6÷2(1+2)

SmartmanApps , 2 months ago (edited 2 months ago)

FACT CHECK 5/5

most people just dismiss that, because they “already know” the answer

Maths teachers already know how to do Maths. Huh, who would've thought? Next thing you'll be telling me is English teachers know the rules of grammar and how to spell!

and a two-sentence comment can’t convince them how and why it’s ambiguous

Literally NOTHING can convince a Maths teacher it's ambiguous - Maths teachers already know all the rules of Maths, and which ones you're breaking

Why read something if you have nothing to learn about the topic that’s so simple that you know for a fact that you are right

To fact check it for the benefit of others

At this point I hope you understand how and why the original problem is ambiguous

At this point I hope you understand why it isn't ambiguous. Tip: next time check some Maths textbooks or ask a Maths teacher

that one of the two shouldn’t even be a thing

Neither of them is a thing

not everybody shares your opinion and preferences

Facts you mean. The rules of Maths are facts

There is no mathematically true

There absolutely is! You just chose not to ask any experts about it

the most important thing with this “viral math” expressions is to recognize that

...they are all solvable by following the rules of Maths

One could argue that there should also be a strong connection between coefficients and variables (like in r=C/2π)

There is - The Distributive Law and Terms

it’s fine to stick to “BIDMAS” in school but be aware that that’s not the full story

No, BIDMAS and left to right is the full story

If you encounter such discussions in the wild you could just post a link to this page

No, post a link to this order of operations thread index - it has textbook references, proofs, memes, worked examples, the works!

SmartmanApps , 2 months ago (edited 2 months ago)

FACT CHECK 4/5

a solidus (/) shall not be followed by a multiplication sign or a division sign on the same line

There's absolutely nothing wrong with doing that. The order of operations rules have everything covered. Anything which follows an operator is a separate term. Anything which has a fraction bar or brackets is a single term

most typical programming languages don’t allow omitting the multiplication operator

Because they don't come with order of operations built-in - the programmer has to implement it (which is why so many e-calculators are wrong)

“.NET IDE0048 – Add parentheses for clarity”

Microsoft has 3 different software packages which get order of operations wrong in 3 different ways, so I wouldn't be using them as an example! There are multiple rules of Maths they don't obey (like always rounding up 0.5)

Let’s say we want to clean up and simplify the following statement …
o×s×c×(α+β)
… by removing the explicit multiplication sign and order the factors alphabetically:
cos(α+β)
Nobody in their right mind would remove the explicit multiplication sign in this case

This is wrong in so many ways!

1. you did multiplication before brackets, which violates order of operations rules! You didn't give enough information to solve the brackets - i.e. you left it ambiguous - you can't just go "oh well, I'll just do multiplication then". No, if you can't solve Brackets then you can't solve ANYTHING - that is the whole point of the order of oeprations rules. You MUST do brackets FIRST.
2. the term (α+β) doesn't have a coefficient, so you can't just randomly decide to give it one. It is a separate term from the rest
   Is there supposed to be more to this question? Have you made this deliberately ambiguous for example?
3. if the question is just to simplify, then no simplification is possible. You've not given any values to substitute for the pronumerals
4. (α+β) is presumably (you've left this ambiguous by not defining them) a couple of angles, and if so, why isn't the brackets preceded by a trig function?
5. As it's written, it just looks like a straight-forward multiplying and adding pronumerals except you didn't give us any values for the pronumerals meaning no simplfication is possible
6. if this was meant to be a trig question (again, you've left out any information that would indicate this, making it ambiguous) then you wouldn't use c, o, or s for your pronumerals - you've got a whole alphabet left you can use. Appropriate choice of pronumerals is something we teach in Maths. e.g. C for cats, D for dogs. You haven't defined what ANY of these pronumerals are, leaving it ambiguous

Nobody will interpret cos(α+β) as a multiplication of four factors

7. as originally written it's 4 terms, not 1 term. i.e. it's not cos(α+β), it's actually oxsxxx(α+β), since that can't be simplified. And yes, that's 4 terms multiplied!

