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Re: Please remind me why -3^2 = -9
Posted:
Nov 16, 2012 3:29 PM
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> On Nov 14, 2012, at 11:25 PM, Beth Hentges wrote:
>> The reason we choose -3^2 to mean -(3^2) is because we do powers
>> before multiplication and because -a = -1*a.
On Nov 15, 2012, at 2:36 AM, Clyde Greeno wrote:
> # And the reasons for that rule and for that equation are ????
> It seems as though you are trying to *conclude* what is actually a
> grammar-rule of the written language. The "powers before
> multiplication" is just such a grammar rule, and it gains no
> rational support from the equation.
(1) Indeed, the grammar we use is not the only possible one, e.g.
(Reverse) Polish notation. Or, as someone mentioned a while ago on
this list, the grammar used by Excel.
(2) A much more general difficulty encountered by the students is that
linguistic issues receive very little attention in textbooks---if any.
For instance, three most important linguistic issues are:
A. The fact that symbols are ALWAYS context-dependent. For
example
i. In 2+3, + calls for the procedure "start at 2 and
count 3 UP"
ii. In 2 apples + 3 bananas, + is to be read as AND
iii. In 2 thirds + 3 fifths, + is to be read as AND
just as in ii. and for exactly the same reason.
iv. in +5, + means positive when signed numbers are
being introduced to code things that "can go either way". (Money on
the table is coded by plain numbers while money changing hands is
coded by signed numbers.)
v. In (-2)+(+3), the middle plus calls for the fairly
complicated procedure by which we add two signed whole numbers.
vi. In 2x7^+3, the + codes for 2 being multiplied by 3
copies of 7 while in 2x7^-3, the - codes for 2 being divided by 3
copies of 7. In both cases, x is only intended to separate the
coefficient, 2, from the base, 7.
viii, When we start with plain (unsigned) numbers, the
place-holder, "x" or "a" or ,,, , stands for any plain number. But
then, when we move to signed numbers, we should logically use a two-
places place holder such as "s,pn" where s stands for + or - and pn
stands for a plain number. It is rarely emphasized that, instead, the
same place holder "a" being in a new context, that of signed numbers,
now stands for a signed number, that is for BOTH the sign and the
plain number.
This lack of emphasis can cause a difficulty: When signed whole
numbers are being introduced, -5 is read "negative 5". But then,
later, when place-holders are introduced, students naturally read -a
as "negative a"---which is of course quite correct as long as "a"
stands for a plain (unsigned) number.
The fact that OPPOSITE -5 is equal to +5 comes from the definition of
signed numbers, namely immediately after the definition of OPPOSITE
where OPPOSITE is short for "Number with the OPPOSITE SIGN of". In
particular, -(-5) should be read as "OPPOSITE of -5"--and thus the
meaning of -(-5) does not involve or depend on any operation. And
then, -a is naturally read as "OPPOSITE of a".
B. There are behind the scene "default rules" to deal with
"missing symbols".
In -2 +3, there are two ways to read + but, either way,
there is something that GOES WITHOUT SAYING:
-- The + can be seen as saying that the second signed number is
positive in which case the fact that the operation is addition (of
signed numbers as in e.) goes without saying
-- The + can be seen as saying that the operation is addition (of
signed numbers as in e.) in which case it goes without saying that the
sign of the second sign number is +.
Unfortunately, these default rules remain behind the scene all too much.
C. As Greeno pointed out, syntactic "rules" are often
confused with semantic "rules" that is relationships among the
contents---described by theorems. For example:
The fact that (-1) ? (-5) = OPPOSITE -5 ---and that -a = -1*a---is a
semantic relationship between OPPOSITE and multiplication of signed
numbers. To see it as a definition would seem to stem from a lack of
consideration for the linguistics of mathematics.
Regards
--schremmer
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