UKH

No calculators allowed.

6 ÷ 2 (1 + 2)

Strange to ask(?).

Surely it's one, or am I missing something?

Depends on whether you read this as:

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

or

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

The correct answer can be sought by replacing / with * thus:

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

9 by the usual Bidmas convention.

It isn't a tricky maths question, it's a slightly ambiguous maths question.  An attempt to start a bunfight about BODMAS and mathematical 'grammar' no better than the numerical equivalent of a Peak/Peaks troll. 

Usually when you're doing a sum like that you're trying to solve a problem, so if you genuinely don't clearly understand what the question means the answer is probably to go back and look at the original problem again.

Eh?

Brackets first.

Edit: thought as much ... https://www.quora.com/What-is-the-solution-of-6-2-1-2-There-are-2-answers-T...

You're saying the maths is ambiguous. Blasphemy.

Comparing maths to grammar. Wash out your potty mouth this instant.

Yes. 1+2=3

And add some more brackets to make it obvious to whoever is working on it next (even if thats you in six months time).

Mathematical notation needs its own grammar/conventions in order to avoid ambiguity.

The fact that this argument over the conventions comes up so often just shows that the conventions have passed their sell by date for usefulness. Anyone wanting to avoid confusion would not write the expression as the OP has done.

6/2(1+2)

B => 6/2(3)

D =>3(3)

M => 9.

Yes, but division before multiplication under BODMAS. So, under this convention,

6 ÷ 2 (1 + 2) (which means  6 ÷ 2 x (1 + 2) ) = 6 ÷ 2  x 3 = 3 x 3 = 9 

School maths teachers are the best people to ask about this.

I suspect the author of that quora question had a broken scientific calculator.

The convention is that you do D and M in the order they come. Likewise A and S. 

In practice (and interestingly) this gives the same answers as doing D before M, but reminds you that 24/4/2 = 2 (not 12)

But it really does matter with A and S. 

10 - 2 + 3 = 11 (not 5)

Kids often get this wrong after being taught Bidmas having got it right before!

You may want to reconsider that!

Oops!

3

Ths might explain why most maths departments I've been in recently have stopped using Bidmas. 

So what have they replaced it with? Lots of brackets? And, if so, do they retain all these brackets for algebra? a+(bc) ?

They still teach order of operations but without reference to the likes of Bodmas, bidmas, bops or pemdas

It matters for D and M also.

10 x 6 ÷ 2

• If done left to right you get (30)
• If done right to left you get (30)

Now flip the question

10 ÷ 6 x 2 (should have picked easier numbers)

• If done left to right you get (3.333)
• If done right to left you get (0.833)

It's the same with A and S. If the S is first, order matters. If the A is first it doesn't.

Take your example:

10 - 2 + 3

Flip it and the order no longer matter:

10 + 2 - 3

• If done left to right you get (9)
• If done right to left you get (9)

So exactly the same conventions but just without the mneumonic?

I think so, I got told off by the kids for referring to bidmas yesterday. I'll inquire next time I'm in. I'm thinking relying on  mnemonic, that can be misleading, doesn't fit with their move towards a more mastery based approach. 

See? Including recently retired ones.

My point was that, in both cases, doing D before M gives the same answer as doing them in the order they come (left to right as you put it).

Surely the question itself is wrong - at least in part.

Shouldn't there be something between the "2" and the "(" - such as a multiplication symbol for example.

It's implied.

It is why you are better off using a denominator if you want to divide one expression by another expression. No ambiguity about what is being divided by what.

It does seem like a reasonable agument for abolishing the use of ÷, apart from anything else kids with poor vision struggle distinguishing between ÷ and +

Yes, I would ban the division symbol* and the use of a slash for division. Only a horizontal line permissible. All these ambiguities then go away.

*Except, perhaps, when dividing one fraction by another.

Due to the conventions at GCSE that would probably result in more confusion. A question requiring kids to simplify an expression wouldn't acvept 6 x a if the answer was 6a.

Although pre (2015) for a question where the answer is y an answer of 1y would be acceptable, but shouldn't be now?? There was also a difference between what is in the specification and the guidance for those marking the exam??? 

I did my Maths A-level in the early 70's. Conventions have evidently changed.

I agree. I never needed a mnemonic. Basically, if you want to be clear you need to use brackets.

Before calaculators? 

Yes, but probably less than in other subjects. You'll be claiming the difference between 2 and 9 is 7 next 🤔

It has made me think about the validity of some research that has looked at the standard of A level Maths over the decades. The general consensus seems to be that it has been pretty constant over the last 30 or so years, but declined* from the 1960's to early 1990's. This is often done by looking at how l current students perform on "old' papers, but if conventions have changed is that fair?

*probably to be expected given the increase in candidates. 

It's th sort of ambiguous question that people ask when they want to garner comments on social media platforms.

Only a fool would calculate anything of real importance without sufficient understanding of the question to safely disregard theoretical rules.

Or it's just a fun little Maths question. Lighten up.

which attempt is to be welcomed, as BODMAS is clearly a hierarchy of colonial value whereas the answer should be derived from a multicultural and decolonised view informed by the student voice, instead of methods from the overseas expansion of British and European empires and justified using narratives of white, able and heteronormative superiority.

