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There is a huge debate on the internet on the value of $48\div2(9+3)$.

I believe the answer $2$ as I believe it is part of the bracket operation in BEDMAS. Mathway yields the same answer. I also believe that if $48\div2\times(9+3)$ was asked it would be $288$ which Mathway agrees with as well.

However, WolframAlpha says it is $288$ either way.

A friend of mine (who is better at math) told me that there is no such thing as 'implicit multiplication', only shorthand so that is in fact done after the division (going left to right, not necessarily because division occurs before multiplication. But he didn't explicitly give a reason)

What is the answer and why?

Aryan
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    I have never understood why one would want to establish arbitrary conventions on the order of operations. Just place appropriate parentheses and remove any ambiguity. Why would one ever want to write something like $5+8\cdot 6:4+3-2:5$? – wildildildlife Apr 09 '11 at 15:34
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    @wildildildlife: Well, the reason we put a convention on the order of operations is to make our life simpler. Not for problems like this, mind you, but for general expressions. E.g., if you had to write all the necessary parentheses every time you write out a polynomial, it would obscure things rather than clarify them. Instead of $2x^2+7x+1$, you would have to write$$\Bigl(2\bigl(x^2\bigr)\Bigl) + \Bigl(7x\Bigr) + 1.$$ – Arturo Magidin Apr 09 '11 at 19:36
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    «There is a huge debate on the internet»?! – Mariano Suárez-Álvarez Apr 15 '11 at 21:27
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    @Mariano You beat me to it. – Adrián Barquero Apr 15 '11 at 21:29
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    haha and personal friends as well. But most of us agree on 2. However the 288 guys refuse to change their mind. Some of them said being drunk and doing math is not a good idea (as we were at a bar at the time) –  Apr 15 '11 at 21:29
  • http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics#Ambigious_or_not.3F – GeoffDS Apr 15 '11 at 21:29
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    I don't think this is a place for questions which google can answer (http://www.google.com/search?q=48%C3%B72%289%2B3%29). – Fabian Apr 15 '11 at 21:31
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    @Numth: Good link, but nothing conclusive. @Fabian: Your assuming google is always right. Mathway says 2. Do i believe a search engine or a math site...... –  Apr 15 '11 at 21:33
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    @Fabian: Half the questions on SE Google could answer, admittedly not as simply; also with precedence matters, BEDMAS is a standard to avoid confusion, as opposed to a cast iron law. There is scope for the calculation to be interpreted differently. Maybe the question would be better phrased "Why is BEDMAS the "correct" order?" rather than an example. – Orbling Apr 15 '11 at 21:35
  • @Orbling: The debate is more about if the multiplication is part of bracket or is it a normal multiplication. Some people throw in PEMDAS but i ignore that. –  Apr 15 '11 at 21:38
  • I agree. Everybody can define the expression like he/she likes. I still believe that the question is not suitable for math.SE... – Fabian Apr 15 '11 at 21:38
  • @Cplayer: Is there nothing that says if 2( is part of the bracket when doing order of operation? @Fabian: How would you feel about BEDMAS VS PEMDAS discussion? But i dont feel that is relavent or resolve the question as both multiplication and division take the same precedence. –  Apr 15 '11 at 21:41
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    @acidzombie24: I don't understand why this has anything to do with BEDMAS VS PEMDAS. In my understanding $2(something)$ is just a shortcut for $2×(something)$ used such that does not have to write a lot (and if it is unambiguous). Otherwise, you would have to define a function $2(x)$... – Fabian Apr 15 '11 at 21:45
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    See also http://math.stackexchange.com/questions/16502/do-values-attached-to-integers-have-implicit-parentheses – Bill Dubuque Apr 15 '11 at 21:53
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    I must be crazy. I wasted 5 min of my life on this ... – GWu Apr 15 '11 at 21:59
  • @acidzomb The point of my link was not to conclude anything. If this is between you and your friends, it shows that you're asking the same question at a bunch of different sites. – GeoffDS Apr 15 '11 at 22:14
  • @Numth: Its all over the internet. I only heard about it a few days ago. It started over a week ago. I only asked this once. –  Apr 15 '11 at 22:25
  • @Fabian: I would have thought questions on mathematical notation are appropriate here. – Orbling Apr 15 '11 at 23:03
  • @Lovre: I've merged the other question into this one, since they are absolutely identical in intent. – Willie Wong Apr 16 '11 at 17:29
  • Mathway spit out $288$ for me, not $2$. There is nothing ambiguous about $2( \cdots )$. Now I do have a problem every time I see an expression like $\sin 3k\frac{\pi}{2}$ – Joshua Shane Liberman Apr 29 '11 at 18:19
  • http://knowyourmeme.com/memes/48293 This should be deleted. @Bill @Adr @Mar etc – The Chaz 2.0 May 02 '11 at 21:56
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    I voted to reopen. I don't see why this is not a real question. The fact that someone picked it up as a meme on the net and wrote "there's a huge debate on the internet" doesn't make the question whether division or multiplication takes precedence here invalid; this is a perfectly legitimate question about mathematical notation. – joriki Feb 25 '12 at 17:22
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    @joriki: I think that you should start a meta thread about that, instead of bringing this into discussing in these comments. – Asaf Karagila Feb 25 '12 at 17:34
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    Note $ $ This question was recently reopened based on discussion in this meta question. Please join the discussion there before voting to close again. – Bill Dubuque Jul 28 '12 at 01:29
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    The most amazing thing is that guys asking these questions are, according to StackOverflow, good programmers! No wonder many programs crash once in a while... – JMCF125 Jun 13 '13 at 22:23
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    I would rewrite it as $48÷2(9+3) = 48\cdot \dfrac 12 \cdot (9+3)$ – Steven Alexis Gregory Jun 19 '16 at 18:25
  • https://i.stack.imgur.com/gYLWw.jpg – sergiol Mar 04 '17 at 02:09
  • @jmcf125 Either that, or they were the first to ask the really dumb questions. – GregRos Nov 06 '17 at 01:03
  • In Wolfram Alpha: enter these 2 equations 6÷2(2+1) & 6÷2x3 and then replace the ÷ with / . something is not right! in the first the equation = 9 then = 1, in the second the equation = 9 and then = 9 this appears to be more a fault of Wolfram Alpha. but it definitely demonstrates – TheArchitecta Nov 26 '22 at 11:23

