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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.