6

I saw this photo on my social network.

The ambiguous expression $6\div2\times3$ yielded 2 different answers.

The Picture

The difference is the order of operations. If the division's done first then the answer is 9. If the multiplication was first then the answer would be 1.

What is correct?

There are rules for the order of operations like "BODMAS" for example (I use it) but it doesn't say what is correct in such situation.

Is it possible that both answers are correct unless "brackets" are specified? Or is there one correct answer? Is there a rule for the "direction" in which those must be done? "left to right" for example?

edit: It looks like it's known that it should be done from left to right but the question now is whether:

  • It's just a convention.

  • It's a definite rule that makes us say that one of the answers is wrong.

tag should be "operator-precedence" for example. Help me with the tag please :)

842Mono
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    It's a matter of convention. – user2345215 Mar 04 '14 at 13:19
  • http://math.stackexchange.com/questions/33215/what-is-48293 – Najib Idrissi Mar 04 '14 at 13:20
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    Odd that they have the same manufacturer. – David Mitra Mar 04 '14 at 13:22
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    @DavidMitra odd indeed. could be photoshop for all we know. but aside the fact it is the same manufacturer, this is a legit question. I don't like that someone downvoted, if non math people want to know, we should make them feel welcome. – Guy Mar 04 '14 at 13:24
  • @Sabyasachi exactly :) ...Although that makes me a non math guy which makes me very sad XD XD – 842Mono Mar 04 '14 at 13:25
  • Without calculators: division and multiplication have both the same stand in preceedence (when no parentheses are there), and one must first do the first from the left. In this case thus, the first operation is division and then multiplication and the result is nine. – DonAntonio Mar 04 '14 at 13:35
  • ...okay I'm going nuts! Is it "from left to right" or is it "both are correct unless specified"???????????? – 842Mono Mar 04 '14 at 13:40
  • @DonAntonio this is generally taken as the way to resolve ambiguity should that ever arise. But there is no reason that the other "non general" behavior be considered wrong. Bottomline, you should not write ambiguous equations. Both are not correct unless specified, but rather, not specifying is very very wrong. – Guy Mar 04 '14 at 13:42
  • Do those calculators have a way to explicitly insert the times (X) symbol before the parentheses? – Govert Mar 04 '14 at 13:44
  • @Govert you mean like $6\div2\times(1+2)$? If so then yes. – 842Mono Mar 04 '14 at 13:45
  • A related question might be: What is your initial interpretation of the ambiguous expression 6/2(1+2)? – Govert Mar 04 '14 at 13:46
  • @Mina I think 6÷2×(1+2) can only be read as 9. Without the × I can see it being ambiguous. – Govert Mar 04 '14 at 13:49
  • It is ambiguous. I know that but I just want to know what to do if I face such thing in a quiz or something. (Although it's unlikely because they should put brackets!) – 842Mono Mar 04 '14 at 13:51
  • @Mina You'd be at the mercy of the questioner... If you can write more than just the answer, it would be good to show how you're interpreting the ambiguous question, at the start of your answer. You'd still try to interpret it in the most natural way (maybe 9 in this case, but I see the point about a/bc vs. ac/b in the comments to another answer). – Govert Mar 04 '14 at 13:55
  • What does the left-hand calculator show when you add the ×, so for 6÷2×(1+2)? – Govert Mar 04 '14 at 13:57
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    @MinaMichael In a quiz, I advise taking the questioner to court. – Guy Mar 04 '14 at 13:59
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    @Govert I have the same model calculator 6÷2×(1+2)=9 and 6÷2(1+2) = 1. – Warren Hill Mar 04 '14 at 14:05
  • Those calculators aren't mine but I have one similar to them. (from the same manufacturer) – 842Mono Mar 04 '14 at 14:10
  • @Sabyasachi, in two countries at least (Mexico and Israel), kids are taught about operations preceedence in 6th-7th year: multiplication and division first, sum and substractions second, and among two operations with the same stand, the first one from the left is first. It is not simply "a way to resolve ambiguity" but rather a pretty clear, unambiguous rule . – DonAntonio Mar 04 '14 at 14:22
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    @Warren - That suggest this calculator is interpreting the multiplication-by-juxtaposition as a higher-precedence operator than regular multiplication (see http://www.purplemath.com/modules/orderops2.htm). That's not entirely crazy, given how we'd interpret $5x/30x^2$. – Govert Mar 04 '14 at 14:22
  • No mathematician would write 6÷2×3, so I think you are asking in the wrong forum. – GEdgar Mar 04 '14 at 14:45
  • Now I agree with @GEdgar...though kids that are taught the preceedence rule are given this kind of exercises. – DonAntonio Mar 04 '14 at 14:48
  • Some of this discussion reminds me of the linear algebra joke, if we define a matrix by $a_{ij}=i+j$, what is the value of $a_{123}$? – Barry Cipra Mar 04 '14 at 15:10
  • @GEdgar I know it's implicit but what if you encounter it? Should we say that both are correct or is there some definite rule?

