Definition discussed the operation of empowerment
The intention of this short series of posts is to expose the operation definition of empowerment as it is universally accepted by the mathematical community, adding some comments that are usually omitted in texts . I clarify that the limitations of Blogger forced me to indicate the operation under the symbol "^", although we all know that usually indicated by setting the exponent in size small and slightly elevated above the writing line.
The first point to consider is that empowerment is an operation is defined by subsequent cases, according to the numerical set which is the exponent. Will initiate a very basic case:
Case 1: a ^ 2 = aa (The number to can be any real number.)
[René Descartes was the first, or at least one of the first, in using this notation.]
Question: Why is aa ^ 2? Could it be a ^ 2 = a + a?
Answer: That a ^ 2 equals aa is only a convention notation which obviously was chosen because it was useful (we can surmise that the product of a number by itself appeared many times in the calculations and that justified the use of a specific notation for the operation. The answer to the second question is clearly yes. The mathematical notations often depend on arbitrary choices that could have been very different.
Question: Is it an abuse of notation using the same symbol ^ (in fact, the exponent notation) for ( -2) ^ 2 or 3 ^ 2.
Answer: Obviously, no. In both cases we speak multiplying a number by itself so why would it be an abuse of notation? If you would have raised that question to Descartes, probably would have thrown a draft by the head, like a couple of centuries later another Frenchman would, Galois, with an examiner who questioned him in that style (of course, that draft was not Wood like ours, but it was a sponge).
Case 1 (extended): a ^ n = aa ... a (Where to repeats n times.)
The number can to be any number real. The value of n can only be a strictly positive integer, ie 1, 2, 3, 4, ... because these values \u200b\u200bcan only speak of "number of times." Could occur to give an inductive definition of this case, but it's just more elegant formalization of the same idea.
Properties:
Important: So far we have defined the calculation of powers with integer exponent strictly positive and, therefore, the properties refer only to that case. In Section 3 the value of a must be nonzero in the other can be any real number.
1. (a ^ n). (A ^ m) = a ^ (n + m)
2. (a ^ n) ^ m = a ^ (nm)
3. a ^ n / a ^ m = a ^ (n - m) . This property is worth (for now) only if n is strictly greater than m , because otherwise the member would have a power law have not yet defined. Obviously to be nonzero.
Question: What happens with a ^ 0?
Answer: We have not yet defined, will appear in the following case.
continued ...
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