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Dear internet - May 2, 2011
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Spiked Math Comic - Dear internet



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134 Comments

How could we possibly get a result other than 9? I don't get it.

Maybe: 6/2(2+1) -> 6/2(3) -> 6/6 [Mistake] -> 1

I think that I'd just call this a malformed input. There's a reason mathematical papers write out divisions and multiplications explicitly or use plenty of parentheses. Computer scientists do the same thing in their code; of the three programs I tried this in, two of them complained and exploded.

It't because multiplication and division have the same priority, so what is actually "hiding" between 2 and (1+2) is another multiplication sign. Therefore, 6:2(1+2)=6:2*(1+2)=6:2*3=3*3=9
This is a very common mistake that beginners in programming make because they forget that the computer doesn't think the same way we do.

Funny thing is, I got this wrong, but if it had been 6/2(1+2), I'd have gotten it right. Weird.

same here! I didn't recognize the division sign.
My husband had shown this to me the other day on Yahoo Answers too.

6/2 = 3
(1+2) = 3
3x3 = 9

You guys, the answer is 7. How many of you forgot to distribute?!

6÷2(1+2) => 6÷2+4 = 3+4 = 7

:)

This is hilarious. Sometimes that "new math" gets me so confused.

The division would still come before the multiplication
6/2=3
3(1+2)
3+6=9

why 7??

because if that's the answer,
then what's the answer to this..

6÷2(2+1)= ?

The TI-84 plus gives the answer 9.

It's been a long time since I've even seen that division symbol; it does make it ambiguous. If you'd said 6/2(1+2) I'd have fairly confidently assumed everything to the right of the / was the denominator (up to the next + or -, of course), and yet, I suppose the two symbols are supposed to mean the same thing. And yet, if it had been 6/2*3 I'd have asked for clarification, and if it had been 6ø2*3 I'd have just carried out the two operations in the order they appear without a second thought.

Um... just pretend that's the division sign in the last equation. Off-by-one error caused by reading a table of html entities from Google rather than the page itself. :)

The issue is that the implicit multiplication: "2(1+2)" actually has same priority as parenthesis, so what's going on here is this:

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

this is known as juxtaposition.

Wrong. Division takes two operands a/b . What you are saying would be correct if you were doing the parsing from right to left. But since the convention si a/b where a is a multiple of b and not the other way round

its funny, if you had written it as 6/2x(2+1) i would have gotten it right, but without that x sign, i read it entierly differently

I just think we have learned to be "bothered" by parantheses, so we have a tendency to want to remove it first. It's only a heuristic.

BEDMAS still works for your particular example though so perhaps if you had given 2 example questions so that it wouldn't.

Thanks spikedmath! I have been arguing with FaceBook friends about this!!

The value is undefined because the statement is ambiguous.
A / B * C is no more meaningful than P AND Q OR R, in that both require additional Brackets to define the precedence.

Incidentally, C-based programming languages will parse both, because they DO have absolute order of precedence.

no, AND has high priority than OR: in binary code, AND corresponds to x and OR corresponds to +, so P AND Q OR R has no ambiguity.

AND corresponds to x and OR corresponds to +
WHAT?
This is utter bollocks.
Boolean operators don't remotely "correspond" to mathematical ones, even in programming languages.

As stated, AND & OR have defined relative precedence in coding, but only in coding. In a mathematical or formal logic context A AND B OR C is not a syntactically valid statement.

Of course they do. From the truth table of AND and OR, it is easily seen that they correspond to operations x and + in F2, the field of two elements. Since it is a field, you can talk about the priority of x and + just like Q or R or C, and of course, all other arithmetic laws, such as:
A AND (B OR C) = (A AND B) OR (A AND C)

Actually, AND and XOR correspond to multiplication and addition in F_2. OR does not have a natural arithmetic analogue (unless you want to talk about 1-(1-x)(1-y))

You can distribute either operation. Precedence b/w AND and OR doesn't matter.

they're both correct.
(mod 8)

But Bourbaki said it is a meaningless expression!

