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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?

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    $\begingroup$ Don't write the expression like that if you want to avoid ambiguity. $\endgroup$
    – mrf
    Mar 30, 2015 at 8:33
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    $\begingroup$ 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. $\endgroup$
    – goblin GONE
    Mar 30, 2015 at 8:39
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    $\begingroup$ 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. $\endgroup$
    – Nick
    Mar 30, 2015 at 8:46
  • $\begingroup$ 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 $1$? $\endgroup$
    – JiK
    Mar 30, 2015 at 8:55
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    $\begingroup$ @Nick. It's not quite that easy. Some operators have right-to-left associativity. The usual convention is to interpret a^b^c as $a^{(b^c)}$. $\endgroup$
    – mrf
    Mar 30, 2015 at 9:21

6 Answers 6

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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.

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    $\begingroup$ 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. $\endgroup$
    – Gereltod
    Mar 30, 2015 at 8:45
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    $\begingroup$ @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. $\endgroup$
    – 5xum
    Mar 30, 2015 at 8:48
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    $\begingroup$ @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? $\endgroup$
    – 5xum
    Mar 30, 2015 at 8:54
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    $\begingroup$ I just read about PEMDAS from here: mathsisfun.com/operation-order-pemdas.html. It says Multiply and Division have same priority. Do it from left side. $\endgroup$
    – Gereltod
    Mar 30, 2015 at 9:03
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    $\begingroup$ @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. $\endgroup$
    – Nick
    Mar 30, 2015 at 9:03
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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.

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  • $\begingroup$ 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. $\endgroup$
    – user63985
    Aug 2, 2019 at 21:35
  • $\begingroup$ 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. $\endgroup$
    – user63985
    Aug 2, 2019 at 21:43
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    $\begingroup$ @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. $\endgroup$
    – David K
    Mar 24, 2021 at 3:33
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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 $4$.

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

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  • $\begingroup$ What could be well defined one? 8 / 4 * (4-2) is it? $\endgroup$
    – Gereltod
    Mar 30, 2015 at 8:34
  • $\begingroup$ @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) $". $\endgroup$
    – SamC
    Mar 30, 2015 at 8:54
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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.

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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.

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