From those 7 points, we can see this is not a real Maths problem. You deliberately made it ambiguous (didn't say what any of the pronumerals are) so you could say "Look! Maths is ambiguous!". In other words, this is a strawman. If you really think Maths is ambiguous, then why didn't you use a real Maths example to show that? Spoiler alert: #MathsIsNeverAmbiguous hence why you don't have a real example to illustrate ambiguity

Implicit multiplications of variables with expressions in parentheses can easily be misinterpreted as functions

No they can't. See previous points. If there is a function, then you have to define what it is. e.g. f(x)=x². If no function has been defined, then f is the pronumeral f of the factorised term f(x), not a function. And also, if there was a function defined, you wouldn't use f as a pronumeral as well! You have the whole rest of the alphabet left to use. See my point about we teach appropriate choice of pronumerals

So, ambiguity really hides everywhere

No, it really doesn't. You just literally made up some examples which go against the rules of Maths then claimed "Look! Maths is ambiguous!". No, it isn't - the rules of Maths make sure it's never ambiguous

IMHO it would be smarter to only allow the calculation if the input is unambiguous.

Which is exactly what calculators do! If you type in something invalid (say you were missing a bracket), it would say "syntax error" or something similar

force the user to write explicit multiplications

Are you saying they shouldn't be allowed to enter factorised terms? If so, why?

force notation that is never ambiguous

We already do

but that would lead to a very convoluted mess that’s hard to read and write

In what way is 6/2(1+2) either convoluted or hard to read? It's a term divided by a factorised term - simple

providing context that makes it unambiguous

In other words, follow the rules of Maths.

Links about various potentially ambiguous math notations

Spoiler alert: they're not

“Most ambiguous phrases and notations in maths”

e.g. fx=f(x), which I already addressed. It's either been defined as a function or as pronumerals, therefore nothing ambiguous

“Absolute value notation is ambiguous”

No, it's not. |a|b|c| is the absolute value of a, times b, times the absolute value of c... which you would just write as b|ac|. Unlike brackets you can't have nested absolute values, so the absolute value of (a times the absolute value of b times c) would make no sense, especially since it's the EXACT same answer as |abc| anyway!

In-line power towers like

Left associativity. i.e. an exponent is associated with the term to its left - solve exponents right to left

People saying "I don't know how to interpret this" doesn't mean it's ambiguous, nor that it isn't defined. It just means, you know, they need to look it up (or ask a Maths teacher)! If someone says "I don't know what the word 'cat' means", you don't suddenly start running around saying "The word 'cat' is ambiguous! The word 'cat' is ambiguous!" - you just tell them to look it up in a dictionary. In the case of Maths, you look it up in a Maths textbook

Because the actual math is easy almost everybody has an opinion on it

...and any of them which contradict any of the rules of Maths are demonstrably wrong

Most people also don’t know that with weak and strong juxtaposition there are two conflicting conventions available

...and Maths teachers know that both of them are made-up and not real things in Maths

But those mnemonics cover just the basics. The actual real world is way more complicated and messier than “BODMAS”

Nope. The mnemonics plus left to right covers everything you need to know about it

Even people who know about implicit multiplication by juxtaposition dismiss a lot of details

...because it's not a real thing

Probably because of confirmation bias and/or because they don’t want to invest so much time into thinking about stupid social media posts

...or because they're a high school Maths teacher and know all the rules of Maths

the actual problem with the ambiguity can’t be explained in a quick comment

Yes it can...