Better to use non-ambiguous expressions unless you hope to be misunderstood. For, say, ring theorists, it is a more natural interpretation to read it as a fraction x/y, where x and y are integers and y just happens to be given by an algebraic expression.

Personally, to describe your ambiguous operation, I would use the much simpler "6 2 ÷ 1 2 + ×".

no, AND has high priority than OR: in binary code, AND corresponds to x and OR corresponds to +, so P AND Q Or R has no ambiguity.

sorry, double posts from above...

I am one of those who agree that unspoken multiplication/divison is at the same priority as parenthesese.
I surmise that the author knew that already and made this comic a joke. But this comic is more confusing more than fun to a person with ordinary Math background...

It's not a joke... or rather, the joke is that this shouldn't be confusing because the convention is defined to be that implicit multiplication is NOT the same priority as parentheses. I agree that it seems sensible that it could be the other way, but it is not. Check Google, Mathematica, a T1-84 plus, or Wolfram|Alpha... the comic is correct.

Haha, that was a good one, I would have never expected anyone to get any other result than 9, but googled this and actually there are people who get it wrong. Anyway, I much prefer the airplane-conveyor-belt problem, since it actually does contain a lot of ambiguity and you can have a nice discussion over what the result will be under different assumptions.

As stated above, the issue is implicit multiplication. If one saw 6/2X, then one would assume it is 6/(2X). Because the leading 2*() format is used, it is ambiguous as to what the author intended. I think convention is that if the 2 is not a coefficient of a variable expression, then it is treated as multiplication and the answer is indeed 9. But if it was 6/2X, where X=1+2, then it would be 1.

*Because the leading 2*() is NOT used, sorry for the typo

Intriguing. I *do* see 1/2X, with X=(1+2), so I computed the result to be 1.

You know what happened? Osama died! I think I now know the reason...

What? He divided by zero? He used an ambiguous formula??

I was having a discussion on this problem recently. I stayed out of it to say that the problem is written poorly, and it makes a mockery of the order of operations and common sense. When you leave out a multiplication sign, you have an implicit set of parentheses.

Multiple divisions in infix notation are headache. For example a/b/c/d equals (a/b)/(c/d) (double fraction) not ((a/b)/c)/d. And I never got clear answer from math teachers how to evaluate a/b/c.

This is simply because a/b/c makes no sense. We can do it with multiplication because of a theorem stating that the result for (a*b)*c or a*(b*c) is the same.
Hence, either one shouldn't use the notation at all, or explain beforehand what they mean.

a/b/c evaluates to (a/b)/c = a/(b*c)

The left-to-right evaluation of operators is universally assumed unless you're using reverse Polish notation or something to that effect (in which case you'd need to specify that).

This comic (and the heated discussions!) indicate on the contrary that there is no clear and universal convention when using division. So, better to make sure and use parentheses.

Plus, as a left-notationist, I tend to evaluate strings of operators from the right. For example, f \circ g means "apply g first, f afterwards". This is the more commonly used convention in mathematics, but it is not universal enough to go without saying.

Actually the comics should read:
What is 6÷2(1+2)?
Dear internet... the answer is NaN.

But what if you solve it using the distributive rule of multiplication? Solving it that way, 2(1+2) becomes [(2x1)+(2x2)] = 2+4 = 6. Then it's 6 / 6 = 1

But the whole point is that the associativity doesn't work that way. Division and multiplication have equal precedence and associate to the left. Fully parenthesising the formula, you get
((6÷2)(1+2))
and then the distributive law gives
(6÷2) + 2(6÷2) = 3(6÷2) = 3x3 = 9

Thumbs up to the person who so concisely pointed out the major point here. It just as we read - left to right.

Associativity doesn't work that way because it doesn't work at all when divisions are involved.

HP50g gives 9 too.

Apparently Mike missed this day in the school of xkcd.

+1 like.