Parklife

That’s where you vary the length of the horizontal line to make clear it’s one fraction divided by the other 

Yes, of course, but for the purposes of teaching kids, it can be confusing.

Quite so - lucky that it does.

Or that too many people don't know them or can't apply them.

Yes but somebody wanting to create confusion for clicks would write it exactly like that.

How about looking at 2 fundamental techniques and what stage/year they've been taught at over the years:

• Differentiation
• Integration

I understand that even Differentiation isn't taught until A/S level now - is that correct?

That doesn't really work, other things may be taught instead.

I think there is a genuine issue with BIDMAS highlighted by this example. The convention of leaving out multiplication symbols makes it look intuitively as if the multiplication is a single "object" which it is natural to do first. The problem does not arise when children first learn BIDMAS because with basic arithmetic you can't leave them out  (32 obviously won't mean 3×2) but then there are problems later with leaving them out with algebra and, as here, multiplying a bracket.

Yes.

That hardly seems fair if the content or emphasis has simply changed.

Calculus is fundamental to everything. I'm very surprised to read what Michael Hood has written here, assuming it's true.

What kinds of other things do you think might be being taught instead?

Differentiation wasn't in standard O Level when I did it in 1979. It was just in the Additional Maths which few pupils took.

I certainly didn't do calculus for my 'O' Level maths in 1982 - they came as a nasty shock the next year for 'A' Levels!

As I remember, there was a lot of emphasis on matrices, presumably on the basis that they were needed for those new fangled computer things.

Well calculus has been A/S or A level since at least the early 90s and probably much longer, so nothing new there 

Computing is an obvious one. 

It was in 1977 when I did O levels. Just goes to show the rate of change!

Not sure that is a reliable indicator. GCSE maths was certainly made 'harder" in 2015, but I'm not convinced it did anything to raise standards. Intially the grade boundaries were lowered and subsequently the papers got easier, similar when AS was introduced in 2001, it took a decade to undo the damage done by the the Core 2 paper that year. I could teach multiply by the power and reduce the power by 1 to primary school kids, but what's the point.

Which is probably why, on the other hand, hardly any pensioners have a clue how to answer KS2 questions. 

If anyone did their A level pre 1960, I'm confident it didn't include Dijkstra's Algorithm. 

Sadly this may not be the case.

It wasn't long ago (5 years ago maybe) I caught an apprentice using excel and calculating each row with a physical calculator and typing the answers in by hand.

Which is the same at GCSE level now, including the bit about very few kids doing Additional Maths. 

I know! I have had many conversations with students along the lines of

 "Can you use Excel"

"Yes"

"Err..."

But presumably schools do some stuff with computing that wouldn't have been there 20, let alone 50nyears ago.

I just had to look that up. It was first published in 1959, so you are probably right!

It was just an observation. Lighten up

I too did Maths O level in 1979 and was not exposed to differentiation until the start of A levels.

At the time there was a big push towards a 'New Maths' syllabus, meaning we did stuff like matrices, calculations in bases other than 10, set theory, and even topology, often working in groups rather than in a more traditional teacher-facing structure.

I, for one, thought the newer ideas were far more interesting, and it was disappointing when A levels focused more on calculus. That probably explains why I was hugely more keen on pure maths than applied maths when it came to undergraduate study.

I did O-level in 74, differentiation definitely in. Don't think integration was, but it was certainly early in first year of A-level.

But I did SMP maths which really went off into lots of theory rather than drilling practice, A-level further maths was tougher than anything in 1st year degree; mechanics (IIRC centre of percussion of irregular objects was tough; e.g. cricket bat), group & ring theory, boolean algebra & logic, differential equations, integrating to find volumes etc. Edit: forgot about all the matrices stuff, inverses and all that.

However, you didn't have to remember formulae (although it was quicker), the SMP log tables (remember those) had all the equations etc and you were allowed those in the exams, which really weren't what you could remember, but properly testing what you could actually do.

I think posters different experiences illustrates that this was pre national curriculum which gave schools much more freedom to choose what they taught. I'd also point out that the fact you took O levels put you very much in the minority (top 15-30%?). Now virtually every student takes GCSE maths and over 70% at the higher level. 

So exactly the same point as it is today 

Not 70 %. And falling up to 2019.....

https://support.ocr.org.uk/hc/en-gb/articles/360038360392-GCSE-Mathematics-...

I didn't realise it had changed so much since 2015. Obviously my retirement has resulted in a dramatic drop in standards. 

From (very sketchy) memory, some exam boards did have calculus on the syllabus in 82, - Cambridge iirc. SMP (what I took) didn't. What you were taught depended on the school you went to.

I was in an early comprehensive, I think most took O-level maths but some might have taken CSE. Certainly the school had a mix of O-levels and CSEs, and we were streamed in most (but not all) subjects.