Forgotten rules of Maths - The Distributive Law (e.g. a(b+c)=(ab+ac)) applies to all bracketed Terms, and Terms are separated by operators and joined by grouping symbols

Bam! Done! Explained in a quick comment

SmartmanApps , 2 months ago (edited 2 months ago)

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:

https://programming.dev/pictrs/image/e95fab48-fc6f-4bb4-bf65-59d259f03283.png

The first is a fraction

The second is a division

The third is also a fraction

The last is a multiplication by a fraction

https://programming.dev/pictrs/image/528b1a25-398d-49e2-819f-36826077ab30.png

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)

SmartmanApps , 2 months ago (edited 2 months ago)

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...

https://programming.dev/pictrs/image/7d8c3846-3cd5-4d21-ad51-1debb41f0745.png

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

SmartmanApps , 2 months ago (edited 2 months ago)

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

dm319 , 3 months ago

@wischi "Funny enough all the examples that N.J. Lennes list in his letter use implicit multiplications and thus his rule could be replaced by the strong juxtaposition".

Weird they didn't need two made-up terms to get it right 100 years ago.

SmartmanApps , 3 months ago

Indeed Duncan. :-)

his rule could be replaced by the strong juxtaposition

"strong juxtaposition" already existed even then in Terms (which Lennes called Terms/Products, but somehow missed the implication of that) and The Distributive Law, so his rule was never adopted because it was never needed - it was just Lennes #LoudlyNotUnderstandingThings (like Terms, which by his own admission was in all the textbooks). 1917 (ii) - Lennes' letter (Terms and operators)

In other words...

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

...Terms/Products., as we do today in modern high school Maths textbooks (but we just use Terms in this context, not Products).

SmartmanApps , 3 months ago

Starting a new comment thread (I gave up on reading all of them). I'm a high school Maths teacher/tutor. You can read my Mastodon thread about it at Order of operations thread index (I'm giving you the link to the thread index so you can just jump around whichever parts you want to read without having to read the whole thing). Includes Maths textbooks, historical references, proofs, memes, the works.

And for all the people quoting university people, this topic (order of operations) is not taught at university - it is taught in high school. Why would you listen to someone who doesn't teach the topic? (have you not wondered why they never quote Maths textbooks?)

#DontForgetDistribution #MathsIsNeverAmbiguous

cypherpunks Mod , 3 months ago

I'm curious if you actually read the whole (admittedly long) page linked in this post, or did you stop after realizing that it was saying something you found disagreeable?

I’m a high school Maths teacher/tutor

What will you tell your students if they show you two different models of calculator, from the same company, where the same sequence of buttons on each produces a different result than on the other, and the user manuals for each explain clearly why they're doing what they are? "One of these calculators is just objectively wrong, trust me on this, #MathsIsNeverAmbiguous" ?

The truth is that there are many different math notations which often do lead to ambiguities.

In the case of the notation you're dismissing in your (hilarious!) meme here, well, outside of anglophone high schools, people don't often encounter the obelus notation for division at all except for as a button on calculators. And there its meaning is ambiguous (as clearly explained in OP's link).

Check out some of the other things which the "÷" symbol can mean in math!

#MathNotationsAreOftenAmbiguous

SmartmanApps , 3 months ago

did you stop after realizing that it was saying something you found disagreeable

I stopped when he said it was ambiguous (it's not, as per the rules of Maths), then scanned the rest to see if there were any Maths textbook references, and there wasn't (as expected). Just another wrong blog.

What will you tell your students if they show you two different models of calculator, from the same company

Has literally never happened. Texas Instruments is the only brand who continues to do it wrong (and it's right there in their manual why) - all the other brands who were doing it wrong have reverted back to doing it correctly (there's a Youtube video about this somewhere). I have a Sharp calculator (who have literally always done it correctly) and most of my students have Casio, so it's never been an issue.

trust me on this

I don't ask them to trust me - I'm a Maths teacher, I teach them the rules of Maths. From there they can see for themselves which calculators are wrong and why. Our job as teachers is for our students to eventually not need us anymore and work things out for themselves.

The truth is that there are many different math notations which often do lead to ambiguities

Not within any region there isn't. e.g. European countries who use a comma instead of a decimal point. If you're in one of those countries it's a comma, if you're not then it's a decimal point.

people don’t often encounter the obelus notation for division at all

In Australia it's the only thing we ever use, and from what I've seen also the U.K. (every U.K. textbook I've seen uses it).

