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Re: Please remind me why -3^2 = -9
Posted:
Oct 18, 2012 8:58 PM
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So, what is this unary operation of negation? If I write the number 3, have
I written an operation? To me, I have not. I have written the symbol for a
negative number. I understand it in the context of, say, ³square 3, then
³negate² the answer². The common interpretation would be 3^2 of course, but
as I have said, the abstract version is, I believe, too ambiguous to safely
use. I think an equally valid statement like ³write down a negative 3, then
square it² could produce the same picture, at least to those who cannot
remember the other interpretation.
>From Wikipedia I note ³There exist differing conventions concerning the
unary operator - (usually read "minus"). In written or printed mathematics,
the expression -3^2 is interpreted to mean -(3^2) = -9,[3]
<http://en.wikipedia.org/wiki/Order_of_operations#cite_note-2> but in some
applications and programming languages, notably the application Microsoft
Office Excel <http://en.wikipedia.org/wiki/Microsoft_Office_Excel> and the
programming language bc
<http://en.wikipedia.org/wiki/Bc_programming_language> , unary operators
have a higher priority than binary operators, that is, the unary minus
(negation) has higher precedence than exponentiation, so in those languages
-3^2 will be interpreted as (-3)^2 = 9.[4]
<http://en.wikipedia.org/wiki/Order_of_operations#cite_note-3> In cases
where there is the possibility that the notation might be misinterpreted,
parentheses are usually used to clarify the intended meaning.²
It does occur to me that stressing the ³correct² interpretation of 3^2 with
students develops no understanding of operations that I can see, and it does
consume an amount of time and create more consternation than it is worth. If
I only saw it once in a while in texts, I wouldn¹t have commented, but every
text makes a big, special deal about this unholy sequence of symbols, and
every teacher makes sure it¹s on the final. :-)
Phil
On 10/18/12 2:38 PM, "Guy Brandenburg" <gfbrandenburg@yahoo.com> wrote:
> Well, lets think of a nice way to remember it!
>
> How about this:
> Please! escape! No! Medical Doctor! Ambulance Surgeon!
>
> Guy
>
> On Oct 18, 2012, at 14:24, "Collinge, Peter (Mathematics)"
> <pcollinge@monroecc.edu> wrote:
>
>> Part of why students are confused about 3^2 is that our standard teaching
>> about order of operations is incomplete. We don¹t generally specify where
>> negation fits into the order of operations. In fact, PEMDAS (if not banned)
>> should be taught as P.E.N.MD.AS, with negation inserted after exponentiation
>> and before multiplication/division. Then it would be clear that in -3^2, the
>> exponentiation operation is performed before the negation. (I realize that
>> some would say that negation is just multiplying by -1, so would have the
>> same precedence as any multiplication, but clearly many students don¹t see
>> that.) But unfortunately P.E.N.MD.AS just doesn¹t make a nice,
>> easy-to-remember mnemonic.
>>
>>
>> Peter Collinge, Professor
>> Department of Mathematics
>> Monroe Community College, Rochester NY 14623
>>
>> E-mail: pcollinge@monroecc.edu <BLOCKED::">mailto:pcollinge@monroecc.edu>
>> <mailto:pcollinge@monroecc.edu>
>> Voice: (585)292-2943
>> <image001.jpg>
>>
>>
>> From: owner-mathedcc@mathforum.org [mailto:owner-mathedcc@mathforum.org] On
>> Behalf Of Jack Rotman
>> Sent: Thursday, October 18, 2012 12:39 PM
>> To: Wayne Ford Mackey; Guy Brandenburg; john.peterson20@comcast.net; Philip
>> Mahler
>> Cc: mathedcc
>> Subject: RE: Please remind me why -3^2 = -9
>>
>> Phil and all:
>>
>> The expression ³-3^2² deals with the order of operations; the most advanced
>> operations are always done first unless a grouping symbol forces¹ a lower
>> prior operation to be done first. Since exponentiation is more advanced
>> than the sign of a number, the exponent only applies to one symbol (the 3)
>> when there are no grouping symbols.
>>
>> Instead of banning this type of problem, I believe that we should ban
>> ³PEMDAS² or anything like it. The use of overly simplistic rules (often
>> stated as a sequence of nouns) discourages learning and encourages
>> memorization. If a large rate of correct answers is the only criteria, just
>> have students use a calculator and train them on use of parentheses. If we
>> are teaching mathematics, we should focus on understanding the priority of
>> operations. What Phil internalized was this understanding; saying PEMDAS¹
>> does not provide any of the understanding to our students [I normally spend
>> about a tenth of my time in class trying to undo the damage of PEMDAS.
