocal of that is 2.
Working the problems backwards like this (or is it sideways?) and demonstrating there is no confusion about what the answer is demonstrates that the more sophisticated take on Order of Operations that engineers, physicists, and other math-heavy professionals use daily is 100% correct. They understand instinctively that the juxtaposition means you don’t separate the terms by an extant operational sign. After all, they passed arithmetic and moved on to algebra where that principle is prominent (i.e., 8 ÷ 2a = 4/a, NOT 4a).
If you want to see my other articles discussing various aspects of this debate, please see my summary page that has links to all of these articles: The PEMDAS Chronicles: Confronting Social Media Ignorance of PEMDAS’s Theoretical Foundation | Sunday Morning Greek Blog
I’m going to brag on myself a bit too here. When I posed this concept to Copilot (Microsoft’s AI), I got the following response: “It’s not commonly taught, but it’s mathematically sound and elegant.” Copilot even used my own Facebook post that I’d just put up less than a half-hour earlier, as a source for its analysis.
Until my next inspiration,
Scott Stocking
My thoughts are my own.
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