Check out some of the other things which the “÷” symbol can mean in math!

Go back and read it again and you'll see all of those examples are worded in the past tense, except for ISO, and all ISO has said is "don't use it", for reasons which haven't been specified, and in any case everyone in a Maths-related position is clearly ignoring them anyway (as you would. I've seen them over-reach in Computer Science as well, where they also get ignored by people in the industry).

cypherpunks Mod , 3 months ago

Has literally never happened. Texas Instruments is the only brand who continues to do it wrong [...] all the other brands who were doing it wrong have reverted

Ok so you're saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today, and then also admit you know that it did happen with some other brands in the past?

But, if you had read the linked post before writing numerous comments about it, you'd see that it documents that the ambiguity actually exists among both old and currently shipping models from TI, HP, Casio, and Canon, today, and that both behaviors are intentional and documented.

There is no bug; none of these calculators is "wrong".

The truth is that there are many different math notations which often do lead to ambiguities

Not within any region there isn’t.

Ok, this is the funniest thing I've read so far today, but if this is what you are teaching high school students it is also rather sad because you are doing them a disservice by teaching them that there is no ambiguity where there actually is.

If OP's blog post is too long for you (it is quite long) i recommend reading this one instead: The PEMDAS Paradox.

In Australia it’s the only thing we ever use, and from what I’ve seen also the U.K. (every U.K. textbook I’ve seen uses it).

By "we" do you mean high school teachers, or Australian society beyond high school? Because, I'm pretty sure the latter isn't true, and I'm skeptical of the former. I thought generally the ÷ symbol mostly stops being used (except as a calculator button) even before high school, basically as soon as fractions are taught. Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?

SmartmanApps , 3 months ago (edited 3 months ago)

Ok so you’re saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today

You asked me what I do if my students show me 2 different answers what do I tell them, and I told you that has never happened. None of my students have ever had one of the calculators which does it wrong.

that both behaviors are intentional and documented

Correct. I already noted earlier (maybe with someone else) that the TI calculator manual says that they obey the Primary School order of operations, which doesn't work with High School order of operations. i.e. when the brackets have a coefficient. The TI calculator will give a correct answer for 6/(1+2) and 6/2x(1+2), but gives a wrong answer for 6/2(1+2), and it's in their manual why. I saw one Youtuber who was showing the manual scroll right past it! It was right there on screen why it does it wrong and she just scrolled down from there without even looking at it!

none of these calculators is “wrong”.

Any calculator which fails to obey The Distributive Law is wrong. It is disobeying a rule of Maths.

there is no ambiguity where there actually is.

There actually isn't. We use decimal points (not commas like some European countries), the obelus (not colon like some European countries), etc., so no, there is never any ambiguity. And the expression in question here follows those same notations (it has an obelus, not a colon), so still no ambiguity.

i recommend reading this one instead: The PEMDAS Paradox

Yes, I've read that one before. Makes the exact same mistakes. Claims it's ambiguous while at the same time completely ignoring The Distributive Law and Terms. I'll even point out a specific thing (of many) where they miss the point...

So the disagreement distills down to this: Does it feel like a(b) should always be interchangeable with axb? Or does it feel like a(b) should always be interchangeable with (ab)? You can't say both.

ab=(axb) by definition. It's in Cajori, it's in today's Maths textbooks. So a(b) isn't interchangeable with axb, it's only interchangeable with (axb) (or (ab) or ab). That's one of the most common mistakes I see. You can't remove brackets if there's still more than 1 term left inside, but many people do and end up with a wrong answer.

By “we” do you mean high school teachers, or Australian society beyond high school?

I said "In Australia" (not in Australian high school), so I mean all of Australia.

Because, I’m pretty sure the latter isn’t true

Definitely is. I have never seen anyone here ever use a colon to mean divide. It's only ever used for a ratio.

Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?

All my textbooks use both. Did you read my thread? If you use a fraction bar then that is a single term. If you use an obelus (or colon if you're in a country which uses colon for division) then that is 2 terms. I covered all of that in my thread.