Intresting, have you got a current name for it, just me being nosey. I went to a comp. in a market to a decade later, I was very much in the minority (only top set did O level)  recently the (equivalent) schools results have always been above national average. Friends your age had to take the 11+ my understanding was that about 15 to 20% passed and went to the High School, but even then some of them took CSE maths rather than O level. I'm not sure if anyone who went to the secondary modern did O levels 

I did my maths O level in 1978, a year before I did the rest, in a comprehensive that had only recently been a secondary modern. I am impressed that you can remember what you did.

Over the years I have realised that anything of any substance is interesting, even statistics and number theory.

I hadnt realised that either.....when I was a maths teacher (2011 - 14) more than 1/2 of the cohort would take higher paper; over the last few years whilst tutoring, it appeared that even those in set 3 of a 6 set cohort were taking foundation level. The last couple of years have been somewhat buggered up by the pandemic, but I didnt realise the split was so biased toeards foundation until i googled it as a result of reading your previous post......

Anecdotally, several of the students I've been tutoring at foundation level have stated that (parents too...) "I dont want to do higher as there's no point because I dont want to do maths at A level so I just need a level 5". The inevitable question i ask then is "oh, what do you intend studying at A level?" has in one extreme case elicited the answer "Physics, Biology and English". There seems to be a certain communication issue within the school if a potential A level physics student thinks level 5 maths is going to be enough, especially when she should have targeted a 7 and aspired to an 8.......

I went to university in 1974.  I think it was during the first year that some of the science students bought Sinclair calculators as they had become cheap then, particularly if you bought the kit version.  We even went round to someone's room after evening meal to see one it was such a novelty!

Amazing that 5 years later I was able to program a moon landing game on a Casio scientific by 1979.  It could even store programs on an audio cassette and "play a little melody".

Beauchamp College, Oadby, Leicester.

Leicestershire went comprehensive (must have been 1971 at latest) whilst the city stayed Grammer & Secondary Modern.

Well, I retired as a maths teacher in 1984 (to take a different career path, of course - I'm not 100). My answer is clear - blame the OP for setting an ambiguous question.

Later in my career, when drafting guidance for teachers on the organisation of educational visits (school trips) - amongst other things - I learnt one important rule. If two readers interpret the same statement differently, it's almost always the writer's fault.   

Martin

That makes a change from the opposite request I used to often get "I want my son/daughter to change tier because it's easier to get a 5 on the higher“

I'd be reluctant to let them on with a 6. My figures might be out of date, but even a 7 indicates they got about half the GCSE questions wrong. I always though the talk of a jump between GCSE and A level was misplaced, the jump was for those students who didn't understand (at least half) GCSE maths. A you say this has been exacerbated by covid and not help by the inflated grades that were then awarded. 

Ah, if the intake has remained similar that goes some way to  explaining your experience. I'll admit to being skeptical that any comprehensive school had a majority of students sitting O levels, but seeing their recent results it makes sense. As a rough indicator I think the current EBaac % is a reasonable indicator of historic O level results. So national average is currently mid 20`s, Beauchamp's is 67% (there is only one selective/grammar school in Cumbria and they can only manage 54%). 

Is it still one of the few areas that still has middle schools? 

I don't believe there's any ambiguity. The answer is 1. The 2 right next to the bracket means you double what's in the bracket and then divide 6 by that amount. The use of the divide symbol is convenient when typing, but if you were writing this with a pen, you'd put 6 on top of a horizontal line and the rest under it.

for that to be so (ie to have the 6 as the numerator and the rest as the denominator it would have to be written 6 ÷ (2(1+2)).

You are correct in that there is no abiguity. The order of operations is to sum the 2 numbers in the bracket; the sum then becomes 6 ÷ 2 x 3. As opposed to  6 ÷ (2x3).

Not really a maths question though is it, other than in a trivial sense?  It's a personal bugbear of mine when people refer to simple arithmetic as "maths".  It's a bit like someone refering to electric engineering as changing a fuse in a fuse box.  Actual maths has pretty complex and (for me) difficult to parse and understand symbolic relationships that express something about the world in some way.  A very simple example might be this activation function (hyperbolic tangent) for a neural network:

tanh(z) = (e^z - e^-z)/(e^z + e^-z)

arithmetic ≢ mathematics

arithmetic ⊂ mathematics

And a small one too.

Tell me more about how tanh(z) is an activation function for a neural network

https://www.desmos.com/calculator/jwjfuz4g9v

I teach hyperbolic functions at A Level Further Maths and am always on the look out for modern usage cases to start the lesson with 'Whats the point of ...'

This is a good point that there isn't really a jump between GCSE and A Level if the students understand all of GCSE maths, my experience is that almost all prospective A Level students don't understand most of GCSE Maths, so there is a jump.

We let GCSE grade 6 students onto the A Level Maths course, and I have one who is currently in Y13 who just got an A* in a very tough mock paper so it is possible.  Students get lower grades at GCSE for all sorts of reasons, and some of those can be fixed with hard work and good teaching.  The problem is what do you do with those students who don't make the leap from GCSE (bumbling along doing a bit in lessons and nothing else) to A Level (really hard work and hours of independent study just to keep up).  We move them to an AQA Level 3 Maths Studies course, which is also called Core Maths, but not all colleges have this option.

