Mathematics III: PEMDAS

[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.

[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.

[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.
 

[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:

[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.