I miss the old (good) xkcd

THANK YOU! That was exactly what popped in my head when I saw it on facebook. As mentioned above the expression is basically meaningless because of the ambiguity of the expression. So 1 or 9 or apparently 7 are legitimate answers. This is why we don't use that division symbol. And even with a / instead I would really prefer parentheses to make explicit when its argument. On a board this isn't an issue because you can draw the fractions with horizontal division signs and so everything under is clearly part of what you are dividing by. Here it just isn't clear at all.

Even though it requires unnecessary application of conventions, I don't think it qualifies as "ambigous" as the conventions used to solve this are valid and used through out Maths?

I'm not going to argue with the convention, but I disagree with the statement that it's 'used throughout maths' because the ÷ symbol is not used throughout maths. I've barely seen it since primary school, where it was only used for equations much simpler than the one above. People probably only get the answer wrong because ÷ is almost never used in such situations, so they're not familiar with or not used to applying the convention. And frankly, I'm happy with it that way. Using ÷ in equations like that doesn't help anyone.

Oh, and let's stick to Leibniz notation for calculus, eh? :)

Hear hear.

Ah, I thought of precisely that xkcd strip after reading today's spikedmath!

thanks for the link, xkcd expresses my sentiment precisely (i have been reposting that link in countless facebook comments)

RPN for the wizzle =p

Strictly speaking, according to PEMDAS etc., the answer would be 9. But nobody who intends the answer to be 9 would ever write the expression like that. They would write 6(1+2)/2. If the expression reads 6/2(1+2), I understand that they intended to write 6/[2(1+2)]. So the understood answer should be 1.

That's my opinion too.

If I'm ever a math teacher (which, for the sake of humanity, I hope never happens) I'm putting this on every test I ever give just to irritate the fuck out of everyone.

PEMDAS! I MISS U!

When written like 6 ÷ 2(1 + 2) it is actually translated to 6/(2(1+2)), which comes out to 1. For you to get 9 the equation must be written like this 6/2 (1+2). It is common mistake and this is considered a troll equation in the world of mathematics because people over look the uniqueness of ÷ symbol which changes a lot when analysed properly. We just tackled this equation in class because it has been going viral on the internet.

aww mike, you should've posted the *wrong* answer, and let everyone go nuts for a day or two before revealling your troll

Maybe he did just that!

What the heck does the I stand for in BIDMAS? Google says indices, but since when are those an arithmetic operation?

Wrong. It is not a matter of convention, because the convention is not to use that division sign.

With that sign, and that much space, there's an implied parentheses. In other words, 6 divided by 2x = 3/x

I agree, though, that 6 ÷ 2 x (1+2)=9

You're quite mistaken. Take a second to go look up what the obelus means. Heck, wikipedia it, if you must. The obelus is equivalent to the solidus. There are a few countries that use the obelus for subtraction, and a few that might use it for ranges (like 40÷50 meaning forty to fifty), but otherwise, the solidus and the obelus are equivalent.
tl:dr you're either a troll or misinformed.

I don't get the message in "Multiplication and division are of equal precedence, as are addition and subtraction".

In order to have a convention that produces unambiguous answers, a hierarchy must be defined. It wouldn't make sense to make the order of multiplication and division open to arbitrary choice, because then we would have situations like 6/2*3 which could be 9 or 1 depending on the arbitrary choice. So also with addition and subtraction.

The thing is, that addition and substraction is the exactly the same thing, since a-b := a +(-b). The same holds for multiplication and division. therefore the mnemonics are misleading. For the definition of a field you just "need" two operations (e.g. multiplication and addition).

You could also do what computers do and just use addition for everything...
Ask a CS major how today!

No...ALU's do not use addition for everything, they have different methods for multiplication and division. Unless you're referring only to addition and subtraction in which case disregard my comment.

Here it is not a matter of precedence but a matter of operator associativity. Since mulitplication and division have left associativity so the order of calculation is:
1> 1+2
2>6/2
3>3*3

Here it is not a matter of precedence but a matter of operator associativity. Since mulitplication and division have left associativity so the order of calculation is:
1> 1+2
2>6/2
3>3*3

I also recall that you should always switch division to multiplication by the reciprocal. Then the expression becomes 6 times 0.5 times (1+2) = 9 Presto!