Yes, most of my A level teaching was from 1997 to 2010. At that time the argument used to about kids who had got a C at GCSE. I remember one getting an A at A level, but they had been home educated previously and their parents didn't believe in the use if calculators.even for the exam! A handful of others went from a C to a good A level grade, as you say, the common denominator there was consistent  hard work. Does this imply a C from that era is equivalent to. 6 not 4/5? 

AS Accounting was my preferred option for those kids who liked doing calculations, but couldn't cope with the more abstract elements of  A level maths. Not the most exciting course to teach though. 

Now you mention it, I think that's what I remember too. I did SMP 'O' Level then switched to a very traditional board for Pure and Applied Maths 'A' Levels and found the first term very difficult as I adjusted.

Ok, I've let this run long enough. 

The prize goes to Robber Durran. I think he was the only response that gave both the right answer without a mistake in their rational. Sorry if I missed somebody.

So we solve as follow.

BODMAS

6 ÷ 2 (1 + 2)

Multiplication is the default operator and can be written as x or a . or nothing. It means the same thing however it's written. Good explanation here:

https://www.google.com/url?sa=t&source=web&rct=j&url=https://te...

So we have, Brackets first.

6 ÷ 2  x 3

Then ÷ and x have equal priority so we solve in the order they are written.

3 x 3 = 9

Or that’s it’s over 50 years since they were taught the subject and much of it will be forgotten even if taught that area of mathematics to start with.

Where as I didn’t do anything with matrices at O level (1982) or A level (1984).  First encountered them in my Maths degree.  Our exams board was JMB.

I’m not having that, you can’t give an ambiguous problem then choose a solution and declare a winner.

How would you evaluate 1/2pi ?

By your logic it is half times pi. But it just isn’t.

I completely agree. Let's make it maths instead of arithmetic. The OP's expression is a/b(c+d). To any mathematican that evaluates as a/(bc+bd), not (a/b)(c+d).

Although several posters on here can remember their maths lessons from decades ago? It does however highlight a concern of mine that while last minute cramming into students short term memory is effect at improving exam results, it's otherwise pointless as they will have forgotten almost everything 5 weeks later. 

That was the point I was trying to make. Mr Duran had made the point that it was unfair to make current students take "old" exam, I was trying to make the point that the opposite was true as well. 

But it might be. This is why a slash should never be written for division (or, if it is, brackets should be used to avoid any ambiguity).

If someone meant half pi they would write 0.5pi, not 1/2pi

Neural networks are the basis of some of the most impressive achievements to date of machine learning. In particular, AlphaFold from DeepMind has solved, better than anyone else, the problem of predicting how proteins fold (their so-called tertiary structure), given the sequence of amino acids from which they are built (their so-called primary structure). The way proteins fold determines their function, and proteins are the basis of the molecular machinery of life, having been created through a process of transcription and translation from DNA.

It is a mind-blowing achievement of tremendous utility, for example for the development of drugs that enhance or inhibit biochemical processes. At the biomolecular level, shape is a major contributor to function in that it will determine what can interact with what, get discarded or hang around along enough to bring about an action.

You can think of a neural network as a prediction/classification system. Given, say, an image of a digit, it will try to classify it as one of the options 0-9. Given loads of labeled images like this, where it can compare its prediction with the actual known truth, it can 'learn' how to make accurate  predictions more often with unlabelled images.

A neural network comprises layers of nodes, with an input layer, an output layer, and some number of 'hidden' layers in between. Signals are passed from every node in one layer to every node in the next layer, at a level determined by, among other things, an activation function. Ultimately, at the output layer, depending on the level of the signal at each node there, the network will classify the input as one of a prescribed collections of things, such as the digits from 0 to 9in the example above.

In order to learn ie to be better able to make a correct classification of an unlabelled input, the network needs to be able to feed back the level of discrepancy between what it predicted and what was the actual truth. This requires it to do something called back-propagation. For this to lead to learning, the activation function needs to be non-linear, so that its derivative can be related to particular inputs. 

The tanh function meets this criterion. It also ensures that, whatever this inputs, the outputs of any layers are constrained to the range -1 to +1 an d will be centred around zero, which is useful.

It is not the only option. Several other non-linear functions could be used. Like any of them it has disadvantages, in its case that for large absolute values of x its derivative does not change much, which does not help learning in that it cannot discriminate between any such inputs.

For a general idea of what neural networks are about, try the neural network channel of 3blue1brown: 

https://www.youtube.com/playlist?list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi 

For AlphaFold, see:

https://www.deepmind.com/research/highlighted-research/alphafold 

The division sign in the OP is no better though, and ablackett's argument still holds.

If you wrote it as 1÷2π, again by montydon's logic it would be half times pi and again it clearly isn't.

I wouldn't. 

How would you write 1/7pi ?

But it might be. And this is why neither the slash nor the division sign should be used. ALWAYS a horizontal line!

1 

-

 7 pi 

Obviously better spaced, or 1/(7pi)

Which is fine if writing by hand but not in Excel or most programming languages 

Yes, but that missed the point I was making!

I've never used Excel or a programming language! 

