Mathematics III: PEMDAS

[⛄]

6 ÷ 2 (1 + 2) =

[True Federalist (진정한 연방 주의자)]

9 by the rules I was always taught.  Incidentally, in the US : is not generally used to indicate division.  Either / or ÷ is used.

[H.E. VOLODYMYR ZELENKSYY]

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

[⛄]

9 by the rules I was always taught.  Incidentally, in the US : is not generally used to indicate division.  Either / or ÷ is used.

You mean the obelus.
At elementary school we always used the slash, later at high school we only wrote in fractions, but in this case both would be misleading.

[Flake]

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

[Arturo Belano]

Following PEMDAS, the answer is 1.

[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.

[MaxQue]

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

[⛄]

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

That's the question.

[MaxQue]

Well, is there is a definite way to have a final answer (the right one)?

[Smid]

Well, is there is a definite way to have a final answer (the right one)?

Yes. Ask Muon2.

[dead0man]

9

[🐒Gods of Prosperity🔱🐲💸]

Multiplying what's in parenthesis by the thing adjacent to it is what makes sense to me even if that's not the standard.
With PEMDAS order of operations, I guess you would need to write such an equation as  "6 ÷ (2(2+1))" right?

[Јas]

http://en.wikipedia.org/wiki/Order_of_operations

[tik 🪀✨]

Were I writing this out I'd probably write 6÷2 as 6/2 with (1&2) next to it. I'd then distribute 6/2.

((6/2)*1) & ((6/2)*2)
(6/2) & (12/2)
(18/2)
9

I am obsessive with parenthesis, though. Programmer's trauma. This is a poorly written problem.The right answer is both  because the order of operations is arbitrary.

I could very easily be wrong. I'm no mathmagician

[MasterJedi]

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.

This is the way I was taught. Anything within the parentheses first, then division/multiplication from left to right.

[⛄]

Well, is there is a definite way to have a final answer (the right one)?

Yes. Ask Muon2.

Is Muon2 a mathematician?

[Grumpier Than Uncle Joe]

Well, is there is a definite way to have a final answer (the right one)?

Yes. Ask Muon2.

Is Muon2 a mathematician?

He has a PhD in physics so one would assume he's adept at math.

[DemPGH]

Not exactly a math genius, but I interpreted the colon as a division sign and rather quickly arrived at 9.  I can't really manipulate it to arrive at 1.

If the colon indicates that it's a fraction, you'd still arrive at 9, wouldn't you?

[Cincinnatus]

Following PEMDAS, the answer is 1. 9

Come on guys..

[angus]

6 ÷ 2 (1 + 2) =

In algebraic logic, the answer is 9.  On any calculator (e.g., TI89) that follows algebraic logic you will get 9.  On some low-end calculators, such as those you buy at the Dollar Tree you might get a 1.  Interestingly, on my Casio fx260solar I actually get a 2 as the answer.  Neither 1 nor 2 are the correct answer in the formalism taught in US schools.  Only 9 is correct.

Punch in the sequence 5+3x2= on your calculator.  If your calculator follows algebraic logic, it should show 11 as the answer.  If it follows arithmetic logic, it will show 16 as the answer.  11 is correct here.

[⛄]

6 ÷ 2 (1 + 2) =

Interestingly, on my Casio fx260solar I actually get a 2 as the answer.

Two? Really? Made in China?

[angus]

Malaysia.

Actually that's sort of ironic.  None of this actually requires advanced math to follow.  US students would have been taught something called the "order of operations" probably around the eighth grade.  We had little mnemonics like Please excuse my dear Aunt Sally to help us remember the order.  Basically, multiplication and division are of the same order; addition and subtraction are of the same order.  Absent parenthesis, you take them like that.  In this case, there are parenthesis on the last sum so your problem becomes 6 divided by 2 multiplied by 3.  The six gets divided by 2, then that quotient gets multiplied by 3, which yields 9.  

The irony lies in the fact that in places like China and India and Malaysia, the students are taught such things at a much earlier age than the US students are.  So the fact that the cheap sweatshop-produced calculator made in the Far East gets it wrong is funny, while my US-made Hewlitt Packard calculator gets it right.  To be fair, the HP calculator uses neither algebraic nor arithmetic logic.  It uses reverse-Polish notation logic (RPN).  When using a RPN logic calculator, you stack it up first, then press the appropriate operator keys.  It expects the user to know the order of operations.  You enter then problem in such a way as to get a 9 only if you already know what you're doing.  

I also have a TI in my office.  On that one I get a 9 as well.

[Grumpier Than Uncle Joe]

6 ÷ 2 (1 + 2) =

In algebraic logic, the answer is 9.  On any calculator (e.g., TI89) that follows algebraic logic you will get 9.  On some low-end calculators, such as those you buy at the Dollar Tree you might get a 1.  Interestingly, on my Casio fx260solar I actually get a 2 as the answer.  Neither 1 nor 2 are the correct answer in the formalism taught in US schools.  Only 9 is correct.

Punch in the sequence 5+3x2= on your calculator.  If your calculator follows algebraic logic, it should show 11 as the answer.  If it follows arithmetic logic, it will show 16 as the answer.  11 is correct here.

I forgot to mention angus knows a thing or two about math. 

[angus]

the order of operations is arbitrary.

Not at all.  Algebra was invented by the Arabs long ago, but it did not come into general use in Europe till the 17th century.  Of course the symbols used were somewhat arbitrary, but not the logical order.  Here's an excerpt from Frans van Schooten's 1646 edition of Vieta:

      ________________
B in D quad. + B in D

We would express this as B(D^2 + BD).  Van Schooten uses a horizontal bar (viniculum) over a sum whereas we use parenthesis.  (Canadians use brackets.)  All of these symbols are equally arbitrary, but the logic behind them is not.  What is important here?  The fact that Van Schooten even felt the need to use the vertical bar suggests that even as long ago as 1646 the idea that Multiplication/division was of a higher order than addition/subtraction!  

I'm no historian, but I do like history and have actually looked into this subject a number of years ago. It seems to me that the basic rule that multiplication (and division) has precedence over addition (and subtraction) arises naturally and without much disagreement in the 17th-century literature as algebraic notation was being developed and the need for such conventions arose.  Even though there were numerous competing systems of symbols, forcing each author to state his conventions at the start of a new book, they did not have much to say on the subject of order of operations.  This is because the distributive property implies a natural hierarchy in which multiplication is more powerful than addition.  

(Note the absence of a separate vinculum upon "B in D" above.  This seems to indicate already that it is assumed by Van Schooten that the reader would understand that multiplication happens at a higher order than addition.)  

Also, there may be something in the grammar of Arabic that logically helped with the development of algebraic formalism.  I do not read or understand Arabic well enough to parse that.  (My studies are limited to the "Teach Yourself Arabic" paperback with cassette tape I purchased in anticipation of a trip to the Pyramids, and that was a very long time ago and I have forgotten most of what I learned then.)  Or it may be that the language of mathematics, independent of symbolism, transcends the spoken idiom, and that all learned men in all cultures would naturally assign a higher order to multiplication than to addition.  

Ernest knows a whole bunch of stuff about a whole bunch of stuff.  Maybe he knows whether the we think multiplication trumps addition because the Arabs taught us that, or whether the Europeans and Chinese figured that out independently because it just feels so right.