I am right, therefore you must be wrong

I am right, therefore you must be wrong: An underground peat fire …

There is an underground peat fire on social media about sums like 8÷2(2+2). Some argue the one and only answer must be ‘16’, others are adamant that it can only be ‘1’. Most involved in this ‘underground peat fire’ will take up battle stations defending the one and only proper order of mathematical operations. Sometimes arguing that there is or can be only one order of operations and that it is like an universal mathematical law. Missing the point that orders of operations are nothing more or less than important conventions, like driving on the right hand side of the road, or the left hand side. 

What is interesting is that most are unwilling or incapable of looking at the ‘meta’ issue and fail or are unwilling to accept that conventions can vary and change. What they have been taught and have embraced must be right, and therefore others producing different answers to equations must be wrong. 


Maybe this ‘relativity of conventions of order of operations’ is easier to see and accept for Dutch people that have been taught mathematics before the 1990’s. In Dutch schoolbooks and exams, up into the 1990s, a rather ‘eccentric’ convention for the order of operations was used following the mnemonic: “Meneer Van Dale Wacht op Antwoord” (Mister Van Dale is waiting for an answer) Which implied that Power precedes Multiplication, which precedes Division, which precedes Roots, which precedes Addition, which precedes Subtraction. (With parenthesis preceding everything) 


Making 8 ÷ 2 x 4 +9 = 10

instead of the international more likely PEMDAS outcome of  8 ÷ 2 x 4 +9 = 25


With the rise of desktop computers and pocket calculators in daily life and education this local convention became rather impractical and was swapped for the international PEMDAS order or operations.


8÷2(2+2) and the order of operations

8÷2(2+2) =   1

8÷2(2+2) = 16


In some mathematical conventions, multiplication when noted as a juxtaposition like A(B+C) is seen as an integral part of the parenthetical term — making that multiplication part of step ‘P’  in the PEMDAS order of operations. Other mathematical conventions say that A(B+C) is nothing more or less than A×(B+C) which means that the multiplication takes place in the MD phase of PEMDAS.


A ‘fun’ thing is that some calculators (or spreadsheets) give precedence to implied multiplication, and some don’t. Some have an option in the settings to select your preference to match the context you are using your calculator. Some calculators give ‘syntax error’ when entering A(B+C), some change the input explicitly into Ax(B+C)), some do so implicitly. 


So when entering 8÷2(2+2) in your calculator (or spreadsheet) you might end up with 

8÷2(2+2) =   1

8÷2(2+2) = 16

8÷2(2+2) ⇒  change of input by calculator  ⇒ 8÷2x(2+2) = 16

8÷2(2+2) = Syntax Error 


See also: https://en.wikipedia.org/wiki/Order_of_operations#Mixed_division_and_multiplication