How would they interpret 1/2pi ?

Not at a PC but I am pretty sure Excel would interpret it as (1/2)*pi. I.e. strictly by operator precedence. I would always add brackets to avoid things going wrong 

R says

Excel is a pain and says #NAME?

If you write it as =1/2*PI()  then, as you supposed, it gives 1.57....

I assume =1/2*pi() gives the same?

Yes (and could UKC enable reaction emojis like I see elsewhere eg on Teams, so we don't need to clutter up threads with replies like this).

So computers do Bidmas. That's a relief!

Much safer to go back to Reverse Polish notation as implemented in Sinclair calculators 😁

Removes ambiguity IIRC (but I can't remember how it works, I'd need to look it up 😂)

And yet I notice you're not doing that in your posts here.  So I guess that isn't always an option.

If you mean one over seven pi, probably just like that. 
Or if I wanted to be completely clear possibly 1/(7π).

If you mean a seventh of pi I'd write it π/7.

Only because I don't know how to on my phone or laptop. I would put brackets in to avoid ambiguity or misunderstanding.

Yes, but I want rif's answer!

Who remembers reverse Polish as used on 1970s HP calculators?

I would use brackets depending on what was intended.  If language is constrained in functionality then break up over multiple lines.

(Last studied maths in 1972 but still use my O level trigonometry etc)

You can't do it in a post on here.  If you wanted to include a 'correct' algebraic formula in a post you'd probably be best posting it as a photo, or muddle through with crude 'AscII art' like:

1
_

7π

same as deepsoup: if you mean one-seventh of pi, write pi/7

So 0.5pi but pi/7 ?

Well, I do, because I still use a 1980s HP. RPN is great for complicated calculations, though I wouldn't fancy trying to teach it to a mixed-ability class.

Hate to disappoint you but "computers" dont.

The majority of programming languages do however some dont. Especially when you extend Bidmas to the more esoteric computing logic options.

Brackets work in most cases although there are a few weird ones which just go for the ultra simple each in turn at which point its time for separate variables.

With Windows, you can key in 'Windows Key + .' 
This gives you access to loads of extra symbols such as: ⅐.  
So, it's possible to have '⅐π' on UKC relatively easily.
 

I was more on about Roberts "you must use a horizontal line" thing, but that's a top tip thanks.

You've never used Excel?

After a neuron has summed the product of the inputs and weights (and added any bias) it applies a simple activation function to produce the final output value that's forwarded to the next layer.  Various flavours of activation function might be applied depending on what you're trying to accomplish, but the tanh one is quite popular because it (a) can help produce a solution for non linear problems (those that can't categorise values into classes that can be separated with a simple straight line, such as high-dimensional problems like image recognition) and (b) it limits the possible output values such that a small change in input produces a small change in output (makes the learning process more refined).  Another common activation function that behaves similarly is the  sigmoid function f(x) = 1/(1 + e^-x). 

There's lots of good resources that describe activation functions and their use in ANNs better than I ever could via a quick Google.  

I use the desmos tool quite a bit too.  It's pretty awesome!

[Edit] Thinking on it, a simpler explanation might be something along the lines of "a neuron takes a value as an input, multiplies the input value by a weight to produce an output value.  The activation function is applied to that value to determine if it should "fire" or not.  The learning process consists of fine-tuning what the weights are.". This is probably as simple a description of the essence of ANNs I can come up with.

I see mbh has given a fuller explanation (I added mine without reading the subsequent parts of the thread).

In the spirit of example "what's the point of..." things, ANNs have a good one for the uses of differential calculus.  ANN algorithms go through iterations of tweaking the weights in the network and seeing what effect it has on the eventual output (comparing with expected output of the training data).  It uses differential calculus to come up with the gradient of the network error to determine when to stop the learning process.  At which point there is hopefully a model which can be used to accurately predict against new inputs that the ANN hasn't seen before.

They would write pi/2, surely ?

Never to do anything arithmetical or mathematical. I used to make a list very occasionally.

The can remember the broad brush strokes of the subject areas covered , not the actual lessons.

I can't remember at all what we are taught.  

It's not ambiguous. The agreed mathematical conventions remove any ambiguity.

It's half of pi.

You're thinking of (best I can do on a phone).

1 

_

2.pi

Not the same thing.

The thing is it is regularly written and understood as half pi when written by hand and in standard text s- e.g. Kreseig, particularly when space is limited such as on integral limits 

Both C# and Excel give you an answer of 1.57.....

But both are from Microsoft which isn't really a reliable ultimate arbiter of what's correct.

Personally, written as a one-line expression like that would mean one-half of pi to me. If I wanted it otherwise, I'd write it as 1/(2pi).

Seems like an awful lot of people disagree with you there.  As a general rule, if some people think something is ambiguous, they’re right and stating otherwise arrogantly refuses to recognise that other people have different ways of interpreting the world, and that one needs to consider that when putting pen to paper in order to avoid confusion.

Can you show me an actual agreement on the convention?  An ISO standard for infix notation for example that includes implicit multiples?  Frankly, if you can’t you’re taking out of your arse and need to accept that.

