Order of operations / PEMDAS

PEMDAS does not cover matrices...
Photo courtesy of Stuartpilbrow

Someone asked me recently...

Could you please share with me your opinion of the "Please Excuse My Dear Aunt Sally" simplifying expressions. Any feedback could you give will be appreciated. Thank you.

This "PEMDAS" rule is a mnemonic for order of operations:

Please = Parenthesis
Excuse = Exponents
My = Multiplication
Dear = Division
Aunt = Addition
Sally = Subtraction

There's nothing wrong with using a mnemonic to remember the order of operations. However, one has to bear in mind that

1. This rule is not all-inclusive. It omits for example square roots. But the rule is good for all elementary grades. (Square roots would be on the same level or rank with exponents, by the way.)
    
2. The rule doesn't spell out the fact that in reality multiplication and division are "on the same level" or rank. This means that if you have several multiplications and divisions, you do them from left to right, and not "multiplication first, then division".
   
   For example: 60 ÷ 5 × 4. You go from left to right, and first do 60 ÷ 5 = 12. Then you multiply 12 × 4 = 48.
   
   If you want 5 × 4 to be done first, it needs to be in parenthesis: 60 ÷ (5 × 4). Here, first do 5 × 4 = 20, and then 60 ÷ 20 = 3.
   
   Similarly, addition and subtraction are on the same level: if both exist in an expression, they are to be done from left to right.
   
   An example: Simplify the expression 2 × 5 − 6 + 8.
   
   1: Multiply 2 × 5 = 10.
       The expression is now 10 − 6 + 8.
   
   2. Subtract. 10 − 6 = 4.
       The expression is now 4 + 8.
   
   3. Add. 4 + 8 = 12.
   
   So, perhaps it's more illustrative to lay out the PEMDAS rule like this:
   
   Please = Parenthesis
   Excuse = Exponents
   My Dear = Multiplication & Division
   Aunt Sally = Addition & Subtraction
   
   ...and say it with little pauses at the commas: Please, Excuse, My Dear, Aunt Sally.
   
   
   
3. Fraction line as a division symbol generates implicit groupings or parenthesis in the fact that anything written on top or bottom of the fraction line is to be done before the division. It's as if the whole numerator and denominator were inside parenthesis.
   
   So...
   
   
   
   

   4 × 32
   --------
   7 − 22
   

   
   means the same as (4 × 3²) ÷ (7 − 2²). Simplifying this is left as an exercise for the reader. The answer is 12.
   

See also a detailed lesson on order of operations at Purplemath and Wikipedia's note on mnemonics.