Interesting. In kalgebra (libqalculate), if I use 6/2(1+2) the result is 1. But if I write 6/2*(1+2) the result is 9.

may I add, that 6÷2(1+2) is different than 6/2(1+2), and is also different than 6/(2.(1+2)) where additional brachets represent a "-".
EVEN THOUGH, I still read the ÷ as "-"... It's pretty weird though :)

Bad input, therefore undefined. I will brook no other answer. Assigning a convention to these types of situations just confuses matters. Much simpler to ask, "What the heck did you mean here, Jim?"

I'd like to ask a question...
What does BEDMAS, BIMDAS, BODMAS etc. mean?
(sorry for asking guys... I am not a native English-speaker)

To everyone that asked: The B in BEMDAS and all the others probably stands for brackets.

Interesting. On Qalculate! (advanced calculator for linux), 6/2*(1+2) returns correctly 3, but 6/2(1+2) returns 1. Obviously the "not written" multiplication is implemented with higher priority. But this can be useful, when writing long denominators, one doesn't have to write a lot of parentheses.

Are you sure about "correctly"? If 40% of the peolpe are unable to calculate the equation correctly, I guess there are some programmers among them :).

Yes 9 right

I still don't get it

Let (1+2)=a
Then
6÷2(1+2)
=6÷2a
=6÷6
=1

Why would the person write such if (s)he wanted to talk about
6(1+2)÷2

because you divide first...
Maybe this could help a bit better:
6/2*(1+2)
=6/2*3
=3*3
=9

See this is interesting, and the question becomes a lot more confusing when you say what is 6÷2x where x=(1+2)? The answer is still nine, yet one someone writes 2x, we generally think of that as all one term, and group it together wherever it goes (so here we'd think of it as 6/(2x)).

But all this proves is writing stuff on the computer and with ancient division symbols is difficult to understand sometimes.

I have to say that I'm a little disappointed in this comic, since you're not saying why it's 9, just that the convention has been adopted.
A choice for getting loads of comments/a choice with another reason?
or just a brain fart?

He did: "Multiplication and division are of equal precedence..." is the explaination! 6 / 2 * 3 = (6 / 2) * 3 != 6 / (2 * 3). It's no magic...

Dear internet,

This is a malformed question. There is no "math standard" that would allow you just to key in the numbers to your calculator, and expect to get the "correct" answer. You have to understand what you are doing, be aware of the pitfalls, and write your formulas in an unambiguous way.

And this from a guy who will fight you if you believe in Texas Instruments rather than me on the meaning of 5^5^5.

There is a standard, which allows the calculator to give you the "correct" answer, but there's a very good chance that this is not the correct question (assuming somebody actually wanted to calculate something rather than troll the internet.) The real correct answer depends on what the question was supposed to be, if there even was a question.

Dear internet,
6÷2(1+2)=42

Wait. Isn't 42 equal to 6x9 ?

Wait a minute! Isn't 42 equal to 6x9 ?

Sorry for the double post, blush.

Yeah, it's 42.

But did we find the question to the great answer? :D

yeah the answer is definitely 1

This irritates me... firstly if you apply the BEDMAS or what ever you get 9. When the info-graphic implies that bedmas doesn't work. The difficulty arises with the difference between a*b and ab does ab have higher precedence than A÷b which it should. But I cant find reference which bothers me.

similarly what is a a÷bc if a=4 b=2 and c=5 ? 10 or 0.4? I say 0.4

if I wrote a÷b*c then sure 10.

Late to the party, but if we view this as 6(2^-1)(1+2) - which is a more literal way of expressing an equivalent expression - it becomes painfully obvious. "Malformed input," as Vebyast put it, is the best way to describe what's going on here.

There is somehting called multiplication by juxtaposition, which would state that the multiplication indicated by the parentheses takes precedent, making the answer 1
Some ppl accept this, other laugh at it

THERE IS NO ORDER OF PRECEDENCE! EVERYTHING SHOULD HAVE PARENS.

