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What is answer for 8 / 4 (4-2) = ?

My answer is 4. But some says it's 1. And arguing each others. They even using some calculators for prove them. Even those calculators showing both 1 and 4 as result. What should I tell to those who saying 1? Or my answer is wrong?

iadvd
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Gereltod
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    Don't write the expression like that if you want to avoid ambiguity. – mrf Mar 30 '15 at 08:33
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    Guys, please don't downvote this too heavily. We've all asked a question like this at some point in our lives, and this is a new user we're talking about. – goblin GONE Mar 30 '15 at 08:39
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    On the contrary, this a is a good question as far as bracket rules on a calculator are concerned. I mean, how does a calculator respond to such ambiguity? I'm sure many are interested. – Nick Mar 30 '15 at 08:46
  • A good question in SE should give some more original work: You mention you are arguing, yet you don't explain what your arguments are. Why do you think the answer is $4$, and why do some think it is $1$? – JiK Mar 30 '15 at 08:55
  • I think its 4 because multiply and divide have same priority. So you do which one is first. Others giving result as 1, because they think they must do first operand after you do operand inside () – Gereltod Mar 30 '15 at 08:56
  • @JiK: I'm sure he understands that there are different conventions that can be adopted. He's simply asking what the preferred one is. – Nick Mar 30 '15 at 09:06
  • @Gereltod: Calculator rule of thumb for inline ambiguous expressions, do the first one (of either multiplication or division) you encounter Left to Right – Nick Mar 30 '15 at 09:14
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    @Nick. It's not quite that easy. Some operators have right-to-left associativity. The usual convention is to interpret a^b^c as $a^{(b^c)}$. – mrf Mar 30 '15 at 09:21
  • @mrf: I was just talking about multiplication and division but that's a really nice heads up. – Nick Mar 30 '15 at 09:25
  • It's rather a duplicate of this one: https://math.stackexchange.com/questions/33215/what-is-48%C3%B7293 – Hans Lundmark Aug 02 '19 at 14:18
  • Very similar to this question – Maximilian Janisch Aug 02 '19 at 19:19

6 Answers6

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Don't get insulted by my next sentence, as I promise I elaborate after writing it:

This is a stupid question.

You are not stupid for asking it, and I guess it must be asked sometime, but I hope the continuation of my answer explains how irrelevant and pointless questions like these are.

The "perfect answer" to this question depends completely on the order of operation you have in mind, and obviously, the first thing to do is to perform the subtraction (because it is in brackets), meaning $$8/4(4-2)=8/4*2.$$

The next step is where it gets weird. Using the incredibly annoying PEMDAS rule, you need to first perform multiplication, then division, so $$8/4*2=8/(4*2)=1.$$

However, that's if you went to an American school. If you went to school in Slovenia (central Europe), you were taught that division is equivalent to multiplication, so you would get $$8/4*2=(8/4)*2=4.$$

In the end, the answer is completely ambiguous and depends on the conventions you were taught.

Now, my main point:

You may well ask why this question is "stupid" in my opinion. I mean, why would it be stupid just because the answer is "depending on convention"?

Well, the point is that knowing the answer to this question is completely meaningless. Even when you know the answer, you also know that, since conventions differ, you will in future use parentheses to make sure your meaning gets across.

The only true purpose of questions like this is to stir up "controversy", and many schools waste hours and hours of lessons to imprint either PEMDAS or some other rule into the skull of young kids. The result is that 10 year olds, instead of being excited about mathematics, end up thinking that mathematics is an algorithmic process in which you perform tasks a robot can perform much faster, and the result of these tasks is some number that the teacher then grades.

Then, you encounter someone that was taught a different set of conventions, and you find a problem (like the one here) in which the two conventions yield different results, and often times, people then conclude Huh, those silly mathematicians, they can't even decide on the rules they preach.

The end result of questions like this, therefore, is that mathematics gets a bad rep.

