Negative vs. Subtraction in Order of Operations

Date: 10/30/2002 at 13:03:36
From: Eric Hamilton
Subject: Order of operations - negative versus subtracting

I am trying to find some definitive source for treatment of leading 
negative numbers in simplifying with order of operations. I have 
looked through FAQs on different sites. It's a simple problem. What if 
the leading term of an expression is a negative number with no 
parenthesis? Consider this example:

   -5^2 + 2 * 2 - 8^2.   

Is this -25+2*(-62) = -149  (the leading "-" is a subtraction)
or is it 25+2*(-62) = -99   (the leading "-" is a negative sign) ?

If the leading "-" before the 5 is taken as a sign indicator, it seems 
that the first square term has to be come out to 25. Otherwise, order 
of operations could interpret the "-" in front of the 5 as a 
subtraction - but from what? Why is it wrong to consider this to be a 
negative sign rather than a subtraction? I find written materials 
amazingly vague on this issue; they always give examples of leading 
"-" signs embedded in parentheses (in which case the problem is 
trivial) or else don't give any examples of leading negatives raised 
to an even power. Can you suggest the proper simplification for the  
above expression and any text or definitive statement of convention of 
how leading negative numbers are to be treated in order of operations 
convention?

Date: 10/30/2002 at 13:35:11
From: Doctor Peterson
Subject: Re: Order of operations - negative versus subtracting

Hi, Eric.

We've covered this issue numerous times. Here are two answers from 
our archive, with slightly different perspectives:

   Negative Numbers Combined with Exponentials
   http://mathforum.org/library/drmath/view/53240.html 

   Squaring Negative Numbers
   http://mathforum.org/library/drmath/view/55713.html 

I agree that it is not mentioned clearly often enough. A few texts 
explicitly teach that -x^2 means -(x^2) and -3^2 means -(3^2), but 
others just avoid the latter and only use the former in examples, 
without teaching it. The former is unequivocally true; the latter is 
most often avoided, so it is harder to prove it is standard practice.

Here's a summary of the issue:

When we write -x^2, we can interpret negation equivalently as either 
subtraction from 0:

    -x^2 = 0 - x^2 = -(x^2)

or as multiplication by -1:

    -x^2 = -1*x^2 = -(x^2)

The normal order of operation rules cover it, and give the same result 
either way. I wish that the rules were stated more clearly, that 
negation, multiplication, and division have the same precedence; but 
that is standard practice whether it is taught or not. Note, by the 
way, that the negative sign is still a negative sign; the confusion 
arises only over what it is negating.

Your issue is that when we write -3^2, it looks as if the negative 
sign were part of the number being squared. This is why such a form 
tends to be avoided; rather than making and enforcing rules that are 
easy to get wrong, real mathematicians just choose to write clearly, 
using parentheses where anyone might misread what they write. But it 
seems clear that the same rule ought to apply to -x^2 and -3^2.

One complication here is that some people get into trouble by 
evaluating x^2 when x = -3 as

    x^2 = -3^2

and miss the fact that (if the standard rule is followed) what they 
wrote is not what they meant, because the negation now applies to the 
whole expression rather than just to the 3. It is important to 
maintain a practice of putting a variable's value in parentheses when 
it is substituted, in case the order of operations might change the 
meaning:

    x^2 = (-3)^2

The same problem arises if we replace x with an expression like a+b:

    x^2 = a+b^2

means the wrong thing, while

    x^2 = (a+b)^2

can't go wrong.

I should add that you evaluated your example expression incorrectly:

    -5^2 + 2 * 2 -8^2

is not

    -25 + 2*(-62) = -149

(where you evaluated 2-8^2 first), but

    -25 + 4 - 64 = -85

(where 2*2 and 8^2 come before the subtraction). Be careful with the 
order of operations!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/