If there is no order of precedence, how would adding extra operations help? Assuming your statement that "there is no order of precedence" is true, parentheses wouldn't have any precedence either...

maybe I misunderstood your use of "order of precedence". i meant that functions don't(read shouldn't) have some before others by default, as it should always be specified in the expression. does anyone know if the "function argument argument" can have similar ambiguities? anyway, things simply should not be written in an ambiguous way.

also: whoah you actually responded to me.

Maybe I'm just tired, but I'm not entirely understanding what you're getting at. Also, I'm not a famous Mike or anything. The author posts as Spiked Math :)

IF we use the order of operation of BODMAS (bracket,orders (exponents), division, multiplication, addition and subtraction)
6/2(1+2)
6/2(3)
3(3)
=9

This thing was discussed on the forums of a cricket managment website called battrick (no plug intended)

"Multiplication and Division are of equal precedence."

Then why do you have to divide before you multiply to get to your solution?

This equation uses a faulty and misleading way to describe division, that's why there is no real 'solution' to it.
1 and 9 are of equal 'correctness'.

You don't. Do you remember from Algebra 1 that dividing is multiplying by the inverse? So 6÷2 is the same as 6*(1/2). So now, solve the equation
6*(1/2)*(1+2)
Simple enough? There is a right answer. 9 has 100% "correctness", and 1 (and 7 as well) have 0% "correctness", as you put it. That is the real solution to it.
Q.E.D.

I remember Algebra 1. Nobody used that division sign.

I think the point is that since they're of equal precedence, the operations are carried out in the order that they appear, and that's why you divide before you multiply.

I would say that (apart from the fact that nobody who cared about the answer would write it that way) it's not the order of precedence of multiplication and division that confuses people, but the idea that implicit multiplication (without a * sign) has the same precedence as explicit multiplication. It looks like it should have higher precedence because all those numbers are right next to each other without even anything in between them to suggest they're separate, unrelated terms in the equation. Apparently it doesn't have higher precedence, so one shouldn't write 6÷2x unless one means 3x. But that's okay, because nobody would write 6÷2x anyway. Not only because anyone doing algebra is unlikely to use ÷, but also because anyone doing algebra is unlikely to use a literal number as high as 6. (One time, we ended up with a 12 in an equation, but it turned out a Greek letter had been mistaken for a 6 earlier on.)

Although division and multiplication have equal prevalence, the first 2 in this equation is part of the brackets. That equation is saying 6 divided by 2 '1+2's. Not 6 divided by 2, times '1+2'.

The answer is 1, teachers are right, the author is wrong.

Surprisingly.

The 6 is not part of the brackets, the 6 is next to the brackets. Go back to your algebra again, since you're not getting it.
3x = 3*x = 3÷(1/x) = 3*(1x)
All those are exactly the same. Multiplication by juxtaposition has NO PRIORITY over multiplication or division. All multiplication by juxtaposition says is that if you have a digit next to a variable, they are supposed to be multiplied together.
And just so you know, my teachers are right. But that might be because I'm a math major so all my professors are smart. And they taught me well which is why I am getting the right answer.

As a highshcool math teacher, I can't believe all the discussion that followed...usually folks on here are fairly mathematically inclined, and I was surprised by the number of responses that indicated some confusion. Should multiplication and division, equal precedence, left to right, not be straightforward? YES, THE ANSWER IS 9, FOR CRYING OUT LOUD!

How did my calculator end up with 3 anyway?

I am kind of tired of seeing this. regardless of all this mathematical order stuff, the issue is the inline notation and connection of the 2(1+2). This is either intended as the divisor of 6 as a whole, or just the 2 is the divisor. Sure, we all know order of operations and 6÷2(1+2) is technically exactly the same as 6/2*(1+2), and that the equation equals 9. However people associate a connected multiplication as a bracketed number grouping, so that 6÷2(1+2) is interpreted as 6/[2(1+2)]. Wikipedia also awknowleged this ambiguity, stating that "a/2b, which arguably should mean (a/2)b but would commonly be understood to mean a/(2b)". These people still follow order of operations, but they have first made an assumption based on common usage and get an end result of 1. I blame the author for poorly formatting their equation, leading to easy misinterpretation.