EDITED TO ADD: If you don't use both then how do you write to divide by a fraction?

jordanlund , 5 months ago

Interesting that Excel sees =6/2(1+2) as an invalid formula and will not calculate it (at least on mobile). =6/2*(1+2) returns 9 because it's executing the division and multiplication left to right (6/2=3*3=9).

Google Sheets (mobile) does't like it either and returns an error. =6/2*(1+2) also returns "9".

SmartmanApps , 3 months ago

Excel and Google are both wrong. In fact, Microsoft excels 😂in this area, with Excel, the Windows calculator, and MathSolver all getting it wrong in different ways! dotnet.social/@SmartmanApps/111164851485070719

doctorn , 5 months ago

I don't see the problem actually.

1. Everything between ()
2. Exponents
3. multiply and devision
4. plus and minus
5. Always work from left to right.

==========

1. 1+2= 3

2. No exponents

3. • 6 devised by 2 (whether a fraction or not) is 3
   • 3 times 3 is 9

4. Nothing remains

Th4tGuyII , 5 months ago

The meme refers to the problem of handling implicit multiplication by juxtaposition.
Depending on what field you're in, implicit multiplication takes priority over explicit multiplication/division (known as strong juxtaposition) rather than what you and a lot of people would assume (known as weak juxtaposition).

With weak juxtaposition you end up 9 just as you did, but with strong juxtaposition you end up with 1 instead.

For most people and most scenarios this doesn't matter, as you'd never encounter such ambiguous equations outside of viral puzzles like this, but it is worth knowing that not all fields agree on how implicit multiplication is handled.

SmartmanApps , 3 months ago

The meme refers to the problem of handling implicit multiplication

There's no such thing as implicit multiplication.
dotnet.social/@SmartmanApps/110925761375035558

SmartmanApps , 3 months ago

I don’t see the problem actually.

Everything between ()

You recreated the problem right there - ignored The Distributive Law. a(b+c)=(ab+ac). i.e. 2(1+2)=(2x1+2x2). After step 1 - solving brackets - all that's left is 6/6. dotnet.social/@SmartmanApps/110819283738912144

Donebrach , 5 months ago

Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma. At the end of the day it seems mathematical notation is just as flexible as any other facet of written human communication and the real answer is “make things as clear as possible and if there is ambiguity, further clarify what you are trying to communicate.”

Th4tGuyII , 5 months ago

Pretty much. While it's worth knowing that not everyone agrees on how implicit multiplication is prioritised, anywhere that everyone agreeing on the answer actually mattered, you wouldn't write an equation as ambiguous as this one in the first place

SmartmanApps , 3 months ago (edited 3 months ago)

It's not ambiguous. People who say it is have usually forgotten The Distributive Law or Terms, or more commonly both!

SmartmanApps , 3 months ago

Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma.

Not even remotely similar. Maths rules are fixed. The order of operations rules are at least 400 years old.

mathematical notation is just as flexible as any other facet of written human communication

No, it isn't. The book "A history of mathematical notation" is in itself more than 100 years old.

Donebrach , 3 months ago

Wow neat, and yet the thread was full of people going back and forth about how the equation can be misinterpreted based on how the order of operations can be interpreted. Thanks for your months later input though.

SmartmanApps , 3 months ago

I only just found the thread yesterday. There's only 1 "interpretation", and the only back and forth I've seen about interpretations is about implicit multiplication, which isn't a thing, at all - it's people conflating The Distributive Law and Terms dotnet.social/@SmartmanApps/110925761375035558

Donebrach , 3 months ago

So you are saying exactly what I said; people can misinterpret things that other people have written. Good job. Thanks again for stopping by a 3 month old thread about a dumb meme.

