Why Is There Ambiguity Here?

[ToraToraTora]

https://www.msn.com/en-gb/news/world/mathematician-gives-answer-to-simple-sum-that-has-been-dividing-the-internet/ar-AAFfILw?li=BBoPWjQ&ocid=mailsignout
under BODMAS rules it's 16, even if you say D and M are same priority the left to right rule still makes it 16.

[ToraToraTora]

sorry for 16 read 1.

[bhg481]

No ambiguity - BODMAS gives 1

[ToraToraTora]

indeed, it has apparently divided the internet!

[jim360]

If you read from left to right it's 16, not 1:

1. (2+2) = 4
2. 8÷2(2+2) = 8÷2x4
3. 8÷2=4
4. 8÷2x4 = 4x4 = 16.

which is the answer that google chrome gives if you type the sum into its address bar.

[jim360]

It's confusing, then, that anybody can say that BODMAS gives 1. Taken literally, in any case, BODMAS says "division before multiplication", which would rule out 1. But more precisely, BODMAS should be read as B,O,D/M,A/S, where division and multiplication have equal priority but are read from left to right.

But really the ambiguity is down to sloppy notation. Occasionally in scientific journals you will see something like a/2b, which is meant to be read a/(2b) rather than the more pedantic reading of (a/2)*b, but context usually resolves this problem (eg because all the experts know what a and b stand for and whether multiplying or dividing them would make more sense). So a secondary rule, and quite common, is to think of a division sign to mean "divide what's immediately on the left of this by everything that is on the right of this until you see a + or - sign".

[Prudie]

BODMAS gives 16

[jim360]

I suppose another reason for the ambiguity is that everybody *thinks* they know the answer, and at least half of them are usually wrong, but since we were all taught this at school we can't possibly be wrong about something so basic so everyone who disagrees with us is clearly thick and doesn't know maths at all.

This is in any case just the latest version of an age-old debate, one we've had on AB multiple times.

[ToraToraTora]

gawd I think I was right the first time! dam vino!

[jim360]

Guess that means you've answered your own question :)

[ToraToraTora]

not really I mis applied the rules the second time, there is no ambiguity as such just me applying the rules first then talking myself out if it then realising I was right to star with.

[jim360]

My own way of seeing it is that there is no ambiguity once you apply the rules consistently, but I don't think the rules are terribly clear sometimes. Here, for example, I think you'd probably find a lot of people who would say that 8÷2(2+2) is 1 but 8÷2x(2+2) is 16, the logic being that "implicit multiplication", ie not writing the multiplication symbol, should take higher priority than explicit multiplication. Then there's the potential difference in handling 8÷2(2+2) and 8/2(2+2), which two different division symbols sometimes being treated as having subtly different rules. To take the example I gave earlier, a/2b is often used to mean a/(2b), but nobody would write a÷2b in a serious scientific journal with the same intention.

There's also some confusion possible between, say, BODMAS and PEMDAS, which are the British and American ways of remembering the exact same rules but which unhelpfully imply that, in Britain, D comes strictly before M, but in the US M would come strictly before D.

And the final thing is that the conventions for handling division and multiplication are more or less arbitrary. There's no provable mathematical justification for why 8÷2(2+2) should be read from left-to-right (giving 16), as opposed to right-to-left. It is only a convention. Heck, even 4+2*3 could be plausibly read as 18 rather than 10 if you just decided that all compound sums should be read from left to right.

And if people don't remember the convention properly, but then get dogmatic about it anyway (or even if they're right but still *** about it), then it leads to heated arguments for no good reason.

[bhg481]

I think the ambiguity arises because, in a real situation, the derivation of the formula would have made it obvious which of the two options was the one to use. On top of that a mathematition would have used an extra set of brackets around the 8÷2 if the division were to be done first. Hence I conclude that the answer is 1.

[Old_Geezer]

I'm just glad I was taught neither BODMAS nor PEMDAS in school but had described to me verbally how it's done by explaining the priority of brackets, powers, ÷ ×, + -, left to right, etc. and so can come up with the right answer. (Of course, one has to spot the implied × where values appear outside of brackets.)

[jim360]

bhg: copy and paste the expression as it appears into any internet address bar, and the answer will be given as 16. Perhaps the answer was *meant* to be 1 in a certain context, and I agree that it's presented in a way that is deliberately meant to flirt with ambiguity, but absent such context it's 16, as given by BODMAS/order of operations.

[fiction-factory]

Many people misinterpret BIDMAS not realising that the D and M have equal precedence (so left to right is the convention), as do the A and S. When I present calculations like these I always put in extra brackets, even though they aren't needed, just to reduce the chances of misinterpretation.
Anyway, giving a question like this out of context is artificial to me. If you were working out a real life problem it should be clear as to the order in which you would do the steps

[spathiphyllum]

So TTT, are you set on 1 or 16 then? LOL

[mibn2cweus]

However . . .
since this -
2(2+2)
equals this -
(2x2)+(2x2)
therefore -
8÷2(2+2) = 6
8÷(2x2)+(2x2) = 6
8÷4+4 = 6
2+4 = 6
6 = 6

[fiction-factory]

^Not sure whether that is exactly what you meant to type- were you trying to show 1=6 or 16=6, mic?
Anyway, when you expand
8÷2(2+2)
shouldn't you insert brackets? . i.e.
8÷((2x2)+(2x2))

[fiction-factory]

^mib not mic

[jim360]

I rather suspect mib was messing with us :)