FACT CHECK 2/5

The behaviour is intended and even carefully documented in the manual

...and yet still a bug (I saw at least one other person point this out to you)

A few years ago, there was a Microsoft feature intended for people in China, but people who weren't in China were getting that behaviour. i.e. a bug. It was documented and a deliberate design choice for people in China, but if you weren't in China then it's a bug. Just documenting a design choice doesn't mean bugs don't happen. A calculator giving a wrong answer is a bug

weak juxtaposition is only used by old calculators

Based on the comments in the above video, the opposite is true - this problem first arose in '96

because they are scientific calculators.

So the person programming it is far more likely to need to check their Maths first - bingo!

TI (Texas Instruments) also has some calculators that use strong juxtaposition and some products that use weak juxtaposition

...and some that use both! i.e. some follow Terms but not The Distributive Law. As I said to begin with, these are 2 DIFFERENT rules, and you can't just lump them together as one

evaluate 1/2X as 1/(2X)

Which is correct, as per Terms

while other products may evaluate the same expression as 1/2X from left to right

What you mean is they evaluate it as 1/2xX, since 1/2X and 1/(2X) are the same thing

it would be necessary to group 2X in parentheses

No, not necessary, since 2a=(2xa) by definition, alluded to in Cajori in 1928...

Sharp is a bit of an exception here, because all their other scientific calculators seem to

...follow all the rules of Maths, always. There's something to be said for making sure you're doing it right. :-)

Google uses the same priority for explicit and implicit multiplication

...and they will actually remove brackets I have put in and replace them with their own ("hi" to all the people who say you can fix any calculator by "just add more brackets" - Google doesn't CARE what brackets you've added!)

Desmos and GeoGebra try to force the user into using fractions (which is a good design decision if you ask me)

It's not, because a ÷ isn't a fraction bar. They're joining 2 terms into one and thus sometimes changing the answer

A lot of other tools like programming languages, spreadsheets, etc. don’t allow implicit multiplication syntax at all

It's not that they don't allow it, it's that it's not provided with the language by default in the first place! Most languages only provide you with some numbers, operators, and a few functions (like round), and it's up to the programmer to implement the rest. Welcome to why there are so many wrong e-calculators

let you choose if you want weak or strong juxtaposition

...which is a red flag to not use that calculator!

This gives you more control about how you like the calculator to behave in these situations

I'm not sure it does. I'd have to switch on "strong juxtaposition" (the only kind there is) and see what else has been disobeyed in Maths. e.g. Google removing my brackets and adding different ones

Wolfram|Alpha only uses strong juxtaposition between named variables, but weak juxtaposition for everything else. This might seem strange and inconsistent at first but is probably the least surprising behaviour for most people

I find any exceptions to following the rules of Maths surprising! No, you can't just make up your own rules

many textbooks, “a/bc” is intended to denote a/(bc)

a/bc=a/(bc) in every textbook

Wolfram Language, it means (a/b)×c

Welcome to "we're gonna add brackets to what you typed in and change the answer"

a multiplication sign has been omitted

...then that means it's not "multiplication" - it's Terms and/or The Distributive Law. The "M" in the mnemonics refers literally to multiplication signs, nothing else

Multiplication and division have the same priority, they are “mathematically speaking” the same operation. This also applies to addition and subtraction. One is just the inverse function of the other

Yep, and The Distributive Law and Factorising are the inverse of each other

no rule about “multiplication before division” or “division before multiplication” they always have the same priority

...and Brackets is always first, so in this case it doesn't even matter

In no way do any of the mnemonics represent any standard or norm in mathematics

Yes they do - mnemonics represent the actual order of operations rules

most children don’t become mathematicians later in life and if they do, they will learn all the other important stuff about the order of operations later

No, they won't. Year 8 is the last time order of operations is taught, and they have been taught everything they need to know about it by then

it’s hard to pump so much knowledge into children and teenagers

...and yet have you not noticed that teenagers almost never get this wrong - only adults do

Using “PEMDAS” to argue about the order of operations in mathematics

...is a totally valid thing to do. The problem is people classifying Distribution (Brackets/Parentheses with a coefficient) as "Multiplication", when there's literally no multiplication sign

Math notations and conventions evolve exactly like natural languages

No they don't. Maths is universal

A lot of it is heavily based on historical thanks and work from previous generations

It's all based on definitions and proofs, which are immutable

There is no definitive norm, standard or convention of notations and order of operations

You can find them in any high school textbook in your country (notation varies by country, but the rules don't)

some words only appear in half of them (like “implicit multiplication by juxtaposition”)

"implicit multiplication" doesn't appear in any Maths textbooks

sentences like “I saw the man with the telescope”, because it’s not clear if you saw him through the telescope or saw him holding (or looking through) a telescope

Yes it is clear (as I think I saw someone already point out here)

I saw the man with the telescope - the man has the telescope

I saw the man, with the telescope - I saw the man through a telescope

I saw the man through the telescope - I saw the man through a telescope

it should also be clear why there are no arguments or proofs for any side

But there are proofs! (There you go again with the "there is no..." red flag) Order of operations proof

FACT CHECK 3/5

It’s only a matter of taste and how widespread a convention or notation is

The rules are in every high school Maths textbook. The notation for your country is in your country's Maths textbooks

There are no arguments or proofs about what definition is correct

1+1=2 by definition (or whatever the notation is in your country). If you write 1+1=3 then that is wrong by definition

I found a lot of explanations online that were either half-assed or just plain wrong

And you seem to have included most of them so far - "implicit multiplication", "weak juxtaposition", "conventions", etc.