Thomas
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5xum
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    This question was just on Facebook by picture (not text). So everyone right now flaming each others and fighting. That's why I interested in more "professional" answer. – Gereltod Mar 30 '15 at 08:45
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    @Gereltod Then by all means, be a beacon of reason in the flame war. Go and tell the people fighting over this that 1) the question is ambiguous 2) mathematitians do not bother with questions like this. – 5xum Mar 30 '15 at 08:48
  • @5xum: So glad that mathematicians aren't the only people using this site. Math-enthusiasts FTW! – Nick Mar 30 '15 at 08:49
  • They don't hear me :) Lets just leave them. – Gereltod Mar 30 '15 at 08:49
  • @Gereltod Maybe post a link to this discussion into the flame war, some people may yet see reason. Or , of course, leave them. That's probably the most rational choice. – 5xum Mar 30 '15 at 08:50
  • BODMAS says Division has more precedence than Multiplication. The truth is they have same precedence. Thanks, you've just confused a lot of kids. Don't ever cite PEMDAS in the way you have again. Truth is Left-to-Right reading is more important and that's how calculators are designed. – Nick Mar 30 '15 at 08:52
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    @Nick Have you even read my answer? The whole wall of text I wrote was a spewing of concentrated rage against ALL conventions that are taught for the sake of convention alone. How was this "confusing" kids? – 5xum Mar 30 '15 at 08:54
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    I just read about PEMDAS from here: http://www.mathsisfun.com/operation-order-pemdas.html. It says Multiply and Division have same priority. Do it from left side. – Gereltod Mar 30 '15 at 09:03
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    @5xum: Yes, I read your answer. A good teacher teaches that the (MD) have equivalent precedence and inline expressions like that are ambiguous. The way a calculator interprets inline expression is the clear and direct answer required. And the reason I said "Don't cite PEMDAS in the way you have" is because it struck a nerve when you used it to say multiplication has higher precedence than division. If a kid infers that from PEMDAS, then it's the fault of the teacher, not the convention. – Nick Mar 30 '15 at 09:03
  • @Nick I wasn't brought up to use PEMDAS, so I guess I made a mistake. Still, there are conventions in which division has priority to multiplication. For example, I heard a rule once that the slash can basically be rewritten as a fractional line, and thus wraps everything to the right of it. The point is that mathematics should not be about teaching any stupid convention. – 5xum Mar 30 '15 at 09:06
  • @5xum: I agree, time has to allocated for lessons on ambiguity of inline expressions and how fractional expressions have invisible brackets. Coupled with unforgettable propaganda on the importance of brackets, conventions such as PEMDAS never have to be taught. But it's more of a school-system heritage thing by now. – Nick Mar 30 '15 at 09:16
  • @Nick Then we agree. It's bad, harmful, stupid heritage. – 5xum Mar 30 '15 at 09:27
  • I think this one have more why-this-happened explained comments. – Gereltod Mar 30 '15 at 09:29
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    You got a lot of stuff wrong here. First, 8/4(4−2)=8/4∗2 should be 8/4(4−2)=8/4(2) which can be further evaluated out to 8/4(2)=8/(4∗2). The answer is then clear. The second major issue is that you seem to think PEMDAS as a rule is different than BODMAS, BEDMAS, etc. They're all the same thing, and if taught correctly, yield the same answers. Perhaps they are bad acronyms, because they don't capture the nuance, but that's not really about math anymore. –  Aug 02 '19 at 21:26
  • And at this point, I mention GEMS as an alternative acronym. Groups, Exponents, Multiplications [which should be understood as including divisions, because 8/4(2)=(1/8)*4(2)], and Summations, [which should be understood as including minuses, because 8-4=8+(-4). Evaluate in this order, then left to right if there is more than 1 operator. –  Aug 02 '19 at 21:30
  • Mistake made first step 8/4(4−2)=8/(4∗2) i.e. you can't remove brackets unless there is only 1 term left inside, which clearly there's two. The "ambiguity" only arises from people prematurely removing the brackets. I'm aware that a lot of textbooks get lazy and do exactly that, but that's precisely why Curwens 10th rule of teaching is "Let the first impression be a correct one; leave no room for misunderstanding". I have precisely 1 textbook which obeys that guidance (a fully worked example from start to finish, not removing brackets until 1 term left inside). – donaldp Jun 01 '24 at 10:59
  • @donaldp You can remove the brackets. Because what is in brackets is $(4-2)$, and you first calculate that to get $8/4\cdot (2)$, after which you can remove the brackets around $2$ to get $8/4\cdot 2$. – 5xum Jun 03 '24 at 06:49
  • @5xum you can't remove brackets until AFTER you have distributed, as per The Distributive Law. The way you did it gives an answer of 4, but the actual answer is 1, hence the need for The Distributive Law. – donaldp Jun 06 '24 at 07:05
  • @donaldp The distributive law has nothing to do with this question whatsoever. – 5xum Jun 06 '24 at 11:46
  • @5xum You know The Distributive Law applies to EVERY instance of brackets right? Hence the name. Is there brackets in this expression? Yes there is, apply The Distributive Law literally first, as per my working out – donaldp Jun 07 '24 at 09:42
  • @donaldp The distributive law does nothing to address the problem. The problem is whether you first apply division or multiplication. If you apply multiplication first, then the distributive law gives $8/4\cdot (4-2) = 8/(4\cdot 4 - 4\cdot 2) = 8/8=1$. If you first apply division, then you get $8/4\cdot (4-2) = 2\cdot (4-2) = 2\cdot 2 = 4$. So yes, sure, distributive law applies here, but it does nothing to clarify the quesiton. – 5xum Jun 07 '24 at 10:02
  • @donaldp In other words, the problem being asked is the ordering of operations, and the distributive law says nothing about the ordering of operations. – 5xum Jun 07 '24 at 10:04
  • @5xum "apply division or multiplication" - there ISN'T any multiplication! 8 divided by 4 outside of 4 minus 2 - division, distribution, subtraction, and that's it.

