this post was submitted on 20 Jul 2023
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Pretty much all of those rely on the fact that PEMDAS is ambiguous with actual usage. The reason why is it doesn't differentiate between explicit multiplication and implicit multiplication by placement. E.G. in actual usage "a*b" and "ab" are treated with two different precedence. Most of the time it doesn't matter but when you introduce division it does. "a*b/c*d" and "ab/cd" are generally treated very differently in practice, while PEMDAS says they're equivalent.
I see your point. When those expressions are poorly handwritten it can be ambiguous. But as I read it typed out it's ambiguous only if PEMDAS isn't strictly followed. So I guess you could say that it might be linguistically ambiguous, but it's not logically ambiguous. Enter those two expressions in a calculator and you'll get the same answer.
You actually won't. A good graphing calculator will treat "ab/cd" as "(a*b)/(c*d)" but "a*b/c*d" as "((a*b)/c)*d" (or sometimes as "a*(b/c)*d") and actual usage by engineers and mathematicians aligns with the former not the later. You actually can't enter the expression in a non graphing calculator typically because it won't support implicit multiplication or variables. While you can write any formula using PEMDAS does that really matter when the majority of professionals don't?
Actual usage typically goes parentheses, then exponents, then implicit multiplication, then explicit multiplication and division, then addition and subtraction. PEI(MD)(AS) if you will.
Interesting, I decided to try it with a few calculators I had laying around (TI-83 plus, TI-30XIIS, and Casio fx-115ES plus), and I found that the TI's obeyed the order of operations, while the Casio behaved as you describe. I hardly use the Casio, so I guess that I've been blissfully unaware that usage does differ. TIL. I don't think I've ever used or heard of a calculator that supports parentheses but not implicit multiplication though. Honestly though, the only time I see (AB)/(CD) written as AB/CD in clear text (or handwritten with the dividend and divisor vertically level with each other visually) is in derivatives, but that doesn't even count because dt and dx are really only one variable represented by two characters. I'm only a math minor undergrad though who's only used TI's so maybe I'm just naive lol
Or you take HPs approach and just sidestep the entire debate by using reverse polish notation in your calculators. From a technical standpoint RPN is really great, but I still find it a little mind bending to try to convert to/from on the fly in my head so I'm not sure I could ever really use a RPN calculator regularly.