6÷2(1+2)

FACT CHECK 1/5

If you are sure the answer is one... you are wrong

No, you are. You've ignored multiple rules of Maths, as we'll see...

it’s (intentionally!) written in an ambiguous way

Except it's not ambiguous at all

There are quite a few people who are certain(!) that their result is the only correct answer

...and an entire subset of those people are high school Maths teachers!

What kind of evidence/information would it take to convince you, that you are wrong

A change to the rules of Maths that's not in any textbooks yet, and somehow no teachers have been told about it yet either

If there is nothing that would change your mind, then I’m sorry I can’t do anything for you.

I can do something for you though - fact-check your blog

things that contradict your current beliefs.

There's no "belief" when it comes to rules of Maths - they are facts (some by definition, some by proof)

How can math be ambiguous?

#MathsIsNeverAmbiguous

operator priority with “implied multiplication by juxtaposition”

There's no such thing as "implicit multiplication". You won't find that term used anywhere in any Maths textbook. People who use that term are usually referring to Terms, The Distributive Law, or most commonly both! #DontForgetDistribution

This is a valid notation for a multiplication

Nope. It's a valid notation for a factorised Term. e.g. 2a+2b=2(a+b). And the reverse process to factorising is The Distributive Law. i.e. 2(a+b)=(2a+2b).

but the order of operations it’s not well defined with respect to regular explicit multiplication

The only type of multiplication there is is explicit. Neither Terms nor The Distributive Law is classed as "multiplication"

There is no single clear norm or convention

There is a single, standard, order of operations rules

Also, see my thread about people who say there is no evidence/proof/convention - it almost always ends up there actually is, but they didn't look (or didn't want you to look)

The reason why so many people disagree is that

...they have forgotten about Terms and/or The Distributive Law, and are trying to treat a Term as though it's a "multiplication", and it's not. More soon

conflicting conventions about the order of operations for implied multiplication

Let me paraphrase - people disagree about made-up rule

Weak juxtaposition

There's no such thing - there's either juxtaposition or not, and if there is it's either Terms or The Distributive Law

construct “viral math problems” by writing a single-line expression (without a fraction) with a division first and a

...factorised term after that

Note how none of them use a regular multiplication sign, but implicit multiplication to trigger the ambiguity.

There's no ambiguity...

multiplication sign - multiplication

brackets with no multiplication sign (i.e. a coefficient) - The Distributive Law

no multiplication sign and no brackets - Terms (also called products by some. e.g. Lennes)

If it’s a school test, ask you teacher

Why didn't you ask a teacher before writing your blog? Maths tests are only ever ambiguous if there's been a typo. If there's no typo's then there's a right answer and wrong answers. If you think the question is ambiguous then you've not studied enough

maybe they can write it as a fraction to make it clear what they meant

This question already is clear. It's division, NOT a fraction. They are NOT the same thing! Terms are separated by operators and joined by grouping symbols. 1÷2 is 2 terms, ½ is 1 term

BTW here is what happened when someone asked a German Maths teacher

you should probably stick to the weak juxtaposition convention

You should literally NEVER use "weak juxtaposition" - it contravenes the rules of Maths (Terms and The Distributive Law)

strong juxtaposition is pretty common in academic circles

...and high school, where it's first taught

(6/2)(1+2)=9

If that was what was meant then that's what would've been written - the 6 and 2 have been joined together to make a single term, and elevated to the precedence of Brackets rather than Division

written in an ambiguous way without telling you what they meant or which convention to follow

You should know, without being told, to follow the rules of Maths when solving it. Voila! No ambiguity

to stir up drama

It stirs up drama because many adults have forgotten the rules of Maths (you'll find students get this right, because they still remember)

Calculators are actually one of the reasons why this problem even exists in the first place

No, you just put the cart before the horse - the problem existing in the first place (programmers not brushing up on their Maths first) is why some calculators do it wrong

“line-based” text, it led to the development of various in-line notations

Yes, we use / to mean divide with computers (since there is no ÷ on the keyboard), which you therefore need to put into brackets if it's a fraction (since there's no fraction bar on the keyboard either)

With most in-line notations there are some situations with conflicting conventions

Nope. See previous comment.

different manufacturers use different conventions

Because programmers didn't check their Maths first, some calculators give wrong answers

More often than not even the same manufacturer uses different conventions

According to this video mostly not these days (based on her comments, there's only Texas Instruments which isn't obeying both Terms and The Distributive Law, which she refers to as "PEJMDAS" - she didn't have a manual for the HP calcs). i.e. some manufacturers who were doing it wrong have switched back to doing it correctly

P.S. she makes the same mistake as you, and suggests showing her video to teachers instead of just asking a teacher in the first place herself (she's suggesting to add something to teaching which we already do teach. i.e. ab=(axb)).

none of those two calculators is “wrong”

ANY calculator which doesn't obey all the rules of Maths is wrong!

Bugs are – by definition – unintended behaviour. That is not the case here

So a calculator, which has a specific purpose of solving Maths expressions, giving a wrong answer to a Maths expression isn't "unintended behaviour"? Do go on