This is a good discussion on such a seemingly simple expression.
I don¹t see two operations in 3^2. I see a number, -3, being squared. ... Or, I would see that, if I hadn¹t been told otherwise. Just like 3^2 means 3 being squared.
As someone noted the dash is used for multiple meanings, to indicate we want the additive inverse of say 3, or to say we want to subtract 3 from 10, 10 3, which, using the most common definition, is meaningless without knowing that a b means a + (-b), so the dash really does mean we want an opposite of a value, and is not, in fact, an operation.
I sometimes see students who were taught to use a smaller elevated dash to indicate the negative of a number, so 10 - (-3) would be written 10 - -3, with the second dash small, elevated and closer to the 3. That might disambiguate 3^2, depending on which symbol is used. The smaller one means you are squaring a 3, the larger symbol by the definition above must mean 0 3^2 (a = 0) and so PEMDAS actually helps there.
I also don¹t see the problem with PEMDAS, since it is, as far as I can tell, also arbitrary, and established by custom and not axioms. In 3 + 5 x 2, I don¹t see an axiom that would tell me what to do first. So I wouldn¹t know how to explain it without noting the custom.
Of course maybe I¹m displaying an ignorance of the properties of a field or something.
On 10/18/12 1:26 PM, "Paul Hertzel" <firstname.lastname@example.org> wrote:
> Although I agree with Jack Rotman about the damage inflicted by PEMDAS, I'm > not sure > > . . ."the expression ³-3^2² deals with the order of operations" > > gets to the heart of the problem. The reason is, in this case, the little > horizontal bar in front of the 3 could be a part of the number's name. So > "-3^2" is not two operations, just like "43^2" is not two operations. In the > latter, the 4 is part of the number's name, it is not a multiplier. > > This is the problem, in this case, and so I think Guy Brandenburg is right. > Writing -3^2 is just asking for trouble.