    ...Up to the point, some say both are correct and others say no; It's left to right! and I'm going crazy

    The only thing that's definite is that it's left to right conventionally but we're discussing whether this rule is a must or just a convention but both are correct.

     – 842Mono Mar 04 '14 at 15:39
  • Hey this was asked before!!!!! ...why did he get downvotes? – 842Mono Mar 04 '14 at 16:02
  • ...and still no definite answer since then!!!!!!! One said that it's a convention and another said "left to right" – 842Mono Mar 04 '14 at 16:09

2 Answers2

9

There is no contradiction here. The first calculator interprets it as $$\frac{6}{2(1+2)} = 1$$

The second interprets it as,

$$\frac{6}{2}(1+2) = 9$$

It is a matter of how the calculators are designed. In general, it is bad practice to write ambigous equations like that, since your intention is not always clear.

Guy
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  • Yes I know and agree with all that but I'm asking if there's a specific rule to govern which way it should be interpreted. – 842Mono Mar 04 '14 at 13:24
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    @MinaMichael no there isn't. it is a matter of convention, and a wise person always sticks in enough parenthesis to make it unambiguous. – Guy Mar 04 '14 at 13:25
  • Hmmmm alright that's good enough for my mind! Thanks a lot :D – 842Mono Mar 04 '14 at 13:26
  • @MinaMichael Sure. Anytime. Happy math. – Guy Mar 04 '14 at 13:27
  • I'd say there is a very clear rule of preceedence...read my comment above and the other answer. – DonAntonio Mar 04 '14 at 13:37
  • Is there a rule? Yes: each computer language, and each calculator has its own rule. As you know, they may not agree with each other. – GEdgar Mar 04 '14 at 17:00
  • You could argue that 6÷2(1+2) is 1 because parenthesis have the highest precedence. 6÷2(1+2) and 6÷2×(1+2) are different equations. With the former the 2 is explicitly attached to the parentheses. – Luke Mar 04 '14 at 18:09
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There is a a speciefic rule for this: the order of operations. Multiplication and division apear on the same place in this rule so they must be interpreted from left to right so 9 is the correct answer. The other calculator is just wrong.

This probably has something to do with the fact that most people never write division inline like on a calulator, but rather as a fraction

$$\frac{6}{2\times3}$$

where both the numerator and the denominator need to be evaluated first, as if the where in parenthesis. This is however not the case so it is wrong.