SmartmanApps , 3 months ago

So you are saying exactly what I said; people can misinterpret things that other people have written

No, I'm not. They're "misinterpreting" something that isn't even a rule of Maths. There's no way to misinterpret the actual rules, there's no way to misinterpret the equation. There's no alternative interpretations of the notation. Someone who didn't remember the rules literally made up "implicit multiplication", and then other people argued with them about what that meant. 😂

atomicorange , 3 months ago

You look like a real idiot here. I really suggest you actually read the article instead of “scanning” it. You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up. Implicit multiplication is not the controversial part of this equation, which you would know if you read the article and understood what people in this thread are even talking about. Stop spamming your shitty blog and just. Read. The. Article.

SmartmanApps , 3 months ago

read the article instead of “scanning” it.

I stopped reading as soon as I saw the claim that the right answer was wrong. I then scanned it for any textbook references, and there were none (as expected).

You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up

Funny that you use the word "term", since Terms are ONE of the things that people are referring to when they say "implicit multiplication" - the other being The Distributive Law. i.e. Two DIFFERENT actual rules of Maths have been lumped in together in a made-up rule (by people who don't remember the actual rules).

BTW if you think it's not made-up then provide me with a Maths textbook reference that uses it. Spoiler alert: you won't find any.

Implicit multiplication is not the controversial part of this equation

It's not the ONLY controversial part of the equation - people make other mistakes with it too - but it's the biggest part. It's the mistake that most people have made.

shitty blog

So that's what you think of people who educate with actual Maths textbook references?

Read. The. Article.

Read Maths textbooks.

atomicorange , 3 months ago (edited 3 months ago)

Skimmed your comment and it’s wrong. Let me know if you ever decide to read the article instead of arguing against an imagined opponent.

SmartmanApps , 3 months ago

Skimmed your comment and it’s wrong

So tell me where it's wrong.

Let me know if you ever decide to read the article instead of arguing against an imagined opponent

There's nothing imaginary about the fact that he claimed it's ambiguous, and is therefore wrong. Tell me why I should read a wrong article, given I already know it's wrong.

atomicorange , 3 months ago

Why would I bother to read your comment when I know it’s wrong?

SmartmanApps , 3 months ago

You haven't told me where it's wrong yet. I already said where the article is wrong.

atomicorange , 3 months ago

I’ll read your comment when you read the article. Challenge level: impossible.

SmartmanApps , 3 months ago

I stopped reading it when I found it was wrong., and said what was wrong about it. You have still not said where mine is allegedly wrong. I'll take that as an admission that you're wrong then.

atomicorange , 3 months ago

I stopped reading your comment when I saw you still hadn’t read the article.

SmartmanApps , 3 months ago

I'll take that as an admission that you're wrong then. Bye now.

P.S. someone else just provided me with even more things which are wrong in it. Even more glad I didn't waste time reading the rest of it.

atomicorange , 3 months ago

I’ll take that as admission that you’re too unintelligent to read.

SmartmanApps , 2 months ago

Let me know if you ever decide to read the article instead of arguing against an imagined opponent

Read it, wasn't imagined. In fact it was even worse than I thought it would be! Did you not notice about how a blog about the alleged ambiguity in order of operations actually disobeyed order of operations in a deliberately ambiguous example?
I wrote 5 Fact check posts, starting here - you're welcome.

atomicorange , 2 months ago

Wow you’re a slow reader LOL

SmartmanApps , 2 months ago

you’re a slow reader

I see you like to use made-up "facts", just like the blog does. Is that the best you can come up with after repeatedly insisting I should read it? (which yes, would've been a huge waste of time, exactly as I said, had I not turned it into a positive use of time by writing a fact check about it. Alleged fake news turns out to be... fake - who would've thought? Oh that's right, me :-) )

I’ll read your comment when you read the article

So, did you read it now? Or are you a "slow reader" and I need to wait longer for your responses?

uskok , 5 months ago

I agree with your core message, that the issue is caused by bad notation. However I don't really see why you consider implicit multiplication to be the sole reason. In my mind, a/bc is equally as ambiguous as a/b*c. The symbols are not important.