You either were taught something wrong or you misremember it.

Spoiler alert: It's always the latter

IMHO the mnemonics would be better without “division” and “subtraction”, because it would force people to think about it before blindly applying something the wrong way – “PEMA” for example. Parentheses, exponentiation, multiplication, addition

In fact what would happen is now people wouldn't know in what order to do division and subtraction, having removed them from the mnemonic (and there's absolutely no reason at all to remove them - you can do everything in the mnemonic order and it works, provided you also obey the left-to-right rule, which is there to make sure you obey left associativity)

parenthesis and exponents students typically don’t learn the order of operations through some mnemonics they remember them through exercise

That's not true at all. Have you not read through some of these arguments? They're all full of "Use BEDMAS!", "Use PEMDAS!", "It's PEMDAS not BEDMAS!" - quite clearly these people DID learn order of operations through the mnemonics

trying to remember some random acronyms

There's no requirement to memorise any acronym - you can always just make up your own if you find that easier! I did that a lot in university to remember things during the exam

they also state to “not use × to express a simple product”

...because a product is a Term, and to insert a x would break it into 2 Terms

A product is the result of a multiplication

The center dot also should not be used to mean a simple product

Exact same reason. They are saying "don't turn 1 term into 2 terms". To put that into the words that you keep using, "don't use weak juxtaposition"

Nobody at the American Physical Society (at least I hope) would say that 6/2×3 equals one, because that’s just bonkers

Because it would break the rule of left associativity (i.e. left to right). No-one is advocating "multiplication before division" where it would violate left to right (usually by "multiplication" they're actually referring to Terms, and yes, you literally always have to do Terms before Division)

÷ (obelus), : (colon) or / (solidus), but that is not the case and they can be used interchangeably without any difference in meaning. There are no widespread conventions, that would attribute different meanings

Yes there is. Some countries use : for divide, whereas other countries use it for ratio

most standards forbid multiple divisions with inline notation, for example expressions like this 12/6/2

Name one! Give me a reference! There's nothing forbidding that in Maths (though we would more usually write it as 12/(6x2)). Again, all you have to do is obey left to right

Funny enough all the examples that N.J. Lennes list in his letter use

...Terms. Same as all textbooks do now

and thus his rule could be replaced by

...Terms, the already-existing rule that he apparently didn't know about (he mentions them, and products, but manages to completely miss what that actually means)

“Something, something, distributive property, something ….”

Something, something, Distributive Law (yes, some people use the wrong name, but in talking about the property, not the law, you're knocking down a strawman)

The distributive property is just a property that applies to some operations

...and The Distributive Law applies to every bracketed term that has a coefficient, in this case it's 2(1+2)

It has nothing to do with the order of operations

And The Distributive Law has everything to do with order of operations, since solving Brackets is literally the first step!

I’ve no idea where this idea comes from

Maybe you should've asked someone. Hint: textbooks/teachers

because there aren’t any primary sources (at least I wasn’t able to find any)

Here it is again, textbook references, proofs, memes, the works

should be calculated (distributed) first

Bingo! Distribution isn't Multiplication

6÷2(3). If we follow the strong juxtaposition convention, we must

...distribute the 2, always

It has nothing to do with the 3 being inside parentheses

It has everything to do with there being a coefficient to the brackets, the 2

Those parentheses are only there, because

...it's a factorised term, and the opposite of factorising is The Distributive Law

the parentheses do not force the multiplication

No, it forces distribution of the coefficient. a(b+c)=(ab+ac)

The parentheses are only there to make it clear that

...it is a factorised term subject to The Distributive Law

we are implicitly multiplying two separate numbers.

They're NOT 2 separate numbers. It's a single, factorised term, in the same way that 2a is a single term, and in this case a is equal to (1+2)!

With the context that the engineer is trying to calculate the radius of a circle it’s clear that they meant r=C/(2π)

Because 2π is a single term, by definition (it's the product of a multiplication), as is r itself, so that should actually be written r=(C/2π)

When symbols for quantities are combined in a product of two or more quantities, this combination is indicated in one of the following ways: ab,a b,a⋅b,a×b

Incorrect. Only the first one is a term/product (not separated by any operators) - the last 2 are multiplications, and the 2nd one is literally meaningless. Space isn't defined as meaning anything in Maths

Division of one quantity by another is indicated in one of the following ways:

The first is a fraction

The second is a division

The third is also a fraction

The last is a multiplication by a fraction

Creates ambiguity since space isn't defined to mean anything in Maths. Looks like a typo - was there meant to be a multiply where the space is? Or was there not meant to be a space??

By definition ab^-1^=a^1^b^-1^=(a/b)