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A mathematical operator is simply a mapping -- it takes a "thing" and maps it to another "thing". There is not a requisite of having a specific number of operands, though I've never heard of an operand operating on a null set of operands.
For example, absolute value operates over the Real Numbers on a single operand (and returns a single value).
The root operand, does have 2 inputs (no matter how we choose to write the expression): To take the nth root of X means to find y s.t. y^n = X. Like the absolute operator being applied to real numbers, the inverse of taking a root has 2 answers in the real numbers. For example, the 2nd root of 4 is +/- 2 since (+/- 2)^2 = 4 -- it is only 2 if we are working in the positive integers.
When looking at such functions it is very important to consider the set of number you are working with: positive integers, real numbers, complex number... each set provides a sometimes varying set of answers.
I think of an operator as an algorithm: it simply defines an ordered procedure for one to follow.
Take the operator I had defined earlier: ⊕
Since I had defined x ⊕ x = 6, it made the issue trivial, but what kind of algorithm could represent such an operator? Well, if we look at it as a function, then we have ⊕(x,x) = 6 which represents a line in 3-space. Very similar to the 2-space line represented by f(x)=6 or y=6 which we learned in introductory algebra classes. I could have just as easily defined the 4-space function ⊕(x,x,x) = 6 which would still represent a line in 4-space.
NOTE: It's been awhile since my college abstract algebra classes, so I hope I didn't miss any glaring points!
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Jeremy Miller
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