You don't even consider this in your article, instead you seem to take the position that the operations are resolved from left to right. This idea probably comes from programming languages, as they commonly use this convention, but I haven't seen this defined in mathematics anywhere. I'm open to being wrong here, so if you can show me such a definition from an authoritative source (maybe ISO) I'd be thankful.

As it stands, you basically claim "the original notation is ambiguous, but with explicit × the answer is obviously nine, because my two calculators agree", even though you just discounted calculator proofs. By the way, both calculators explicitly define this left-to-right order in their documentation.

The ISO section 7.1.3 you quoted is very reasonable and succinct, and contradicts your claim that explicit multiplication sign removes ambiguity. There would be no need for this section if a left-to-right rule existed.

SmartmanApps , 3 months ago

a/bc is equally as ambiguous as a/b*c

It's not ambiguous at all. By the definition of Terms - ab=(axb) - a/bc is 2 terms and a/bxc is 3 terms. If we were to write it in fraction form (to illustrate the difference), in the former c is in the denominator, but in the latter it's in the numerator, hence a different answer. dotnet.social/@SmartmanApps/110846452267056791

you seem to take the position that the operations are resolved from left to right... but I haven’t seen this defined in mathematics anywhere

It applies to operators, or more precisely division. When doing the divisions, you have to do them left-to-right, but other than that each of the operators can be done in any order. i.e. it doesn't matter what order you do the multiplications in, as long as you do them before the additions and subtractions. Unfortunately I've seen many people misremember left-to-right as an overarching rule, rather than only applying to division.

Poem_for_your_sprog , 5 months ago

Just write it better.

6/(2(1+2))

Or

(6/2)(1+2)

That's how it works in the real world when you're using real numbers to calculate actual things anyways.

SmartmanApps , 3 months ago

Just write it better.

6/(2(1+2))

If you really wanted extra brackets it'd be 6/(2)(1+2). Of course, since there's only 1 term in the first brackets they're redundant, hence 6/2(1+2) is the fully simplified form, and is the way it's written in Maths textbooks.

MinekPo1 , 5 months ago

I've seen a calculator interpret 1 ÷ 2π as ½π which was kinda funny

wischi OP , 5 months ago

All calculators that are listed in the article as following weak juxtaposition would interpreted it that way.

SmartmanApps , 3 months ago

And they're all wrong dotnet.social/@SmartmanApps/111164851485070719

SmartmanApps , 3 months ago

An e-calculator I'm guessing? (either that or Texas Instruments) Desmos USED TO interpret that correctly, but then they made a change with automatically turning division into fractions and broke it (because if you've specified division then it's not a fraction) dotnet.social/@SmartmanApps/111164851485070719

MinekPo1 , 3 months ago

I believe it was a app , yes

kuneho , 5 months ago

isn't that division sign I only saw Americans use written like this (÷) means it's a fraction? so it's 6÷2, since the divisor (or what is it called in english, the bottom half of the fraction) isn't in parenthesis, so it would be foolish to put the whole 2(1+2) down there, there's no reason for that.

so it's (6/2)*(1+2) which is 3*3 = 9.

the other way around would be 6÷(2(1+2)) if the whole expression is in the divisor and than that's 1.

tho I'm not really proficient in math, I have eventually failed it in university, but if I remember my teachers correctly, this should be the way. but again, where I live, we never use the ÷ sign, only in elementary school where we divide on paper. instead we use the fraction form, and with that, these kind of seemingly ambiguous expressions doesn't exist.

Spacehooks , 5 months ago

It seems Americans are taught pemdas and not bodas.

I Looked up doing factorials and
n! = n(n – 1) is used interchangeably with
n! = n*(n – 1)

So Americans will multiply anything first. This is why I put 6 ÷ ( n*(n – 1)) in excel to avoid confusion.

kuneho , 5 months ago

I Looked up doing factorials and n! = n(n – 1) is used interchangeably with n! = n*(n – 1)

yeah, the way I have been taught is that either you put the multiplication sign there or not, it's the exact same, there's absolutely no difference in n(n-1) and n*(n-1). in the end, you treat it like the * sign is there and it's just matter of convenience you can leave it off.