4 Answers4

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There is no Supreme Court for mathematical notation; there were no commandments handed down on Sinai concerning operational precedence; all there is, is convention, and different people are free to adhere to different conventions. Wise people will stick in enough parentheses to make it impossible for anyone to mistake the meaning. If they mean, $(48\div2)(9+3)$, they'll write it that way; if they mean $48\div\bigl(2(9+3)\bigr)$, they'll write it that way.

Gerry Myerson
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    Recommended reading on this topic: "Order of operations" and other oddities in school mathematics by Hung-Hsi Wu, a Berkeley mathematician who has given much thought to mathematics education. – Bill Dubuque May 04 '11 at 14:30
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    My teacher adheres opposite to the convention I like. And she writes it so that there's ambiguity, resulting in rare "mistakes" on my part. Whenever I see $48÷2(9+3)$, I automatically think $2$. – Mateen Ulhaq May 07 '11 at 00:01
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    "Wise people will ..." I can't agree more. (+1) – user26872 Jul 28 '12 at 03:36
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    @muntoo, your teacher isn't wise. :) – JMCF125 Jun 13 '13 at 22:19
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    I don't agree with this type of answer: even if this expression is written by mixing old style symbols and new style ones, you cannot let the idea that you can evaluate an expression by your mood, or convention. There is ONE convention which should be agreed on by any mathematician: multiplication and division have the same precedence. – Emanuele Paolini Oct 14 '13 at 05:35
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    @EmanuelePaolini: "ONE convention which should be agreed on by any mathematician"? I think the problem is exactly the fact that most mathematicians don't use the $\div$ sign, so (from lack of use) we haven't really agreed on a convention. (In fact, just to write that sentence I had to look up how to write $\div$ in TeX because I never use it.) – Jesse Madnick Mar 04 '14 at 13:58
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    @EmanuelePaolini which one, decree the convention! ;) – mach Mar 31 '14 at 05:41
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    @mach as I said: multiplication and division have the same precedence and evaluate left to right. – Emanuele Paolini Mar 31 '14 at 06:57
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    Comment thread necromancy: I wonder if people would distinguish $48 \div 2(9+3)$ and $48 \div 2 \times (9+3)$. If some would, then order of operations isn't the only factor involved. – Brian Tung Aug 06 '20 at 18:56
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It's ambiguous, there is not one right answer in this case, other than possibly that it is undefined. You may have $$\frac{48}{2(9+3)} = 2$$
or
$$\frac{48}{2}(9+3) = 288$$

Therefore, there is no point in debating this.

