"Order of operations" is not a mathematical concept or conclusion, but a technical convention of non-associative language. A mnemonic device that might help students to decode the language is neither good nor bad mathematics ... merely effective or ineffective pedagogy.
"-3^2" is a string of characters representing some composition of functions ... and shorthand for some formula that makes full use of parens. Some alternative, more clear formulas might be: a.. (-(3(^(2)))) --> ?!# : ...meaning 2 >^2 > 3(^2) > -(3(^2)) [right to left] b.. (-((3)^2)) --> (-9) : ...meaning ? > ? > ? c.. ((-(3))^2) --> (9) : ...meaning ? > ? > ? d.. ((((-)3)^)2) --> ? : ...meaning ? > ? > ? [left to right] It is the rare curriculum or teacher that adequately leads students through the associativity issues of alternative interpretations of the written language.
In this case, the challenge to the student might be: "Below, insert parens to show two reasonable interpretations of '-3^2' ... and circle the one that is most commonly used in mathematical literature: -3^2 or -3^2 "
From: Philip Mahler Sent: Thursday, October 18, 2012 7:33 PM To: mathedcc Subject: Re: Please remind me why -3^2 = -9
This is a good discussion on such a seemingly simple expression.
I don't see two operations in -3^2. I see a number, -3, being squared. ... Or, I would see that, if I hadn't 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 don't 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 don't see an axiom that would tell me what to do first. So I wouldn't know how to explain it without noting the custom.
Of course maybe I'm 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 "-3^2" 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.