I also suggest you’ve gone against the prevailing convention on implicit multiplication, which is that the relevant symbol (x, × or *) is only omitted when placed between the two things it multiplies.

Your question is about notation conventions, not maths or arithmetics.  It’s not tricky, it’s somewhere between pointless and irrelevant.  Your answer is arrogant and un-enlightening.  Troll harder.

Edit:

I can't speak for C# but I have an idea...

For Excel, you are wrong.  Keep in mind your OP deliberately hinges on an implicit multiplication symbol, as does the text from Robert Durham that you're reply to (as per my quote, above).

If I type "=1/2pi" in to a blank cell in Excel, it says "We found a typo in your formula and tried to correct it to: 1/PI2 Do you want to accept this correct?".  In a blank sheet this gives #DIV/0, because "PI2" is an alphanumeric reference to an empty cell.

If I type "=1/2pi()" in to a blank cell in Excel (you do use Excel enough to know that Pi is access as a function, right?), it has a real wobble, and won't let me continue with that text in there, because it's gibberish as far as Excels parser and shunting year algorithm is concerned.

If I type in "=1/2*pi()" I get a result of ≈1.57

So, no, counter to your factually incorrect post, excel doesn't interpret "1/2pi" as 1.56 or as any other number.  You are wrong.  It won't interpret implicit operators, and your OP relies on the implicit operator to garner confusion.

I'm really surprised you're sticking to your claimed correct answer given the incredible ambiguity over an implied operator.

Quite. For computing, strict rules are needed and laid down by the authors of programming languages. In written maths, here's no rule as such because there is no final.authority  - it's just what is understood by people reading expressions . As we've established, that is somewhat context dependent 

C# would need the *.

It would also need the 1 and 2 casting to a type which allows decimals.

eg something like

1d/2d * Math.PI

Not so pretty.

My calculator accepts implied.multiplication and returns 1.57... for 1/2pi. (the slash actually a division sign),l  so while the OP is simply trolling, I think there is a division between computer rigidness and how humans interpret some expressions in a known context.

Nope Excel will do no such thing with 1/2pi

As soon as we got to secondary school in Greece, ÷ was deposited in the bin where it belongs. Writing fractions out as fractions you don't end up with this bullshit ever arising, you don't have to think whether the multiplication applies to the numerator or the denominator, it's rather clear. 

If I was worried about people misinterpreting it, I would write (1/2)pi or pi/2.

But this is purely for "one-line text" expressions, if I was using any of these expressions in Excel or in a programming language, I'd make damn sure I knew how it was going to be interpreted and adjust how I wrote the expression accordingly - which would almost certainly mean making the implied multiplicator explicit for a start.

Computers will interpret however the human who designed the language its using interprets it as.

If the team who wrote c hadnt gone with bodmas then chances are most languages would do whatever they whatever they had been taught and semi remembered from school.

Personally I am a fan of being explicit since when I am looking at the code later it makes it easier to try and decide whether something is an error.

A fun one for sql is is you dont specify the join type then it defaults to inner join so "inner join" and "join" are the same.  Then its play guess did the dev know that or, like many devs, are they crap at sql.

1/2.*Math.PI to be precise, the const is a double, not a decimal.

What’s odd is that, aged 6, Wintertree Jr was introduced to fractions as a precursor to more generalised division in school, and the fractions have a line between the top and the bottom numbers, albeit sloping.  They grok that just fine, then the curriculum pivots to an infix style binary operator for division with an arbitrary convention for precedence instead of brackets.  With the complex mnemonics and daft notation, the school system introduces a massive level of confusion that simply doesn’t exist in the professional worlds.  Very frustrating supporting a child through a random and inflexible curriculum when they’ve grasped logarithms, base 2 and differentiation yet the school won’t ask them to count above 100 because that’s not in this years curriculum.

Do that and you get zero back. You could use 1.0 and 2.0 instead but you cant go with a basic int.

And at least 3 of us are (or have been) maths teachers..........

These days I build my maths expressions using a symbolic algebra system, then have them rendered down to both a LaTeX graphic representation for me to verify as the correct interpretation, and to executable code.  Benefits include verification of the code, a single source for documentation and code, and being able to call .simplify() on the maths to optimise the number of operations before even going to code.  (Jupyter Lab, Python and sympy). It’s the 21st century.  We should be writing maths as it’s meant to be written, and letting the computers turn it in to code.

Why are you arguing over syntax? We're talking about order of operation and multiple juxtaposition. If you calculate 1/2pi in either c# or excel using the correct Syntax with no brackets you get 1.57....

We know this. It was.covered ages ago in the thread.

Because the deliberate confusion in your OP is amplified by the presumably deliberate use of an implicit multiplication operator.  That is not “Syntax”, that is convention.

As I’ve explained and you’ve failed to read/understand, you can’t compute “1/2pi” in Excel.

Its not often a poster doesn’t understand the question they’ve posted, but I’m getting an inkling you might.

I used 1d/2*Math.PI 

As long as the numerator is a double the division will be calculated as a double.

So you used something completely different from what you claimed worked.

Now do you see why its useful to be explicit?

