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Negative Squared, or Squared Negative?
Date: 10/18/2002 at 22:57:56
From: Tom
Subject: Negative Numbers Squared.
After reading your answer in
Exponents and Negative numbers
http://mathforum.org/library/drmath/view/55709.html
it seems to me that you're ignoring an important fact: -3 isn't just
-1*3, but a number in its own right, i.e., the number 3 units to the
left of zero. If that's the case, then shouldn't -3^2 have the value
-3*-3, or 9?
If -3 was intended to mean -1*3, then shouldn't it be written that way
and not implied?
Thank you for your time.
Tom
Date: 10/19/2002 at 20:44:50
From: Doctor Peterson
Subject: Re: Negative Numbers Squared.
Hi, Tom.
I do recognize that it is possible to disagree on -3^2. Dr. Rick's
answer to a similar question,
Squaring Negative Numbers
http://mathforum.org/library/drmath/view/55713.html
mentions this disagreement. (Dr. Rick is my twin brother, by the way!)
Like you, he notes that if you think of -3 as a single number, it
makes sense for the negation to bind more tightly to the 3 than any
operation.
That reasoning makes some sense, though I think other
arguments are stronger. But I do agree that since there _is_ some
reason to read it either way, it is prudent always to include
parentheses one way or the other, to clarify your intent, i.e., to
write either -(3^2) or (-3)^2.
Occasionally people will try to argue the point based on the behaviors
of particular calculators or spreadsheet programs. However, these are
really irrelevant, since they all define their own input formats, and
programmers (of which I am one) are notorious for choosing what's
easiest for them, rather than what is most appropriate for the user.
I've noted in several answers in our archives that some calculators,
and Excel, use non-standard orders of operation without apology. But
calculators in particular just don't use standard algebraic notation
in the first place.
There also seems to be a generational difference, with older people
(including some teachers) claiming that they were taught to interpret
-3^2 as (-3)^2.
I suspect that what has changed is not the rules governing "order of
operation" (operation precedence), but that schools are introducing
the issue earlier, before students get into algebra proper. That means
that they start by looking at expressions for which it is less clear
why the rules make sense. I think you will rarely find examples of
"-3^2" in practice, because there is no need for mathematicians to
write it. You will find "-x^2" frequently.
If you approach the idea starting with numerical expressions like
-3^2, you are thinking of -3 as a number and assuming that the
expression says to square it. If you approach it first using
variables, having first discovered that "-" in a negative number is
actually an operator, then it is easier to see why -x^2 should be
taken as the negative of the square. So I'll start with the latter,
and then it becomes natural to treat numbers the same way we treat
variables.
Now, in an expression like -x, clearly "-" is a (unary) operator,
which takes a value "x" and converts it to its opposite, or negative.
The expression "-x" is not just a single symbol, but a statement that
something is to be done to a value. As soon as we start combining
symbols like this, as in -x^2 or -x*y, we have to decide what order to
use in evaluating them.
The trouble is that the "order of operations" rules as commonly taught
(PEMDAS) don't mention negatives. So if we are going to go by the
rules, we have to figure out how a negative relates to them. Well,
there are two ways to express a negative in terms of binary
operations. One is as multiplication by -1:
-x = -1 * x
Treating it this way, clearly
-x^2 = -1 * x^2 = -(x^2)
That is, since -x means a product, we have to do the exponentiation
first.
The other way to talk about negation is as the additive inverse,
subtracting x from 0:
-x = 0 - x
(This is why the "-" sign is used for both negation and subtraction.)
Using this view, we see that
-x^2 = 0 - x^2 = -(x^2)
So both views of negation produce the same interpretation, which does
exponents first, and it is logical to put negation here in the order
of precedence.
But the fact is that there is no authority decreeing these rules;
just as in the grammar of English, we get the "rules" by observing how
the language is actually used, not by deducing them from some first
principles. The order of operations is just the grammar of algebra.
So the real question is, how do mathematicians really interpret
negatives and exponents combined in an expression?
If you look in books, you will rarely find "-3^2" written out, but
you will often find polynomials with negative coefficients. And you
will find that
-x^2 + 3x - 2
is read as the negative of the square of x, plus three times x, minus
2. I have come to believe that the order of operations is what it is
largely so that polynomials can be written efficiently. If "-x^2"
meant the square of -x, then we would have to write this as
-(x^2) + 3x - 2
to make it mean what we intend. Since powers are the core of a
polynomial, we ensure that powers are evaluated first, followed by
products and negatives (the two ways to write a coeffient) and then
sums (adding the terms).
Since we can easily see that this is how -x^2 is universally
interpreted, it makes sense to treat -3^2 the same way.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
Associated Topics:
Middle School Algebra
Middle School Negative Numbers
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Geez. How difficult is this really?
(-1)^2*5^2 is not the same as -1^2*5^2
