8÷2(1+3)

[dm319]

(12-03-2023 10:31 PM)Thomas Klemm Wrote:  Does this really mean addition before subtraction?

No. It's meant to be a helpful acronym, but it is, in itself, ambiguous in this regard. Addition/subtraction and multiplication/division are equal precedence. In other places it can be called 'BODMAS' for example.

(12-03-2023 10:31 PM)Thomas Klemm Wrote:  But is that reason enough to come up with a special rule about implied multiplication?

I only recently came across this apparent facebook craze. When I first worked it out, I naturally calculated '1'. A few youtube videos in, and I got the impression I was wrong about this. To be fair I'd forgotten about the left-to-right rule. But the explanations didn't sit with me, so I went on a deeper dive.

My conclusion is that BODMAS/PEMDAS etc are just helpful acronyms to remind school children the order of operations. It isn't gospel, and in fact it isn't very clear as you demonstrated above. It doesn't even have the 'left-to-right' rule as part of it.

The video I mentioned above is probably the clearest and most concise explanation to me. She uses the term 'juxtaposition'. I don't think 'implicit multiplication' is a real thing, so I'm glad she doesn't use that term. I would use the term 'coefficient of a term', and I believe that it makes most sense for coefficients to have a higher precedence than the operators. They are not equivalent to the multiplication operator IMO, though can be converted to it with brackets, i.e. 2a = (2 x a).

There is a fair argument that where there is ambiguity, then it is best to be clear with liberal use of parens. To be fair though, you can take this to an extreme. You could write out y = 2x + 36 always as y = (2x) + 36, or y = 2a^2 as y = 2(a^2). I'll admit that these are less ambiguous, but I feel like we should agree on this topic to make this clear. The video demonstrates that something like 8÷2(1+3) was used in maths long before we were limited in typing out algebraic expressions on a calculator or in python on a single line, and we will continue to be typing out expressions on a single line for a while.

Here is someone's thread with quite an interesting take and series of arguments about why the answer is 1.

If we take a step back, and if we accept that what the video and that thread say are true, then how on earth did we get into this situation? It seems like most calculators were doing it right, then TI changed quite abruptly. Casio wavered back and forth. HP were inconsistent. Even Wolfram Alpha is inconsistent. Only Sharp have steadfastly stuck from what I can see. I think programming languages go with this rigid PEMDAS interpretation.