    "distributive law applies here, but it does nothing to clarify the question" - doing brackets before division doesn't resolve the question?? You know doing division before brackets is against the order of operations rules, right?

    "distributive law says nothing about the ordering of operations" - order of operations doesn't say do brackets first?? Do go on...

     – donaldp Jun 08 '24 at 04:45
  • @5xum "8/4⋅(4−2)=2⋅(4−2)=2⋅2=4 So yes, sure, distributive law" - oh my. I only just looked at your working out - that's not distribution! You did division before brackets, which is against the order of operations rules. Go back and read my first comment - the first step of working out is there. I'm not sure you even read it. Also, there's no multiply between the 4 and the brackets - adding it changes the answer. i.e. there is quite literally no multiplication in the original problem, but you went and added it. – donaldp Jun 08 '24 at 05:05
  • @donaldp "there ISN'T any multiplication!"... Ok, then what is the operation denoted by $4(4-2)$? Also, a sentence like "doing division before brackets is against the rule" is a too vague statement. It's perfectly OK to do division before brackets in some cases. For example, $2\cdot (1 + 2/2)$, you do division first, then addition. "Brackets" aren't an operation in and of themselves, they are just a way of denoting ordering of other operations. – 5xum Jun 08 '24 at 05:12
  • @donaldp ". Also, there's no multiply between the 4 and the brackets" Of course there is. It is mathematical shorthand to omit the $\cdot$ simbol in some cases, but this is just shorthand. When we write things like $ab$, what is really meant is $a\cdot b$. See here: https://en.wikipedia.org/wiki/Multiplication_sign ("In algebraic notation, widely used in mathematics, a multiplication symbol is usually omitted wherever it would not cause confusion: "$a$ multiplied by $b$" can be written as $ab$ or $a$ $b$.") – 5xum Jun 08 '24 at 05:18
  • @5xum "the operation denoted by 4(4−2)" - distribution!

    "It's perfectly OK to do division before brackets in some cases" - no, it isn't. That's precisely the point of order of operations rules.

    "(1+2/2), you do division first" - Ah, that's inside brackets! That's part of solving brackets. You have to apply BEDMAS recursively to each nested set of brackets. If there were more brackets inside there then you would have to do them before that division.

    ""Brackets" aren't an operation" - no, they're a grouping symbol, and grouping symbols (BE) are solved before we do any operators (DMAS).

     – donaldp Jun 08 '24 at 07:04
  • @5xum "ab, what is really meant is a⋅b" - no, what we really mean is (axb). axb is 2 terms, ab is 1 term. That's precisely why adding a multiplication changes the answer. If a=2 and b=3, then 1/ab=1/6 but 1/axb=1/2x3=3/2.

    (sigh) Don't refer to wiki for Maths. Look in Maths textbooks. Search for "Smartman Apps order of operations" and you'll find a whole thread I wrote with textbooks, proofs, the works.