To me, conventions of PEMDAS and the like IMPLY their own parenthesis. To apply each operation as they suggest requires grouping off the quantities and operation symbols - typically done with parenthesis. So, to me, anyway, using PEMDAS: 6./.2(1+2) => (2(1+2))(1/6)=(2(3))(1/6)=(6)(1/6)=1. (The final multiplication by (1/6) seems to me to be the only way to follow the multiplication with division according to PEMDAS.)Without the additional parenthesis implied by PEMDAS, the interpretation would seem to be open to other conventions. For example, the left-to-right convention seems to correlate with the "spoken" interpretation: "Six divided by two times the quantity one plus two;" => (6./.2)*(1+2)=(3)*(3)=9. Each convention seems to imply its own additional parenthesis. PEMDAS gets me 1; TI-83 gets me 9.

My godfather! So much misunderstanding of simple conventions and mathematical principles. Ahem, degree in mathematical astrophysics and used to be a maths teacher (1983-1987) ...

The answer is 9. Real simple.
÷ is the EXACT SAME OPERATOR as /
. is the EXACT SAME OPERATOR as *
All you need
÷ and * have the same precedence
- and + have the same precedence
All you need is BOMDAS and left-to-right execution of equal precedence operators.

It can ONLY be 9.
I'm unimpressed that some CASs/calcs get it wrong.
Has anyone tried it with Maple, Mathematica, Octave ...?

(6/2)(2+1) would be 9

6/2(2+1) is 1

2(2+1) is an implied quantity [2(2+10] because the brackets are missing in the first part of the equation.

math is fallible. lack of precision is one of it's flaws.

oh and .999... is less than 1, i can prove it.

You have six cows. You divide them by 2 times (2+3) farmers. How many cows do each of the farmers get?

It's funny that you claim your argument is correct due to convention, and then throw a fit saying that the convention many people have been taught is incorrect! Is this arbitrary method for evaluating arithmetic less popular than the one you propose? Did you take a survey?

Anyway, why not just throw in an extra set of parentheses? Then you'd have '(6÷2)(2+1)', and now we can all shut the fuck up about notation.

Just remember this. There is no such thing as division. Division is an illusion. All it really is is fraction multiplication. 6÷2*3 = 6*(1/2)*3. After that multiplication is commutative and associative. It doesn't matter what order you do it in.

This thread is bs. I say if you know that an expression you write can be ambiguous it's more productive make it clear (i.e. just drop in an extra parenthesis) rather than being a smart asses about it and trolling my internets...

Nice, i saw this question on facebook and alot user made the mistake.

The issue is clarity when forced to write expressions in a line. To avoid excess parentheses which make expressions unambiguous but also unreadable the convention is that spatial grouping takes precedence. I have seen this used at all levels of mathematics - and it has a clear purpose of making things easy to read. With the spaces 6 / 2(1+2) means 6/[2(1+2)]. Wolfram Alpha is wrong to ignore this convention. This is also the case if you use the original divide sign. Without the spaces it becomes less clear and so I think the expression is genuinely ambiguous. However I am more inclined to the view that omitted (implied) multiplication also has an implied higher precedence. Therefore for me the answer is 1.

consider what would happen if you did not allow people to use the priority of implied multiplication:
you would not be allowed to say n! ÷ (x)(x+1)(x+2)(x+3)(x+4)...(x+5)
mathematicians would be forced to always have to resort to lots of parentheses, which detract from clarity.
Using giant fraction bars also is often messy.
you would prefer:
n! / (x(x+1)(x+2)(x+3)...(x+4))

so, what is more CLEAR?
1 ÷ 2(k+1)
1 / (2(k+1))

The notations used in mathematics are designed to help mathematicians.
just amend the standards and rules so that the most clear notations are correct.
(kind of obvious: if someone meant to say 6/2 *(1+2), they would not say 6÷2(1+2) )

You are absolutely wrong, the answer is 3:
2(x) is the constant function returning 2, so 6/2(1+2)=6/2=3

What is 6/2(1+2)?
Dear calculator... the answer is 1.

