I could be wrong but it was my understanding that they define division and multiplication as having equal precedence, IIRC the rule is to do them left to right (I think this applies to addition vs subtraction too).
What's the answer to this very basic maths problem?
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I could be wrong but it was my understanding that they define division and multiplication as having equal precedence, IIRC the rule is to do them left to right (I think this applies to addition vs subtraction too).
actually, checking back, the source is available in the comments... the author is referring to implicit multiplication/multiplication by juxtaposition I've already mentioned in here
@ubersonic your 'old method' of simply applying the division symbol to everything on the right isn't supported by that source, the source merely given precedence to the implicit multiplication (perhaps I should have used that phrase rather than juxtaposition)
you can demonstrate this to yourself on your old calculator quite simply by typing in say 6÷2+3
if your explanation was what was implemented and it was simply that everything to the right was the divisor then you'd get 1.2... I suspect this won't be the case and the issue is the multiplication by juxtaposition
(I have pointed this out a few times now and you've seemingly ignored it)
[citation needed]
I think danlightbulb is correct, but it's only relevant because we have an ugly mixture of high school mathematics and real mathematics. We have implicit operators in one part of the sum "2(3)" and explicit, i.e. chained, operators in another part ("÷"). They really shouldn't be mixed in the first place. Either proper mathematical notation should be used throughout, e.g.
or else explicit chain operators should be used throughout. danlightbulb, imo, identified that by mixing the two, you had to follow both sets of rules. In which case the "2(3)" would have to be treated as a single discrete value.
slow down a bit... can you take another look at what I posted in the full post you quoted from and what I posted in post #68
specifically referring to the snippet you've quoted - no it wouldn't be correct to put implicit multiplication at a higher precedence than powers (if you need an example of why I'll be happy to provide one but I think you may have skim read what I posted) ergo he is incorrect when previously claiming that
he can be correct re: implicit multiplication taking precedence over explicit multiplication - this is simply a question of convention... I already went through this in post #68 and provided two examples of calculators using each convention
In mathematics, division and multiplication have equal precedence. They are, after all, the same thing really. But in education, the British government teaches BODMAS which puts division ahead of multiplication. Which is why I argue that once the job of teaching children is done, the tools used to do so should be put away and proper mathematical notation used that avoids it ever coming up.
A GCSE Mathematics revision text book by CGP, circa 2004. That's the best I have on this. It doesn't match what I was taught in school. Or other people. But I have assumed that it was written in accord with official guidance. I could be wrong.
You're right. I misread your reply. However, I didn't read danlightbulb as arguing everything to the right of the division symbol then took precedence. Only that the evaluation of "2(3)" did. I.e. if they'd further added more operators to the right, including indicies as you say, that would not take precedence. As we seem to have read danlightbulb's replies differently, I'll bow out of arguing defending their position.
I don't think you'd even see things written like that these days using the ÷ operator, presumably back in the day there were issue with printing/type setting ergo there are the two different conventions re: dealing with implicit multiplication - it seems to be the case in the 1917 school book that implicit multiplication was given precedence ditto to (according to wikipedia) some physics journals, but it isn't universal, in the case where the other convention is used you'd need to take care to explicitly specify the * (or use parentheses) in the case of your example above
I already gave an example like this but since the above example is deliberately more confusing and so would not be desirable to write like that under one convention perhaps lets take another:
1/2cos x
this could be read as either (1/2)cos x or 1/(2cos x)... it depends on the convention in place
though with modern journals/text books it shouldn't be an issue as you'd avoid using the ÷ or / operators
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@dowie there is only one correct mathematical convention here. You don't add operators that are not there to make new terms. It really is that simple. As written, the only interpretation is 1/(2Cos(x)). I can understand why someone might read it as the other, but it would be wrong.
If you wanted it the other way, add a space.
1/2 Cos(x), then you would naturally read it as one half of cos x.
I do concede that the way it's written could create comprehension issues. Hence why you have to have the skill to identify the terms. Obviously it's a case of write it right in the first place I agree with that. It's no different to a paragraph of text with missing punctuation really, you would evaluate it and identify the break points yourself.
There is no problem with BODMAS, it's fine and it works. It is people splitting terms that is the issue here.
So coming back to the original question. There is only one right answer to 6÷2(3) and it's not 9. You don't add operators that are not there to change the evaluation order.
If you wanted it the other way, add a space.
1/2 Cos(x), then you would naturally read it as one half of cos x.
I do concede that the way it's written could create comprehension issues. Hence why you have to have the skill to identify the terms. Obviously it's a case of write it right in the first place I agree with that. It's no different to a paragraph of text with missing punctuation really, you would evaluate it and identify the break points yourself.
There is no problem with BODMAS, it's fine and it works. It is people splitting terms that is the issue here.
So coming back to the original question. There is only one right answer to 6÷2(3) and it's not 9. You don't add operators that are not there to change the evaluation order.
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Odd, when I was taught it the rule was that division and multiplication were equal and you did them left to right, same with addition/subtraction. Guess they felt that was too complicated lol.
that's just a screen print with the book repeating BODMAS... which is of course confusing as I've pointed out earlier in the thread, but the book then states that the calculator will do division and multiplication before addition... I don't however think you'd find a problem within the book where they've arrived at an incorrect answer thanks to using one thread of the other. It is just a badly written book.
he didn't, you're still misreading (sorry).. ubersonic did
I think you've missed what I was referring to - it was the previous claim that the precedence was as a result of the B in BODMAS, this is incorrect as it would mean implicit multiplication took precedent over powers, you can read back through the posts if you like but he's already conceded that.