So however excel likes it then?

If you believe most actuaries (had to work on a project to make a pension modeller available online. it really was depressing looking at the starting point in excel).

It's pretty much the same here as well. 

6+4÷2=  was on a recent KS2 Arithmetic paper and I suppose something similar could be on a foundation GCSE paper, but I can't remember writing ÷ on the board very often teaching secondary school students. Perhaps the discussion on this thread indicates we should? 

The question was:

I interpret this question to mean, would it calculate:

1/[2pi] or [1/2]pi

Maybe RObert was asking a different question, but this is what I interpreted the question to be.

I gave an answer of 1.57....

If you interpreted this to mean I just typed 1/2pi into excel of VS and got an answer then you are mistaken and I can inform you that I wrote it in a way that excel and the compiler can understand. 

Otherwise I would have got ##!??^^^^ARGGHH!

I interpreted it the same.  In C, Java, Python and php it gives the answer of  1.57079 (etc). Unsurprisingly.

Try typing the same things into Wolfram Alpha and it'll assume that the implied multiplication should be done first. Of course, it will also rewrite it into a more sensible, unambiguous format involving horizontal lines and no ÷ sign:

https://www.wolframalpha.com/input?key=&i=6+%C3%B7+2+%281+%2B+2%29

For 1/2pi its answer depends on which input style you choose: if you choose "natural language input" it'll give you π/2, but if you click "math input" it changes to 1/(2π).

Programmers would tend to avoid writing 1/2*PI (or language-specific equivalent): I for one ain't got time for remembering whether * and / will evaluate right-to-left or left-to-right, or for assuming that anyone else knows. Also it executes two operations where only one is needed (assuming you mean π/2)... And in C and similar languages it evaluates to 0.

1f/2*PI actually does give you ≈π/2 in C and Java; however...

The Julia language, which is aimed at mathematicians and scientists, actually has an implicit multiply, which has higher precedence than any operator but ^ (power) to match general mathematical usage, and it also defines a constant called literally π. In Julia, the expression 1/2π evaluates to 1/(2π), while 1/2*π gives π/2. OTOH 1÷2π == 1÷2*π == 0, because Julia defines ÷ to be an integer operator (rounding down).

​​​​​​In Julia, 6÷2(1+2) == 1, while 6÷2*(1+2) == 9.

Hmm.  I'd be a bit sceptical of the programming capabilities of a programmer so unfamiliar with the language they're using that they don't know the precedence rules of basic arithmetic.  If you're looking at slightly more esoteric things like bitwise operations then maybe, but adding and multiplication?

That's really interesting. I heard of languages that carry out operations in the order they are written left to right regardless and it's up to tthe user to force order with brackets, but I've never known of one that prioritises juxtaposition multiples. Very interesting.

I don't agree with this. I use several languages. I think I know the order they all use, and I think they are all the same as excel/C# but I would need to test it. And even if I took the time to remember each one (assuming they are different), I'm too lazy to think it through when writing so I would just use brackets anyway.

I'd rather prevent problems when writing rather than fix it later in unit testing. I'm guilty of not doing unit testing at all since stuff I write is used commercially, but not externally so I tend to check on the fly so even more important to be certain it's correct. Also did I mention I'm lazy. Plus, even if I learnt them all today, I'd probably forget in a few months' time.

I think far less important than the difference between Microsoft and others is understanding the difference between retaining precision internally to the evaluation of an expression and forcing an intermediate result to be stored, in which case storing the intermediate result at a lower precision than is necessary.  (I've seen this cause "aircraft" tracked in an ATC radar simulation to split in to two on crossing a quadrant boundary!)  The two tracks went at right angles to each other!

I can remember in the 1980s how in Dimensional Control the owners of "Crappio" calculators were ribbed by those with HP41Cs.

It was routine on the Casios and on computers with Microsoft Basic to have to multiply numbers by 1000 before performing calculations where angular readings in Degrees Minutes and Seconds of Arc were involved as there simply were not enough digits precision available.  No such problem with HP kit whether HP 41C HP9826 or earlier programmable calculators (which actually came with documentation about their "operating system" and enormous internal maths architecture.

At which point you also get into the fun of floating point inaccuracies.

In which case you have already made your own interpretation as to what 1/2pi should evaluate as. It’s your interpretation that’s getting typed into Excel, and you are no longer asking it to interpret 1/2pi.

It's easy: ÷X or /X means *(1/X) so 1/2pi -> (1*(1/2)*pi)

Same with -X this means +(-X)

Apply these rules then the rest of B[O|I]DMAS continues without an ambiguity over the precedence of * / + - or the order in which it was written or entered.

That's not true.

If I type =1/2*PI() into excel, excel could either do

• =(1/2)*PI(), or
• =1/(2*PI())

Excel follows BODMAS so does the former. I didn't tell it to do the former.

When you do that, you decide how to interpret the text in your OP "6 ÷ 2 (1 + 2)" in to the dissimilar text of =1/2*Pi()"; critically you decide that the implicit multiplication shall be translated directly to a standard multiplication with the standard precedence on binding.