Note that the reason you are get different answers from mathway and google calculator is that the algorithms they use for parsing input are different. These algorithms apparently (and understandably) leave it up to the user in this case to give input that can only be interpreted in one way. This is not the case, and is therefore why the two's answers differ.

Justin
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  • Is there nothing that says if "2(" is part of the bracket operation or not while doing order of operation? If not then i'll agree. –  Apr 15 '11 at 21:42
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    @acidzombie24: No, there is not. The standard notation to use in this case if you are going to use brackets is to use "fractions" rather than $÷$, as shown. Hope this clears things up =D – Justin Apr 15 '11 at 21:44
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I would say it isn't even well-defined. In Group Theory or such, you usually pass by a statement that says "associativity means that $(1 + 2) + 3$ is the same as $1 + (2+3)$, so we can write $1 + 2 + 3$ without ambiguity." $\div$ doesn't have this property of being unambiguous.

This is one of the advantages of using $\frac{48}{2}(9+3)$ or $\frac{48}{2(9+3)}$ - it's not quite associative, but it isn't ambiguous. I haven't seen $\div$ since elementary school probably for this very reason.

Aryan
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Michael Burge
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    And personally, I like to think of the obelus as a symbol meaning that "something" goes on top and bottom of the line, and the dots are just placeholders for the things on the left and right. – Joe Z. Apr 02 '14 at 04:00
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    Would it really be any different if ÷ were replaced with /, the symbol most programming languages use for divide? – Alec Jacobson Dec 05 '17 at 03:05
  • forward slash would then put everything following into the denominator. and hence would make it obvious $\frac{48}{2(9+3)}$ – TheArchitecta Nov 26 '22 at 10:05
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There is no order difference between implicit and explicit multiplications. Purplemath suggests that implied multiplication outside of parentheses also gets parenthetical order priority over all other multiplication(division). So they would interpret the implied multiplied parenthetical as $48\div(2\times(9+3)) = 2$.

Alternatively, all implicit-capable calculators I've tried give the same results as Wolfram Alpha which interprets the implication as $(48\div2)\cdot(9+3) = 288$

Some confusion also seems to be the $\div$ division symbol itself as $48/2(9+3)$ visually supports the mutual parenthetical implied multiplication of Wolfram Alpha: $\frac{48}{2}\cdot(9+3) = 288$

The short answer is that the formula as written is too easily misinterpreted and the author should clarify it to ensure its proper calculation.

PS: to further furrow your brow - employing the parenthetical as a variable yields different results. So, $48\div2c$ where $c=9+3$ yields $2$ - but this conflates the distinction from parenthetical to coefficient-variable syntax. $48\div2\cdot c$ where $c=9+3$ yields $288$.

wythagoras
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    it seems like your confusing link to wolfram alpha that yielded 2 was a bug, since now it also yields 288. – Jens Timmerman Dec 21 '15 at 16:17
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    Some calculators parse implicit multiplication differently than explicit. This answer is incorrect. TI calculators do this. http://mathforum.org/library/drmath/view/72166.html – Sue VanHattum Aug 02 '19 at 20:07
  • @SueVanHattum, your link is broken. From what I have seen TI calculators use a forward slash. instead of a division symbol, and yet if you convert to long hand they are getting the answer wrong, so I would suggest TI is computing wrong, because everything after the forward slash should be in the denominator TI calculator is wrong. – TheArchitecta Nov 26 '22 at 09:59
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    Oddly, my link goes to something other than what it says. Here's a link to a good article: https://www.themathdoctors.org/order-of-operations-implicit-multiplication/

    I'm not sure how you can call a calculator wrong. No, everything after the slash is not required to be in the denominator. It depends on interpretation. Amusingly, my TI-30X IIS shows the bar and dots symbol on the keypad, but uses the / on the screen.

     – Sue VanHattum Nov 26 '22 at 21:14