Precedence of Unary Operators
Date: 09/01/99 at 22:51:05 From: Tyler Longman Subject: Unary operator precedence / examples My Dad, who is a programmer, brought up the question of the precedence of unary operators with respect to what I'm learning in math. The PEMDAS rule for order of operations ignores unary operators, and my Dad, despite his trying for several hours, was unable to come up with a problem that displayed the effects of not giving unary operators their proper precedence. His knowledge of unary operator precedence comes from the world of computer programming, not necessarily the same as in the mathematics world (computer software vendors occasionally develop their own precedence schemes). Anyway, one list I found showed unary operator precedence below that of multiplication/division and above addition/subtraction. But this seems to disagree with what my Dad believes he was taught in college, namely that unary operators have a precedence right after that of parentheses. Can you help explain all of this and maybe give an example showing how ignorance of the unary operator precedence can cause an incorrect result? Many thanks, Tyler
Date: 09/02/99 at 11:59:50 From: Doctor Peterson Subject: Re: unary operator precedence / examples Hi, Tyler. Good question. I'm a programmer too, so I appreciate what you're talking about. I can easily think of an example, one that we are asked about all the time: -3^2 The negation operator properly has precedence below the exponential, so that this means -(3^2) = -9 rather than (-3)^2 = 9 We don't usually list unary operators in PEMDAS because they're thought of as being implied by the rules for binary operations. You can think of the minus sign as either subtraction -3^2 = 0 - 3^2 = 0 - 9 = -9 or multiplication -3^2 = -1 * 3^2 = -1 * 9 = -9 and in either case it has lower precedence than exponentiation, so it gives the same result. I can't think of any case where it would matter which of these two latter interpretations we give, that is, any case where giving "-" an additive precedence or a multiplicative precedence will make a difference in the result; commutativity and distributivity seem to take care of that. For example, (-2) * 3 (putting negation at the same level as multiplication) is the same as -(2 * 3) (putting it after multiplication). But certainly negation must not have a lower precedence than addition, because then -2 + 3 would mean -(2 + 3) which is so contrary to sense that it's hard to imagine doing it. When negation is listed in the order of operations, it's commonly put with or just above the multiplicative operators. After all, it is essentially a form of multiplication. You're right that each programming language defines its own precedence; they generally try to follow tradition, but there can be special issues that force them to modify it. Here's the list for C: ()  . -> ("primary expression operators") * & - ! ~ ++ -- (unary operators) * / % (binary operators: multiplicative) + - (additive) >> << (shift) < > <= >= (comparison) == != (equality) & (bitwise and) ^ (bitwise exclusive or) | (bitwise or) && (logical and) || (logical or) ?: (ternary operator: conditional) You see that C does put unary operators right after parentheses; but since there is no exponent operator, this doesn't conflict with my rule. You may be interested in this page I ran across, on the order of operations in Microsoft Excel. It says that they evaluate unary minus before exponentiation, and will not change it though they acknowledge that this is different from the normal order: http://support.microsoft.com/support/kb/articles/q132/6/86.asp - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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