That's not correct, sorry. This is the whole source of the confusion and what I tried explaining in #68
there isn't a single convention that means implicit multiplication alway takes higher precedence than explicit thus we get this confusion - this is why you'd generally avoid using the / or ÷ symbols in the first place (there is no need to use them these days!)
the problem with BODMAS is that it confuses people, it confused you a few posts back for example when you gave a higher precedence than you should have to implicit multiplication under the belief it related to the 'B' in BODMAS which implies it is of higher precedence than a power
You know, that's just as valid. I suspect what happened was that somebody couldn't make a clever acronym out of that and so decided to unilaterally change things because it seemed like it was simpler to them.
You should hear me on power of ten Megabytes and Gigabytes sometime. It makes my comments on the subject of mathematical operators look like the calmness of a Zen monk.
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There is a convention. If there wasn't then we'd have some serious problems right now with anything using maths like engineering or software or whatever.
Look at algebra for the convention. If you substitute variables for the numbers in the example then it becomes obvious what the right evaluation order is. No maths professional would do it any differently because they'd get the wrong answer!
No need to keep bringing back my BODMAS error, I conceded the way I wrote my point was incorrect. My understanding of BODMAS and term identification is just fine thanks.
Look at algebra for the convention. If you substitute variables for the numbers in the example then it becomes obvious what the right evaluation order is. No maths professional would do it any differently because they'd get the wrong answer!
No need to keep bringing back my BODMAS error, I conceded the way I wrote my point was incorrect. My understanding of BODMAS and term identification is just fine thanks.
I think you're misunderstanding the point, I'm not arguing that there aren't conventions I'm pointing out that there are/were different conventions when it comes to implicit multiplication and that this is the source of most of the confusion re: this question. I'm also pointing out that there isn't really much use these days for using those division operators as modern typesetting means we can be much clearer when it comes to division.
I explained and demonstrated this in post #68, two calculators, same manufacturer, using different conventions
there is an explanation on wikipedia too for example:
https://en.wikipedia.org/wiki/Order_of_operations#Exceptions
you say look at algebra for convention but:
1/2x needs a convention for you to say that it is (1/2)x or 1/(2x) i.e. whether the implicit multiplication takes precedent or not
this isn't an issue these days as you just don't use the / or ÷
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Ok I agree with you on everything except for the lack of an implicit operator convention. That article basically says that some software interprets implicit c0nventions in a non standard way. I.e. there is a standard (and the only logical) way (implicit operators first) and some software gets it wrong so has to be used with caution.
Therefore there is still only one answer to the original formula. The correct way is implicit operators first. Software and calculators that don't do this have got it wrong and need to be used with caution.
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That is the whole reason why this question causes so much arguing online.
people citing for example 'BODMAS' from school will get the answer 9 as there is nothing within that framework to give higher priority to implicit multiplication (this is what the 1917 journal article cited by the youtube video ubersonic posted was arguing against.. essentially the author was arguing against how the order of operations when followed from elementary texts doesn't account for implicit multiplication and how it is frequently used)
when writing out mathematics by hand or in LaTeX you'd not need to use the ÷ anyway - I'm presuming part the reason why we used to have to have convention for this related to typesetting in textbooks and journals whereas these days the symbol simply isn't used past primary school
I think the real world issue these days where you'd come across needing to have convention for dealing with it would be in say creating a calculator or programming language and there quite clearly still are different conventions used there for implicit multiplication with some giving it higher precedence and some not... even within the same manufacturer.
So long as the convention being followed is known then I don't see an issue either way, it isn't really an issue in writing mathematical notation anyway these days anyway as division is made much more explicit. The question is a deliberate troll and quite a successful one.
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I agree with you that BODMAS alone is not sufficient but only someone who stopped education at age 12 or something would blindly follow that. To arrive at the right answer you have to then apply some more intelligence to it and to an extent some reading comprehension. It is a question of intelligence. Those arguing blindly for BODMAS are not as far up the scale as you or I @dowie
I don't think it is a question of intelligence at all, it is simply a question of what convention you use - do you give higher precedence to implicit multiplication or not.
for example the people who created Wolfram alpha don't but I wouldn't suppose they stopped education at the age of 12, the founder is a theoretical physicist with a PhD from Caltech
for example the people who created Wolfram alpha don't but I wouldn't suppose they stopped education at the age of 12, the founder is a theoretical physicist with a PhD from Caltech
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I dont think its about choice of convention though. There is a right way and a wrong way. I already demonstrated in a more complex formula what happens if you don't do implicit operators first. The formula becomes a nonsense. What we need is a real formula to demonstrate the convention can only be one way.
of course it is a question of convention
1/2x
1/2cos x
etc.. you need convention to decide whether you're referring to (1/2)x or 1/(2x) otherwise it is ambiguous
you were adamant earlier that the only interpretation is that 1/2cos x = 1/(2Cos(x))
the people behind say Wolfram alpha have decided not to use that convention but would read the above as (1/2)x or (1/2)cos x
https://www.wolframalpha.com/input/?i=1/2x
https://www.wolframalpha.com/input/?i=1/2cos+x
ergo you're using different conventions
so long as you're clear about what you're using/how you deal with implicit multiplication then meh... but this shouldn't be an issue these days as mentioned earlier as you'd make division more explicit by using a fraction bar when writing out mathematical expressions - it is a problem really for calculator manufacturers and programming language creators
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I get what you are saying.
What I would like to do is find a known formula that describes something. Then demonstrate that only one convention works, otherwise you end up with a nonsense formula. I stand to be corrected that it lands the opposite way to what I think lol.
What I would like to do is find a known formula that describes something. Then demonstrate that only one convention works, otherwise you end up with a nonsense formula. I stand to be corrected that it lands the opposite way to what I think lol.