Jens Renders
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  • No. You are adhering to the convention that has this rule. There is no logical reason that the other shouldn't be right. All in all this is undefined behavior, and should be treated as such. – Guy Mar 04 '14 at 13:39
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    No this is perfectly defined and should be done from left to right. http://www.math.com/school/subject2/lessons/S2U1L2GL.html "Working from left to right, do all multiplication and division." – Jens Renders Mar 04 '14 at 13:46
  • This math.com is an authentic source because just about anyone is not allowed to come up with a website. – Guy Mar 04 '14 at 13:48
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    http://www.mathsisfun.com/operation-order-pemdas.html "Otherwise just go left to right." – Jens Renders Mar 04 '14 at 13:49
  • http://www.purplemath.com/modules/orderops.htm "For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first." – Jens Renders Mar 04 '14 at 13:50
  • All I am saying is while it is perfectly okay to have a "tie-breaker" it is not at all advisable for the question formulator to come up with an ambiguous question in the first place. – Guy Mar 04 '14 at 13:50
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    That is correct because it causes confusion, but it is perfectly well defined and the correct answer is 9, not 1 and that's what OP is asking. – Jens Renders Mar 04 '14 at 13:52
  • I would (at first glance) parse it the way you support as well, but their is no meaningful reason the other way should be declared invalid. – Guy Mar 04 '14 at 13:52
  • http://math.stackexchange.com/questions/16502/do-values-attached-to-integers-have-implicit-parentheses – Guy Mar 04 '14 at 13:53
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    the answer with 16 votes makes a very strong case. – Guy Mar 04 '14 at 13:55
  • 17 votes now. :D – Guy Mar 04 '14 at 13:55
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    " Add to this the fact that it makes no sense at all to write a/bc if you wish to convey the meaning ac/b, and one can almost safely say that anyone who writes a/bc means a/(bc)." – Guy Mar 04 '14 at 13:57
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    I'm not trying to discuss the way it should be, I'm stating the way that it is. Btw, most people writing a/bc might mean a/(bc) but most people writing a÷bc probably mean (a÷b)c and this is also the only correct meaning. – Jens Renders Mar 04 '14 at 14:00
  • If I meant ac/b I would write it as such. – Guy Mar 04 '14 at 14:01
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    I see we both concede that a certain bias exists to favor 9 over 1, but within the mathematical community, the bias is negligible and in all cases, it would be expressed unambiguously. I simply refuse to declare 1 invalid due to this bias, because the bias/convention itself makes no sense whatsoever. I would rather not acknowledge the existence of such a convention. – Guy Mar 04 '14 at 14:03
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    How would a convention have to make sense? It's just a rule, made by humans, just chosen, so that everyone does it in the same way. – Jens Renders Mar 04 '14 at 14:09
  • Maths is an otherwise flawless language, one with the ability to perfectly, precisely convey meaning in sharp, short sentences. I am not sure if ancient archaic conventions have a place in mathematics, just because tradition. As for having a convention to make sure everyone does it the same, what about, "Thou shalt not introduce ambiguity or thou shalt be beheaded"? – Guy Mar 04 '14 at 14:12
  • Apart from that I agree with you. – Guy Mar 04 '14 at 14:13
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    ...so it's settling that "there's no rule but conventionally it's left to right"? – 842Mono Mar 04 '14 at 14:19
  • No, there is a rule, but not you should avoid the need for that rule by using parenthesis or fraction notation. – Jens Renders Mar 04 '14 at 14:24
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    I agree with @jenders: the operations preceedence is a rule and, anyway, I think the downvote is uncalled for. – DonAntonio Mar 04 '14 at 14:25
  • @MinaMichael there is a rule in the sense that usually I would write 9. there is no mathematical rule as to why it should be 9. At the end, it just comes down to the situation. Mathematical paper, use parenthesis. I am a teacher and evaluating papers? Use the interpretation that makes most sense in context. Primary school teacher teaching 4th graders? Give them the benefit of doubt. I think at then end, we must remember that maths after all is a language, and what matters is the intention expressed. – Guy Mar 04 '14 at 14:27
  • @JensRenders it was my downvote actually. If you edit your answer(just for sake of editing, maybe add a space somewhere, or shift a punctuation mark, whatever) I will upvote you instead. As of now, my vote is locked. – Guy Mar 04 '14 at 14:28
