

6÷2(1+2)=? - StevenXC
http://stevenclontz.com/blog/2013/03/17/order-of-operations/

======
latimer
According to Google, it's 9

<https://www.google.com/search?q=6%C3%B72(1%2B2)>

------
d0m
>> The issue stems from the incorrect (but often subconscious) belief that
there exists a Bible of Mathematics somewhere

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.

The way I see it is:

    
    
      6/2(1+2)
      6/2(3)
      6/2*3
      3*3
      9
    

If I wanted to apply 2 _3 first, I'd write: 6/(2_ 3). Maybe I thought like
that because that's how all calculators I've used work.

~~~
baddox
I am very surprised that people would evaluate the multiplication via
juxtaposition after the division symbol. According to that method, 6 ÷ 2x
would be evaluated as 3x, which I find extremely unlikely to be the intention
of that expression.

~~~
d0m
Well, 6 ÷ 2x is similar to:

    
    
      6
      __
      2x
    

You've explicitly put spaces around it.

However, if you look at it like:

    
    
      6/2*x  
    

Does 3x seems that unlikely?

~~~
baddox
Again, my point is that my interpretation depends on whether it's
multiplication via the multiplication symbol, like 6/2*x, or multiplication
via juxtaposition, like 6/2x.

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yaakov34
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.

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.

~~~
StevenXC
My (intended) implication was that answering the question

> What is 6÷2(1+2)?

without establishing the axioms of arithmetic is as impossible as answering

> Does CH hold in ZF?

without establishing some axioms of set theory.

Put another way, asking 6÷2(1+2) on the SAT is as ill-advised as asking an
Introduction to Proofs course to tackle the Continuum Hypothesis with just
some naive set theory.

~~~
yaakov34
Again, I think that's a very bad comparison - resolving this difficulty is not
"impossible". You can choose whatever convention you like, and all that
changes is the notation. Of course it's a bad test question, but it is not
"impossible" or even remotely difficult for mathematics - it's just a matter
of choosing a notation.

You seem to think that this question has something to do with the "axioms of
arithmetic" - no, it doesn't, it's a matter of how you write things by
convention. Operator precedence is not and has never been an "axiom". We can
define plus to have the highest precedence, and get the exact same arithmetic
we have now, written differently.

On the other hand, deciding the Continuum Hypothesis is "impossible" in a very
fundamental way - Kurt Godel and Paul Cohen, two of the greatest mathematical
logicians in history, proved that the Continuum Hypothesis is undecidable by
mathematical reasoning as we can best formulate it, i.e. it is independent of
the ZFC axioms. That's not a matter of choosing a notation for it.

~~~
StevenXC
Whether you consider the properties of arithmetic to be axioms or some other
word which describes a fact assumed without proof is a matter of semantics.

The point was that without a precise foundation for mathematics we cannot
proceed - anything further is overanalysis for an article I wrote mainly for
folks without our mathematical background. :-)

~~~
yaakov34
There is a fundamental difference between conventions of notation, and
properties of the thing being notated, in this case arithmetic. I'm frankly
surprised that someone with a mathematical background keeps mixing up the two.
There are a dozen people in this thread of comments posting different
notations - prefix, postfix, what have you. Do you think these define a
different arithmetic? Now of course it's stupid to ask people a question
without defining what you mean by your notation - you might as well mumble
something and expect them to decipher it. But that is in no way equivalent to
asking a question which is undecidable in mathematical logic (which is not
necessarily a stupid thing to do). And features of some particular notation do
not in any way, shape, or form, or by any stretch of semantics, constitute
axioms, or properties, or anything else which belongs to arithmetic itself. Do
you really disagree with that?

------
grannyg00se
(/ 6 (* 2 (+ 1 2)))

The answer is 1.

(* (/ 6 2) (+ 1 2)) would give nine but c'mon...that multiplication way over
on the left came outta nowhere.

~~~
mseepgood
I prefer 6 2 1 2 + * / and 6 2 / 1 2 + * notation.

------
armadillowarran

      6/2(1+2)=6/2*(1+2)=6/2*3=(6/2)*3=(3)*3=9
    

This is pretty well defined, I feel:

\- Expand implicit multiplication to explicit multiplication (e.g. 2(1+2)
becomes 2*(1+2))

\- Left-to-right precedence for operators of equal-precedence (divide and
multiply)

The 2 is no more bound to the parenthesis than it is to the division operator.

Am I missing something?

~~~
gizmo686
When you have 6/2(3), it is not well define if the correct expansion is 6/(2
_3) or (6/2)_ 3\. In my experience with math, the intuitive answer is that
implicit multiplication takes precedence. For example, say instead of 6/2(3),
you had 6/2x. Following your convention, that would expand to (6/2)x, not
6/(2x). You might say that the () around the last number make a difference.
However that would imply that 6/2x!=6/2(x). Which I find deeply unsettling if
true.

Ultimitly, this is why we tend to use notation which uses placement to resolve
these issues unambiguasly, without alot of parentheses.

~~~
armadillowarran
I find it interesting you feel this way.

In my experience, it is pretty well defined:

0\. Evaluate (the inside of!) parenthesis, then

1\. Evaluate exponentials, then

2\. Evaluate division and multiplication operators, left to right

    
    
      e.g. 1 / 2 * 3 / 4 = (1/2) * 3/4 = ((1/2)*3)/4
    

3\. Evaluate addition and subtraction operators, left to right

Your issue seems to stem from the left-to-right concept (in order in which
operators are encountered as you read, left-to-right).