What is 6/2*(1+2)?
Dear calculator... the answer is 9.

Revised edition:


What is 6÷2(1+2)?
Dear calculator... the answer is 1.

What is 6÷2×(1+2)?
Dear calculator... the answer is 9.

I just could not let this one slide: "Dear internet the answer is 9. Everyone with an education, enough to write calculus books, is wrong" and then provide ZERO proofs, identities, properties, etc to show how. Well, I will do it for the author:

First,
if you want to say 0.5x, then you HAVE to write (1/2)x with parentheses or, x "all over 2" with a horiztonal fraction bar, or write x/2. I have never seen (1/2)x before I researched this equation, but since searching online, I HAVE seen fractional coefficients written this way, only because computers are limited to the horizontal typing space.
Therefore:
x/2 = (1/2)x = 0.5x
1/2n = 1/(2n) This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). It was simply 1/2pi, or 1/2e^2.
I have always used ab/cd to mean (ab)/(cd) and I topped almost all of my calculus classes since high school through university.(moot point, I know)
Just to re-iterate, to use 6/2 as a fraction, parentheses are REQUIRED. Every book will tell you this.

Now consider the Identity Law:
a = 1a = 1(a)
We know there is ALWAYS an 'invisible' 1 as a ceofficient of a variable if no other number is there. Therefore:
a/a = 1, and if a is also 1a, then a/1a = 1. Blindly using 'pemdas', some folks would do this:
a/1a = a/1*a = a*a = a^2. I hope this drives home the silliness of this calculation.

Now, on to my second point: consider: factoring, simplifying equations, and the distributive property.

Lets start with the number 6.
6 = (4+2). There is a common factor here: 2. So let's factor it out of both terms.
(4+2) = 2(2+1). The outside 2 remains a part of of the 2 inner terms at all times. It cannot be used in an operation by itself without the rest of (4+2). The reverse of factoring is distribution, so, 2(2+1) = 6. This has to be true always. The argument I have seen to this is that (6/2) can be distributed. This is true ONLY is 6/2 is in parentheses, otherwise, the 6 and 2 are separated by a division slash, and the 2 is a factor of 2+1.

So, let's prove the initial equation:

6/6 = 1
6/(4+2) = 1
6/2(2+1) = 1


the same can be done for other factors:

6/6 = 1
6/(3+3) = 1
6/3(1+1) = 1

Distribution is actually a part of "Simplifying Equations" and is not bound to the order of operations as "multiplication", since it is in fact "removing parentheses by distributing". This can be googled and several references found.

Simplifying 2(2+1) + 3(2+1) = 5(2+1). We "combined like terms" here, by adding, and did not perform the "parntheses" part of order of operations, nor did we multiply, which is also higher priority than adding, because we only simplified.
If we try to prove it is 9, by distributing (6/2), we do the reverse:
6 + 3 = 9
(6/2)2 + (6/2)1 = 9
(6/2)(2 + 1) = 9 (6÷2)(2+1) = 9

Lastly, I hear the argument that "This is strictly numbers and you don't use algebra rules since there are no variables". That is the most asinine arguement I have heard yet. All axioms, laws, and properties use variables, meaning that they hold true for "any number", hence the proofs with variables.

I welcome thoughts on this, in an intellectually formed response. I am tired of the 'flaming' that goes on by imbciles on some other forums with rebuttals like "it is 9. go back to grade 3 you moron", or "google says it is 9", when google changes the equation to (6/2)*(2+1), and wolfram contradicts itself with 2n/2n = 1, and 6/2n = 3/n, but then says 6/2(2+1) is 9. wolframs "terms" state that any answer should be verified with common sense and accuracy should also be verified.

Just because you don't like convention doesn't mean you have the right to force a different convention! If everybody else used this convention, then you'd just force another!!!

This is why people use fraction notation.

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