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/