According to Google, it's 9<p><a href="https://www.google.com/search?q=6%C3%B72(1%2B2)" rel="nofollow">https://www.google.com/search?q=6%C3%B72(1%2B2)</a>
>> The issue stems from the incorrect (but often subconscious) belief that there exists a Bible of Mathematics somewhere<p>Well, yes, I thought so. Most mathematical expressions are pretty standard and I'm surprising that there is no a definitive answer to this question.<p>The way I see it is:<p><pre><code> 6/2(1+2)
6/2(3)
6/2*3
3*3
9
</code></pre>
If I wanted to apply 2<i>3 first, I'd write: 6/(2</i>3). Maybe I thought like that because that's how all calculators I've used work.
I don't understand his idea of comparing the independence of the Continuum Hypothesis from the other axioms of set theory, which is a very deep and important result about (in essence) one of the limits of mathematical reasoning, with this little notational difficulty, which frankly is of very little interest, especially to mathematicians (you do run into this when writing parsers, sometimes...) He actually manages to say that this is "as stupid as" asking for a proof of the Continuum Hypothesis. Reminds me of people who get hung up on whether 1 is a prime number. Yes, no, maybe - people have defined it this way and that, it's a convention you can make without changing anything fundamental about mathematical reasoning.<p>Different systems of axioms lead to very different mathematical objects (compare Euclidean and non-Euclidean geometries). Different conventions lead to the exact same thing written slightly differently - if we say that clockwise rotation corresponds to positive change in angle (the opposite of current convention), we'll exchange plus and minus signs in a bunch of formulas, and nothing else.
<p><pre><code> 6/2(1+2)=6/2*(1+2)=6/2*3=(6/2)*3=(3)*3=9
</code></pre>
This is pretty well defined, I feel:<p>- Expand implicit multiplication to explicit multiplication (e.g. 2(1+2) becomes 2*(1+2))<p>- Left-to-right precedence for operators of equal-precedence (divide and multiply)<p>The 2 is no more bound to the parenthesis than it is to the division operator.<p>Am I missing something?
I have never understood why you would process multiplication and division as separate reduction steps, besides for simplification in elementary-school learning materials. It seems completely arbitrary. Division is just multiplying by an inverse, and subtraction is just adding negative numbers. PEMDAS seems like something that is taught to you, and then never correctly explained.
This is not mathematics, this is symbol manipulation. The author is correct, the answer here is that it is a badly expressed problem - with multiple possible solutions depending on the assumptions the audience makes.<p>In mathematics allowing your audience to make assumptions is very very bad...people are terrible at making and applying assumptions - just look at any studies into eye witness testimony.<p>Machines are just as bad, as they are loaded with assumptions of their programmers.<p>As a Computer Scientist, I absolutely hate coming across badly expressed formulas. Abstraction is key - why are there parentheses around the 1+2, what makes those separate...why not just write 3?? - In fact there are no variables in this problem, it is constant, just give me the constant...or tell me why you have defined it this way.<p>Implicit operations and ordering may save you a couple of characters but generate essays worth of confusion.
Obviously, the author is right that there is no universal objective definition of the order of operations. Still, I find it surprising that anyone with a high school level mathematics education would perform the division via the obelus (÷) before the multiplication via juxtaposition.
> In the real world of mathematics, this is as stupid a question as me asking “Is there a mathematical size bigger than the amount of integers but less than the amount of real numbers?” [1] because you can’t answer it without going through a lot of trouble to specify the “axioms” (basic assumed rules for doing mathematics) you want used on the problem.<p>I don't think this analogy holds very well--a curious student could reasonably and earnestly ask about the continuum hypothesis, since it's not at all obvious from first principles that the answer depends so heavily on obscure set axioms. On the other hand, the order of operations question seems to have been designed to confuse people and stir up meaningless, unresolvable arguments over PEMDAS.
Computers will conclude the answer is 1 as they will prioritise the multiply.<p>Now us humans will get 9 as they will do the 6/2 and then multiply the result with the (1+2) for 3x3.<p>Now given the multiply is implied and computers like to have that symbol in many languages and the divide sign as again a form not overly used in programming languages, then it is clearly expressed in a form for human consumption. With that we imply the 6 divided by 2 is in its own bracket and will think it is 9, then we will think again if we know computers and then think 1.<p>Moral being whilst lots of brackets and braces can look untidy, they do clarify beyond doubt.
I blogged about this nearly two years ago:<p><a href="http://www.solipsys.co.uk/new/AMatterOfConvention.html?HN" rel="nofollow">http://www.solipsys.co.uk/new/AMatterOfConvention.html?HN</a>
The title (on his website) contains the wrong formula. I've skimmed the post a few times, wondering why the correct answer wasn't twelve... Then I saw it :/
It's funny because in other interpretations the "%" symbol evaluates to the modulus operator and this brings up another possible result. 6 mod 2(1+2) = 6 mod 6 ~= 0 mod 6
Wouldn't it have been easier to just express this question as 6/2*3? The addition part doesn't seem to add anything to the point of the article (no pun intended).
just as a data point:<p>as google, wolfram alpha (a mathematica "frontend") interprets it as (6/2)*(1+2).<p><a href="https://www.wolframalpha.com/input/?i=6÷2(1+2)&dataset=" rel="nofollow">https://www.wolframalpha.com/input/?i=6÷2(1+2)&dataset=</a><p>(and even with a variable juxtapositioned it's interpreted like that.)<p>but right, that's no real metric for mathematical use. (i still think the result is 1.)
The way you know this question is dumb? It was written by someone who either doesn't understand how to write math to be readable, or was written by a troll. The use of that division sign, ironically, doesn't make it any easier to determine, although there's surely a subset of FB idiots (the same that would waste time on this in the first place) that wouldn't understand it written as 6/2(1+2).