     – donaldp Jun 08 '24 at 07:10
  • @donaldp Thank you for exposing yourself as either a troll or someone completely uneducated in mathematics. – 5xum Jun 08 '24 at 07:15
  • To anyone else reading this thread, the two takeaways are: 1. $ab$ denotes multiplication, whether there is a multiplication sign in between them or not. 2. The expression $1/2\times 3$ is ambiguous, in the exact same way as the original expression by the OP is ambiguous. Anyone claiming there is a single correct way of interpreting $1/2\times 3$ is lying to you. – 5xum Jun 08 '24 at 07:25
  • @5xum "Thank you for exposing yourself as either a troll or someone completely uneducated in mathematics" - I'm a Maths teacher actually, hence the textbooks in my thread. The troll is the one refusing to learn the facts from a textbook.

    "ab denotes multiplication" - nope, it denotes a product. x denotes multiplication.

    "The expression 1/2×3 is ambiguous" - no it isn't. It's 3/2.

    "Anyone claiming there is a single correct way of interpreting 1/2×3 is..." someone who actually knows ALL the rules of Maths.

     – donaldp Jun 08 '24 at 07:42
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This all comes down to your conventions. If your conventions dictate that $8/4(4-2)$ is shorthand for $(8/4)(4-2)$, then it equals $4$. If you they dictate that $8/4(4-2)$ is shorthand for $8/(4(4-2))$ then it equals $1$. As a general rule, if something looks ambiguous, don't write it without adding in some brackets to help the reader.

By the way, there are systems of rules that disambiguate every such expression; some programming languages implement such things. However, in my opinion, its best not to leverage these kinds of "forced disambiguation conventions." You want to be writing for the reader, not against them.

goblin GONE
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  • Is there any convention that is not related to programming or computer science that says "8/4(4−2) is shorthand for (8/4)(4−2)". My understanding is that programmers do this only because 4(4−2) is the syntax for calling a function. There's virtually no programming language where you can write 4(4−2) without throwing an exception, yet this form is ubiquitous in high school level algebra. –  Aug 02 '19 at 21:35
  • Thus, if you're writing a calc app, you'd have to catch the exception or sanitize the input to never allow x(a) syntax before evaluating. –  Aug 02 '19 at 21:43
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    @63985 "There's virtually no programming language where you can write 4(4−2) ... ." True at least for well-known programming languages, but likewise in the same programming languages xy is the name of a single variable whereas in high school algebra it means $x$ multiplied by $y.$ You're picking at irrelevant differences in notation. Programming languages with C-like expressions will forbid (4-2)4 too, but 4*(4-2) or (4-2)*4 are fine. Now evaluate 8/4*(4-2). Still $4,$ but function notation has nothing to do with the reason. – David K Mar 24 '21 at 03:33
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It is not clear what you mean with the expression $8\ /\ 4\ (4-2)$. Do you mean: $\frac{8}{4(4-2)}$ or $\frac{8}{4}\cdot (4-2)$? The first expression is equal to $1$ and the second equal to $4$.

Karl
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There is no right/wrong answer to the question because the question itself isn't well defined.

John
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  • What could be well defined one? 8 / 4 * (4-2) is it? – Gereltod Mar 30 '15 at 08:34
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    @Gereltod For me no, "it make much more sense" to think $8 / 4 (4-2)$ as $\frac{8}{4 (4-2)}$ since if you want to write $\frac{8(4-2)}{4}$, you can do $8(4-2) / 4$ or $8 / 4 \times (4-2)$. However, the well defined one is what you said "$8 / 4 * (4-2) $". – SamC Mar 30 '15 at 08:54
  • Yes it is. The Distributive Law. a(b+c)=(ab+ac), but in this case b-c. – donaldp Jun 01 '24 at 11:06
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Depending how your read $$8/4~(4-2)$$ can be $$\frac 84 \times (4-2)=4$$ or $$\frac 8{4\times(4-2)}=1$$ As said in answers and comments, the notation you use is more than ambiguous. Use brackets to enclose the expressions.

Claude Leibovici
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I guess the most likely answer is $4$. Almost all programming languages (and some mathematiciens) would agree that

  • multiplication and division are at the same level of operator precedence

  • things on the same level are evaluated from left to right

so $8/4 (4-2)$ is a short hand for $8/4 \cdot (4-2)$, which is evaluated from left to right $(8/4)\cdot (4-2)=2*2=4$

to claim that it should be $1$ is to claim that $$ 8/4 (4-2) = 8/(4\cdot (4-2)) $$ which would be a strange convention.

Of course this problem is boring: in case of a tiny hint of a small possiblity of ambiguity use brackets.

Blah
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