Mathematics III: PEMDAS

[🐒Gods of Prosperity🔱🐲💸]

It is true however that the left-to-right aspect of the order of operations is arbitrary, which is what the problem mentioned hinges upon.

[SWE]

9 is the only logical choice

[angus]

It is true however that the left-to-right aspect of the order of operations is arbitrary, which is what the problem mentioned hinges upon.

That's not entirely arbitrary either.  It must be influenced by the fact that we read from left to right on this forum, and we do that because the Greeks wrote that way, and they wrote that way because the first few writers were probably right-handed and going from left to right reduced the chances of smudging the ink.  It is also likely that right-handed people will hold the chisel in the left hand and the hammer in the right hand, also making left-to-right writing easier in the times before ink and paper.

But the guy who invented algebra actually wrote it from right to left in his first book:

[Grumpier Than Uncle Joe]

Attention posters - no one will out nerd angus in this thread.  Don't try.  Peace be with you. 

[muon2]

One issue that confuses people looking at the expression in the OP is that it isn't very well formed. In particular it mixes expressions from different contexts. Most importantly an expression that uses the division sign would almost always use the matching times sign, not use the implied multiplication by contact. That a consistent use of is the OP should look like

6 ÷ 2 x (1 + 2)

Leaving out the times sign becomes a syntax error if the expression is entered in a spreadsheet cell. For example 6/2(1+2) gives an error while 6/2*(1+2) does not. When typed correctly the answer is unambiguously 9.

Now some writers might be lazy and write 6/2(1+2) instead of 6/[2(1+2)] which would then give 1 as an answer. That's why mixing the types of usage without full use of parentheses and brackets is strongly discouraged.

Another way the OP expression might arise is when someone starts with an expression using a fraction form and tries to convert it to a linear expression. For example

     6
______  might be written as 6/2(1+2), but that misses the implied extra set of brackets
2(1+2)

that would result in the correct transcription as 6/[2(1+2)].

[⛄]

One issue that confuses people looking at the expression in the OP is that it isn't very well formed. In particular it mixes expressions from different contexts. Most importantly an expression that uses the division sign would almost always use the matching times sign, not use the implied multiplication by contact. That a consistent use of is the OP should look like

6 ÷ 2 x (1 + 2)

Leaving out the times sign becomes a syntax error if the expression is entered in a spreadsheet cell. For example 6/2(1+2) gives an error while 6/2*(1+2) does not. When typed correctly the answer is unambiguously 9.

Now some writers might be lazy and write 6/2(1+2) instead of 6/[2(1+2)] which would then give 1 as an answer. That's why mixing the types of usage without full use of parentheses and brackets is strongly discouraged.

Another way the OP expression might arise is when someone starts with an expression using a fraction form and tries to convert it to a linear expression. For example

     6
______  might be written as 6/2(1+2), but that misses the implied extra set of brackets
2(1+2)

that would result in the correct transcription as 6/[2(1+2)].

Thus, you consider 9 the correct answer?

[Chancellor Tanterterg]

9

PMDAS was how I was taught:

1. Parentheses
2. Multiplication
3. Division
4. Addition
5. Subtraction

[True Federalist (진정한 연방 주의자)]

1 by the rules I was taught (multiplication and division in order from left to right).

Which gives 9 here, not 1.

6 ÷ 2 (1 + 2) =
6 ÷ 2 (3) =
3 (3) =
9

Now if you do multiplications before you do divisions, you get 1, but that's not the usual PEMDAS order of operations.

Shouldn't you distribute the 2, since it's in from of a parenthese? The inverse of a factorisation, sort of?

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

I was always taught that the parentheses only grouped what was inside them and not any implied multiplication outside them.

Thus
6 ÷ 2 (1 + 2) =
6 ÷ 2 x (1 + 2) =
6 ÷ 2 x 3 =
3 x 3 =
9

Now obviously, there's no reason why it couldn't also bind the implied multiplication, but as I said, it isn't what I was taught.

If we truly wanted a context free notation, we'd drop infix and use either prefix or postfix.

x ÷ 6 2 + 1 2
or
6 2 ÷ 1 2 + x
are both unambiguously 9.

÷ 6 x 2 + 1 2
or
6 2 1 2 + x ÷
are both unambiguously 1.

[angus]

If we truly wanted a context free notation, we'd drop infix and use either prefix or postfix.

...which is exactly what RPN does, and why some people like it.  Of course, its usage depends upon you knowing the rules of the game from the outset. 

[Foucaulf]

This is third consecutive math post made by the OP in which the question was ill-defined. And this is a problem - these questions do not elucidate, but only confuse. The more this happens, the more people tend to believe math is some chaotic black art of sorts when what matters is that there is a logical process that makes the answer(s) true.

Maybe the OP should read Pólya; it's a classic book, I think.