By my count, three posters have given their opposing view that an implicit multiply behaves differently to an explicit one by binding more tightly - explained in other equally valid ways by the other posts.

Surely you can recognise that there are multiple different opinions out there on this subject?  You have such strong certitude in your interpretation of the convention on implicit multiplies, without having given any supporting evidence.  

Claims that Excel or C# can settle this are clearly a nonsense, as they don't accept an implicit multiply.  The evidence brought to the discussion is from MG and their calculator (presumably the iOS Calculator App?) which tilts one way and jelaby and their interesting insight into Julia which tilts the other way.  

So, it seems clear the ambiguity in your OP derives from the interoperation of an implicit multiply symbol, for which there is no one global convention let alone standard, but where mathematicians lean towards it binding more tightly.

No. Dividing by a double promotes to double, only one of the operands needs to be double in the /.

Edit: ... but I missed a 0 after the '.' in the double literal, replying on a phone keyboard sucks... :-P

Indeed. In fat if you used a slide rule such information had to be written on the exam paper.

I also get the impression from youngsters (at the athletic club) that they are "primed" as to the type of questions they might get - not something that occurred in the 70's, we had to learn the lot apart from formulae that we received on a single sheet of A4 with largish print and big spaces between formulae.

No priming in the 80s either.  I did my O level and A level maths exams with a log table book as both times I forgot my calculator. A friend used a slide rule but only as they’d got one aged 11 and kept using it during lessons.

So my wife asked me this question the other day. She's a Maths teacher. I looked at it for 5 seconds and confidently answered 1. She said nothing. I then thought about my school days and realised my error that I was ignoring BODMAS and said, actually, wait, it's 9. I got this. I know what I'm doing here.

The juxtaposition multiple feels right to calculate after the brackets. That's why the questions works and sparks fun debate and why it is indeed a tricky question. It tricks you. Until the Julia comment, I had never heard of a language that prioritises juxtaposition multiples over a normal multiple (maybe matlab does, but that was too long ago) so I'm happy to soften my view on that. I'm not aware of any other text on this as yet but there are plenty of documents out there that do not differentiate between [ "x", "." or juxtaposition]. 

I would never claim that. There are languages out there that ignore BODMAS all-together.  I was answering a question. I wasn't even 100% sure if excel and C# did follow BODMAS but I was 95% sure they did. incidentally, this is an interesting one: 

https://www.google.com/search?q=6+%C3%B7+2%281%2B2%29&rlz=1C1GCEU_enGB8...

Can't argue with google!

There should be global convention. I'm really not against juxtaposition multiples taking priority as it's what feels natural and it's what I will do if I'm not concentrating. So with that in mind may I present:

BOJMA

    - Brackets

    - Orders

    - Juxtaposition

    - Multiples (includes divide)

    - Addition (includes negative additions)

Just need to think about how to get the entire mathematical community, programmers, and all the education authorities on board. Should be easy.

It was Android but I am sure IOS would do the same.  Now at my desk and just tried with my TI calculator.  It does the same 1/2pi =1.57.  Also looked at Kreseig and he at least sometimes uses implied brackets in such expression (p571 8th edition, for example).  So montyjohn can insist there is only one "right" answer until he is blue in the teeth.  But he's wrong

Yes, I'd be amazed if any students now took an exam now without doing several past papers. I didn't do any in the 80's, although I only had to answer 8 of the 12 questions on the paper. Whether standards have improved is open to debate, but teaching is now certainly more focused on enabling students to perform well on exams. 

I think Gove's plan was to go back further (50's?) so students wouldn't get that formulae sheet and need to memorise them. 

Surprising that nobody so far has referred to https://en.wikipedia.org/wiki/Order_of_operations

Things I noted in it include:

different countries have different conventions for primary teaching

the American Physical Society's author instructions are to proritise multiplication over division

(as is clear from the thread) some calculators and programming languages do prioritise implied multiplication, even though beginner calculators don't. In fact, I'm starting to suspect that BIDMAS had to be introduced once schoolkids were allowed to use calculators -- does anyone know in which year each happened?

The D and M in BODMAS are the same priority. Excel gave you (1/2)*PI() because it evaluates equal priority operators left to right (for * and /).

Apparently Bodmas has been around since the 1920's, it was part of the national curriculum in 1988 so all secondary schools should have been using it then. Scientific calculators early 80's? I certainly used one in 86

Good computer code is readable code (for the most part). In normal life (i.e. dealing with real numbers), multiplication is associative and it doesn't matter what order you execute them in, and there aren't even that many cases of non-associative multiplications even in more advanced maths, and when those do turn up, you disambiguate them with brackets. Assuming that every reader of your code is entirely au fait with the language-specific esoterica like execution order is a good way to encourage unnecessary bugs.

Wow, I’m a bit out of my depth in a post like this but the answer to the OP’s question sprang straight to mind. 
I finished school in 1962. We didn’t have calculators, log tables, slide rules or printed formula, you remembered or not. Obviously something sprang to mind after 60 years.

Will someone put me out of my misery and tell me what BODMASS means?

Brackets

Order (i.e. powers / indices)

Division

Multiplication

Addition

Subtraction