SmartmanApps , 3 months ago

there’s absolutely no difference in n(n-1) and n*(n-1)

There is - the first is 1 term and the 2nd is 2 terms. Makes a difference if it's preceded by a division.

it’s just matter of convenience you can leave it off.

It's a matter of how many terms as to whether it's there or not.

SmartmanApps , 3 months ago

I Looked up doing factorials and n! = n(n – 1) is used interchangeably with n! = n*(n – 1)

Yeah, there's a problem with some lazy textbook authors, which I talked about
here. A term is defined as ab=(axb), and yet many textbooks lazily write it as ab=axb, which is fine if that's the whole expression, but NOT fine if the expression is a/bc (a/(bxc) and a/bxc AREN'T the same thing!), and so we end up with people removing brackets prematurely and getting wrong answers. In other words, in your case, only n!=n(n – 1) and n!=(nx(n – 1)) can be used interchangeably.

Ultraviolet , 5 months ago

The ÷ sign isn't used by "Americans", it's used by small children. As soon as you learn basic mathematical notation in your introductory algebra class, you've outgrown the use of that symbol.

Lazhward , 5 months ago

Children here in the Netherlands use : as a divisor symbol. I don't know whether the ÷ sign is particularly American, but it's not universal.

SmartmanApps , 3 months ago

The ÷ sign isn’t used by “Americans”, it’s used by small children

I don't know where you're from, but it's used universally in Australia - textbooks, calculators, all ages - and from what I've seen the U.K. too.

SmartmanApps , 3 months ago

written like this (÷) means it’s a fraction?

No, that means it's a division. i.e. a÷b. To indicate it's a fraction it would need to be written as (a÷b). i.e. make it a single term. Terms are separated by operators and joined by grouping symbols (such as brackets or fraction bars).

put the whole 2(1+2) down there, there’s no reason for that.

There is - it's a single bracketed term, subject to The Distributive Law. i.e. the B in BEDMAS.

SkiDude , 5 months ago

It’s also clearly not a bug as some people suggest. Bugs are – by definition – unintended behavior.

There are plenty of bugs that are well documented. I can't tell you the number of times that I've seen someone do something wrong, that they think is 100% right, and "carefully" document it. Then someone finds an edge case and points out the defined behavior has a bug, because the human forgot to account for something.

The other thing I'd point out that I didn't see in your blog is that I've seen many many people say they need to evaluate the 2(3) portion first because "parenthesis". No matter how many times I explain that this is a notation for multiplication, they try to claim it doesn't matter because parenthesis. screams into the void

The fact of the matter is that any competent person that has to write out one of these equations will do so in a way that leaves no ambiguity. These viral math posts are just designed to insert ambiguity where it shouldn't be, and prey on people who can't remember middle school math.

wischi OP , 5 months ago

Regarding your first part in general true, but in this case the sheer amount of calculators for both conventions show that this is indeed intended behavior.

Regarding your second point I tried to address that in the "distributive property" section, maybe I need to rewrite it a bit to be more clear.

SmartmanApps , 3 months ago

Regarding your second point I tried to address that in the “distributive property” section,

When I discovered this comment I went to read it, and yes, it's true you discussed the Distributive Property, however, what these people are talking about is The Distributive Law which isn't the same thing (though people often call it the wrong name), and makes the question completely unambiguous. You literally can't move on from the "B", Brackets, in the rules until there are no brackets left - the B is literally short for "solve Brackets" (every letter is "solve (something)"), and so anyone who does the division before solving the brackets has just violated the order of operations rules.

SmartmanApps , 3 months ago

No matter how many times I explain that this is a notation for multiplication

It ISN'T a notation for multiplication - it's a notation for a factorised term, and if you ignore The Distributive Law going back the other way then you just broke the factorised term dotnet.social/@SmartmanApps/110886637077371439

any competent person that has to write out one of these equations will do so in a way that leaves no ambiguity.

This one already does have no ambiguity.