Your example of 6/2x to me is clearly (6/2)x. What is 1/2x ? In my experience,
textbook authors/professors/math teachers tend to be disambiguous and either
use the horizontal line for clarity or use parenthesis.

I attended public schools in Ontario, Canada if it makes any difference.

Edit: Oh yeah, also I have never felt that implicit operators would take
precedence. Interesting!

------
sil3ntmac
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.

~~~
taproot
BEDMAS (what I was taught in school) really throws some confusion in there
concerning PEMDAS. Heh. We've started using PEMA in NZ to lessen confusion. I
assume they go on to explain where division / subtraction went.

------
EllaMentry
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.

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.

Machines are just as bad, as they are loaded with assumptions of their
programmers.

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.

Implicit operations and ordering may save you a couple of characters but
generate essays worth of confusion.

------
baddox
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.

~~~
ulyssesgrant
In high school (and college) I've always been taught that multiplication and
division have equal precedence and should be evaluated in the order in which
they are encountered.

~~~
baddox
Was multiplication by juxtaposition ever mentioned explicitly? I was taught to
give them equal precedence, but that was explicitly for the division symbol
and the multiplication symbol (* or x). Juxtaposition, like 2x or 2(2+1), was
always considered to have greater precedence than the division and
multiplication symbols.

~~~
Semaphor
Nope, never heard that in my 13 years of school in Germany.

------
naftaliharris
> 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.

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.

------
Zenst
Computers will conclude the answer is 1 as they will prioritise the multiply.

Now us humans will get 9 as they will do the 6/2 and then multiply the result
with the (1+2) for 3x3.

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.

Moral being whilst lots of brackets and braces can look untidy, they do
clarify beyond doubt.

~~~
andrewmunsell
No, they won't. Computers will compute it as nine because they evaluate the
parenthesis, then go left to right.

6 / 2 (1 + 2)

6 / 2 * 3

3 * 3

9

------
ColinWright
I blogged about this nearly two years ago:

<http://www.solipsys.co.uk/new/AMatterOfConvention.html?HN>

~~~
ari_elle
Nice, i would have said 9 just because that's how C interprets it.

float x = 6 / 2 * (2 + 1); // x == 9

~~~
beatgammit
Right, but you also had to explicitly add the asterisk for multiplication. I
thought the same thing, but the remarks on juxtaposition do have a point. For
example, if an integer were callable, you'd do the function call--
2(2+1)--independantly of the division.

------
lucb1e
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 :/

------
brodney
Is it 1+3 or 1+2? Title does not match content.

~~~
StevenXC
Thanks for the catch!

------
rubbingalcohol
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

------
Aloisius
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).

~~~
StevenXC
The question is taken from a Facebook post making the rounds. The parentheses
are important however, as they introduce a new ambiguity: does the implied
multiplication from the open parenthesis trump the division symbol on the
left?

Put another way, does it fall into the Parentheses section or the
Multiplication section of the PEMDAS order of operations? (Or whatever
mnemonic your country uses.)

~~~
Aloisius
Why would the multiplication fall in the parenthesis section when it clearly
isn't in a set of parenthesis? 6÷2(1+2) expands to 6÷2*(1+2). The items inside
the parenthesis first and there should be no ambiguity about that.

------
poindontcare
[http://www.wolframalpha.com/input/?i=6%C3%B72(1%2B3)&t=c...](http://www.wolframalpha.com/input/?i=6%C3%B72\(1%2B3\)&t=crmtb01)

~~~
Samuel_Michon
I think you meant: <http://www.wolframalpha.com/input/?i=6%C3%B72%281%2B2%29>

But that's just how WolframAlpha does it. The article is about the ambiguity
in the mnemonic we're taught in school. I thought the correct answer was going
to be '1'.

------
ibotty
just as a data point:

as google, wolfram alpha (a mathematica "frontend") interprets it as
(6/2)*(1+2).

[https://www.wolframalpha.com/input/?i=6÷2(1+2)&dataset=](https://www.wolframalpha.com/input/?i=6÷2\(1+2\)&dataset=)

(and even with a variable juxtapositioned it's interpreted like that.)

but right, that's no real metric for mathematical use. (i still think the
result is 1.)

------
Datsundere
It's one. He himself said parenthesis takes precedence. So:

6 / 2(3) <== Parenthesis still there

6 / 6

1

~~~
taproot
wooosh

\--

It's 9, you've added some invisible brackets in there.

6 / 2 * (1 + 2)

6 / 2 * 3

3 * 3

9

------
Kaivo
When converting it from infix to postfix, it returns 6 2 / 1 2 + * which
returns 9. I prefer this point of view.

------
braum
I was taught to handle parenthesis first before multiplication or division so
the answer is clear to me to be 1.

~~~
andrewmunsell
Well, is the implied multiplication technically part of the parenthesis,
though? I've always been taught the stuff _inside_ the parenthesis takes
precedence, but outside is irrelevant because it's not technically "part" of
the parenthesis statement.

It's like how programming works:

6 / 2 * add(1, 2)

This would obviously evaluate to 9, because the "add" function would simply
return a value and the rest of the statement would be evaluated from left to
right. In this case, the first 2 is not related to the "add" function in any
way.

------
s800
That "divide-by" symbol can suck my left nut.

------
montogeek
1

------
drivebyacct2
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).

~~~
d0m
You wasted time on _this_ by writing this comment.. not sure why you're
discriminating others as idiots.

~~~
drivebyacct2
I'm 96% sure you missed the point of my post.