  • @Sabyasachi I did. Thank you. – Jens Renders Mar 04 '14 at 14:30
  • @JensRenders Done +1. :) – Guy Mar 04 '14 at 14:31
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    No,no. There is a rule not in the sense "that I'd do this or that", but a mathematical rule, period. One could argue this rule is not an international one, not everywhere it is applied, etc., but kids in several parts of the world are taught to apply it. – DonAntonio Mar 04 '14 at 14:32
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    @DonAntonio I would argue that some primary school technical teaching is flawed. Just because people teach it, shouldn't make it right. Being right, should make something right. – Guy Mar 04 '14 at 14:35
  • Although I guess this discussion is moot, since noone here is actually arguing a mathematical point, but much rather the question, are you willing to follow tradition? – Guy Mar 04 '14 at 14:36
  • @Sabyasachi I can't understand why you think this rule isn't right as opposed to other rules. It is just a rule which comes to make things unambiguous and easier to work with. And I can't see how can you be so positive and 100% sure about what is and what isn't, what should and what shouldn't as you seem to be very young and, apparently, guided by what you were told. – DonAntonio Mar 04 '14 at 14:37
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    I think what you, @Sabyasachi, need to take in mind is that this is a rule about notation, not about how multiplication or division works. – Jens Renders Mar 04 '14 at 14:37
  • @JensRenders and Antonio. I understand that this is a rule about notation, I am just expressing my discontent with such a rule, specifically because as Don says, "this rule is not an international one" and it is not hardwired into me, because I wasn't taught as much, and it more so increases ambiguity for me. – Guy Mar 04 '14 at 14:41
  • @DonAntonio More like I am being guided by the fact that I wasn't told anything. I am just dissatisfied with the inconsistency. – Guy Mar 04 '14 at 14:43
  • If you will excuse me, I need to be going now, I have an exam tomorrow, which is not about notations :p. – Guy Mar 04 '14 at 14:47
  • There is no inconsistency at all in this rule, @Sabyasachi, what are you talking about? It is a very clear, concise rule for the four basic arithmetical operations, which can later be expanded to parentheses, powers, etc. Any kid that has learnt this stuff won't have the slightest doubt about what the OP's question's answer is: it is $;9;$ and can't be anything else, period. I can understand some haven't been taught this rule, but you cannot say it is inconsistent just because you don't like it. – DonAntonio Mar 04 '14 at 14:47
  • @DonAntonio I am referring to the inconsistency in enforcement. The fact that is not international. – Guy Mar 04 '14 at 14:50
  • Well @Sabyasachi, by the ammount of sites one gets on this when googling "order of operations" one could think it is more international than what we think...but I really can't be sure. – DonAntonio Mar 04 '14 at 14:52
  • I can testify that it's not enforced where I live, at least. – Guy Mar 04 '14 at 14:54
  • and now I can't be bothered(ironically, considering the fervency of this discussion) because pretty much everywhere it is in fact, unambiguous. – Guy Mar 04 '14 at 14:55
  • who downvoted it now? My upvote is still in place, as I remember it, there were 2 upvotes on this, so 2 people downvoted this? But why? – Guy Mar 04 '14 at 15:05
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    According to your "rule" all those math texts that use “$1/2\pi$” to mean $1/(2\pi)$ are "just wrong". Sorry, but that is just wrong. – MJD Mar 04 '14 at 16:23
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    If it wasn't ambiguous the answer would be 1, not 9. 6÷2(1+2) is 1 and 6÷2×(1+2) is 9. Parentheses have the highest precedence. The 2 is explicitly attached. But you would have to agree on precedence, in that 2( and 2 × ( are different expressions. Unfortunately this is a bit of gray area. – Luke Mar 04 '14 at 18:01
  • @MJD: Another way of looking it is that defining 1/2π as notation for π/2 wouldn't really allow anything to be expressed more concisely or clearly than without such definition, but allowing it to represent 1/(2π) allows the latter expression to be written more cleanly and concisely than would otherwise be possible. I would consider blanks significant; "1/2 π" is (1/2) times π. Also, while it doesn't matter for multiplication and division in ${\mathbb R}$, I would regard ab/c as (ab)/c, but "a b/c" as "a (b/c)", in contexts where it matters. – supercat Mar 04 '14 at 21:07
  • @supercat agreed. although if there are actual numbers involved, I would interpret, 'a b/c' to mean $\large\frac{ac+b}{c}$ – Guy Mar 05 '14 at 10:34
  • @Sabyasachi: With numerals, I would agree. "3 1/2 pints" would bind as "(3+(1/2)) pints". – supercat Mar 05 '14 at 16:17