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21

Oleksandr is correct about the way evaluation works. a/b seems to be interpreted (parsed) directly as Times[a, Power[b,-1]], or more readably: $a\times b^{-1}$. Divide[a,b] is interpreted as is. Evaluation then proceeds from these forms, and the arithmetic is carried out differently for the two cases: either $a\times (1/b)$ or $a/b$. Here are some ...


4

As rasher and the documentation both say, Equal has a certain level of fuzziness. The same is true of SameQ, though it has a more stringent tolerance. The following computations are all done with machine precision numbers. Similar things should hold with arbitrary precision numbers but the analysis might be trickier. (* 12 zeros, difference = ...


3

See the documentation for Equal. There is a tolerance for inexact numbers. The order of operations combined with precision of targets can affect whether things fall "in" or "out" of tolerance. See specifically the "Possible Issues" section in the documents for Equal. As far as why results themselves differ in FP arithmetic, there is no better source than the ...


3

The following gives what you intended: Refine[Expand[P[x, y]^2], (x|y|beta) \[Element] Reals] (* ==> Conjugate[z[y]]^2/E^((2*I)*beta*x) + 2*Conjugate[z[y]]*z[y] + E^((2*I)*beta*x)*z[y]^2 *) In cases where you can live with expansion of complex exponentials into Sin and Cos you can also use ComplexExpand[P[x, y]^2, z[y], ...


2

This is a problem for anything that uses machine precision floats, e.g. Mathematica, Matlab, C, etc. Consider the simpler example $1/10$. In base 10, this fraction has the finite decimal expansion $$ 1/10 = 0.1 $$ But your machine would store this number (and all floats) in binary. The problem is, in binary $1/10$ has the infinite decimal expansion $$ 1/10 ...


2

Coefficient[E^(I a (t - b)) // ExpandAll, E^(I a t)] (* Exp[-I a b] *)


2

An extended comment. I'm not sure if this has been realized, please correct me if it has. The result of the Divide[a,b] operation is not the same as the first 3 which are identical. {a, b} = List @@ RandomReal[{-50, 50}, {2, 1*^7}]; x1 = a/b; x2 = a b^-1; x3 = a/b; x4 = Divide[a, b]; Now... Tally[x1 - x2] Tally[x2 - x3] Both give 10^7 zeros. ...



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