>> Undoing partially correct information is terribly difficult!]
>>
>> Even if we never showed ³-3^2², students would still be evaluating x^2 for
>> x=-3; knowing that this means squaring a negative is a part of basic literacy
>> in mathematics.
>>
>> For those with an interest, I¹ve posted some ³anti-PEMDAS² comments on my
>> blog (www.devmathrevival.net <http://www.devmathrevival.net> ). You can use
>> the search box on the site to find them.
>> Jack Rotman
>> Professor, Mathematics Department
>> Lansing Community College
>> (517)483-1079 rotmanj@lcc.edu <mailto:rotmanj@lcc.edu>
>> www.devmathrevival.net <http://www.devmathrevival.net>
>>
>>
>> From: owner-mathedcc@mathforum.org [mailto:owner-mathedcc@mathforum.org] On
>> Behalf Of Wayne Ford Mackey
>> Sent: Thursday, October 18, 2012 12:11 PM
>> To: Guy Brandenburg; john.peterson20@comcast.net; Philip Mahler
>> Cc: mathedcc
>> Subject: RE: Please remind me why -3^2 = -9
>>
>>
>> It should be read as the opposite of 3 squared. Since 3 squared is 9, the
>> opposite is -9. The "-" sign is used in 3 different ways. In front of a
>> natural number it means negative or minus, in front of anything else it means
>> opposite and between two things it means add the opposite.
>>
>>
>>
>> wayne
>>
>>
>>
>>
>> From: owner-mathedcc@mathforum.org [owner-mathedcc@mathforum.org] on behalf
>> of Guy Brandenburg [gfbrandenburg@yahoo.com]
>> Sent: Thursday, October 18, 2012 6:04 AM
>> To: john.peterson20@comcast.net; Philip Mahler
>> Cc: mathedcc
>> Subject: Re: Please remind me why -3^2 = -9
>>
>> It's a convention. In a case like that, one really ought to use parentheses
>> to make the meaning clear, since a lot of people, not just youngsters, will
>> get confused.
>>
>>
>>
>> If one intends to say (-3)*(-3), then write (-3)^2. If one means - (3)*(3),
>> then write - (3^2).
>>
>>
>>
>> Writing -3^2 is simply asking for confusion.
>>
>>
>>
>> Guy Brandenburg, Washington, DC
>> http://gfbrandenburg.wordpress.com/
>> http://home.earthlink.net/~gfbranden/GFB_Home_Page.html
>> ============================
>>
>>
>> From: "john.peterson20@comcast.net" <john.peterson20@comcast.net>
>> To: Philip Mahler <mahlerp@middlesex.mass.edu>
>> Cc: mathedcc <mathedcc@mathforum.org>
>> Sent: Thursday, October 18, 2012 6:05 AM
>> Subject: Re: Please remind me why -3^2 = -9
>>
>>
>> Phil,
>>
>> -3 means -1 x 3, so -3^2 is (-1)(3^2) = (-1)(9) = -9.
>>
>> John Peterson
>>
>>
>> From: "Philip Mahler" <mahlerp@middlesex.mass.edu>
>> To: "mathedcc" <mathedcc@mathforum.org>
>> Sent: Thursday, October 18, 2012 5:37:07 AM
>> Subject: Please remind me why -3^2 = -9
>>
>> I have been teaching a long time, and I know from experience that 50% of
>> students will tell me that 3^2 = +9 on a test or a final, despite having
>> discussed it a few times in a course.
>>
>> When I first started teaching I taught calculus and precalc. Piece of cake.
>> Then I started with an Algebra I class and couldn¹t connect at all for the
>> first week or so. I was ready to believe I couldn¹t teach. I simply could not
>> explain how I got the right answers when evaluating expressions... Then I
>> discovered the order of operations (PEMDAS to some). A definition of the
>> order of operations which I had so internalized that I didn¹t know there was
>> a rule for it. Since that discovery I¹ve been a wonderful teacher. :-)
>>
>> So... I must be missing something that so many of my students think 3^2 is
>> +9. What is the rule I have never discovered?
>>
>> Full disclosure: I think k^2, k a constant, should be banned from
>> mathematics texts and tests. -x^2, x a variable, evaluated for say 3,
>> absolutely (no pun intended) but not 3^2.
>>
>> Phil
>>
>
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