Grog write left to right.

Gorg is from another country and writes right to left.

Grog read left to right.

Gorg reads right to left

Grog do sum left to right

Gorg does sums right to left

8/2(1+3) even if they technically are meant to be evaluated the same

But 8/2(1+3) isn't a fraction. The / - the computing equivalent of ÷ (which can only be written using Unicode on a computer, so a bit of a pain to use compared to / )- is an operator, which means they're 2 separate terms. A fraction bar is a grouping symbol, which means it's 1 term. In this particular case it doesn't matter, but if it appeared in a bigger expression then it absolutely does matter. The way to write 8/2(1+3) as a fraction inline is to add extra brackets. i.e. (8/2(1+3)) - because brackets are also a grouping symbol.

And as for distributive law vs multiplication, maybe this is just taking for granted a thing that I learned a long time ago, but to me they’re just the same thing in practice

Bu they're not, for the same reason. Firstly, the Distributive Law isn't multiplication at all - which only applies literally to multiplication symbols - it applies to bracketed terms (i.e. is a single term which needs to be distributed) - and secondly it applies to a single term, whereas multiplication applies to 2 terms (one before and one after). Anyone who talks about 2(1+3) needing to be "multiplied" has already made the mistake that is going to lead to a wrong answer (unless they just happen to "multiply" before they divide, which is an accidental way to get the right answer).

if I was factoring something

Indeed, that is the precise reason the Distributive Law exists - they are the opposite operation to each other! Anyone who adds a multiplication symbol has broken up the factorised term, again leading to the wrong answer.

I’m just being a bit lose with the terminology

Yeah, and that's all I was pointing out in the first place - please don't use "implicit multiplication". The term itself - i.e. it includes "multiplication" - leads people to do it wrong (because they treat it as multiplication, not brackets, then argue about the precedence of "multiplication"!). It needs to die!

this can rapidly get unreadable once you nest more than a few parens,

Well that's why the rules of Terms ab=(axb) and The Distributive Law a(b+c)=(a*(b+c))=(ab+ac) exist to begin with - less brackets! :-) Imagine having to write a fraction as (1/(axb)) all the time!

(8)/(2(1+3)) is obviously different than (8/2)(1+3)

Correct, though a lot of people treat it as the latter (yet another way to do it wrong - doing division before brackets) because they figure the 8/2 is "outside the brackets", but in fact only the 2 is, because the slash separates them as being 2 terms.

the rules of math are not set in stone

The rules around order of operations are!

if your notation is ambiguous or unclear to your audience try to fix it

Nothing ambiguous in this expression.

Right idea, but wrong terminology.

There's no such thing as implicit multiplication

xy is a Term - Terms are separated by operators (none in this case) and joined by grouping symbols.

x(y) is a Bracketed Term, and is therefore subject to The Distributive Law, which is the first step in solving Brackets.

And yes, a multiplication symbol is an operator - which therefore separates Terms (which is why ab and axb aren't the same thing - it's 1 term vs. 2 terms), and the "M" in the mnemonics refers literally to multiplication signs, and nothing else.

When I was in school, year 7 was primary school

Oh really? My apologies then. I've only ever heard Year 7 called high school or middle school, never primary school. What country is that in?

multiplication by juxtaposition. Which I’m fairly sure for me at least was in year 6

I've seen some Year 6 classes do some pre-algebra (like "what number goes in this box to make this true"), but Year 7 is when it's properly first taught. Every textbook I've ever seen it in has been Year 7 (and Year 8, as revision).

Also, it's not "multiplication by juxtaposition", since it's not multiplication - it's The Distributive Law - which is Distribution - and/or Terms - which is a product, which is the result of a multiplication.

No, the idea of specifically codifying BIDMAS comes from the early 1900s

The order of operations rules are older than that - we can see in Lennes' letter (1917) that all the textbooks were already using it then, and Cajori says - in 1928 - that the order of operations rules are at least 300 years old (which now makes them at least 400 years old).

If you're talking about when was the mnemonic BIDMAS made up, that I don't know, but the mnemonics are only ways to remember the rules anyway, not the actual rules.

I don’t know why you’re going throughout this thread

I'm a Maths teacher, that's what we do. :-)

a rigid primary school application of BIDMAS will lead you to the wrong answer

Only if the bracketed term has a coefficient (welcome to how Texas Instruments gets the wrong answer), which is never the case in Primary School questions - that's taught in Year 7 (when we teach The Distributive Law).

juxtaposition actually comes before explicit multiplication... I think that’s what you mean when you keep saying “it’s called terms”

Terms come before operators, and we never call it juxtaposition, because The Distributive Law is also what people are calling "strong juxtaposition" (and/or "implicit multiplication"), but is a separate rule, so to lump 2 different rules together under 1 name is where a lot of people end up going wrong. There's a Youtube where the woman gets confused by a calculator's behaviour and she says "sometimes it obeys juxtaposition and sometimes it doesn't" (cos she lumped those 2 rules together), and I for one can see clear as day the issue is it's obeying Terms but not obeying The Distributive Law (but she lumped them together and doesn't understand these are 2 separate behaviours).

Terms and multiplication by juxtaposition can work together

But that's my point, there's no such thing as "multiplication by juxtaposition". A Term is a product, which is the result of a multiplication.

If a=2 and b=3 then...

axb=2x3 - 2 terms

ab=6 - 1 term

In the mnemonics "Multiplication" refers literally to multiplication signs, and nothing else. The Distributive Law is done as part of solving Brackets, and there's nothing that needs doing with Terms, since they're already simplified (unless you've been given some values for the pronumerals, in which case you can substitute in the values, but see above for the correct way to do this with ab, though you could also do (2x3), but absolutely never 2x3, cos then you just broke up the term, and get the wrong answer - brackets can't be removed unless there is only 1 term left inside. People writing 2(3)=2x3 are making the same mistake).

Everything I know is a lie

...including the comment you just replied to. Here is a thread with actual textbook references, historical Maths documents, worked examples, proofs, the works.

denotes it with “/” likely to make sure you treat it as a fraction

It's not the slash which makes it a fraction - in fact that is interpreted as division - but the fact that there is no space between the 2 and the square root - that makes it a single term (therefore we are dividing by the whole term). Terms are separated by operators (2 and the square root NOT separated by anything) and joined by grouping symbols (brackets, fraction bars).

used juxtaposition for multiplication bound more tightly than division

It's called Terms - Terms are separated by operators and joined by grouping symbols. i.e. ab=(axb).

I read the equation and was shocked that anyone would get an answer other than 9

As a Maths teacher, I'm shocked whenever anyone ever gets an answer other than 1. I'm not sure how you came up with 9 when you previously said you've only ever seen strong juxtaposition? You can only get 9 with so-called "weak juxtaposition" (which is wrong).

Yeah, base ten really screws around with programming. You specifically have to use a decimal type if you really want to use it (for like finance or something), but it's much slower.

Can confirm that now working again with the 0.19.3 update - woo hoo! Thank you!