By Jonathan M. Borwein

Thirty years in the past mathematical, in preference to utilized numerical, computation was once tricky to accomplish and so particularly little used. 3 threads replaced that: the emergence of the non-public machine; the invention of fiber-optics and the ensuing improvement of the trendy net; and the construction of the 3 “M’s” Maple, Mathematica and Matlab.

We intend to cajole that Mathematica and different related instruments are worthy understanding, assuming in basic terms that one needs to be a mathematician, a arithmetic educator, a working laptop or computer scientist, an engineer or scientist, or someone else who wishes/needs to take advantage of arithmetic higher. We additionally desire to provide an explanation for find out how to develop into an "experimental mathematician" whereas studying to be higher at proving issues. to complete this our fabric is split into 3 major chapters via a postscript. those disguise hassle-free quantity idea, calculus of 1 and a number of other variables, introductory linear algebra, and visualization and interactive geometric computation.

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This technique works, but turns out to be about twice to three times as slow as using Npar. 64493 What we are seeing is an example of a case where obtaining decimal approximations of previous symbolic computations is simply not feasible. Sometimes we have to work purely numerically, and the reality is that numeric computations aren’t as bad as all that. We should, of course, be quite aware of the fact that numerical approximation can introduce errors in our calculations, and be on the lookout for them, but this should not and does not detract from the usefulness of symbolic computation.

Observe. In[89]:= a ~F~ b Out[89]= F [a, b] It should hopefully be expected, in light of the both pre- and postﬁx notation, that any function that can take exactly two inputs may be used in inﬁx notation. Unfortunately, our previous example is not appropriate for inﬁx notation; it does not have any functions that take two arguments. However, we have already seen a function which lends itself very nicely to inﬁx notation, the Join function. In[90]:= {1, 2, 3} ~Join~ {3, 4, 5} Out[90]= {1, 2, 3, 3, 4, 5} We may chain these notations together, if we wish, and the results are fairly predictable.

This leaves us with two criteria, which we are able to express as a single criterion for the If function by using the && logical ‘and’ operator. We implement this approach using a loop similar to that used above. = n && (Divisors[m] // Total) - m == n, {n, m} // Print ], {n, 1, N} ]; ] Out[194]= {220, 284} Out[195]= {284, 220} Out[196]= {1184, 1210} Out[197]= {1210, 1184} Out[198]= {2620, 2924} Out[199]= {2924, 2620} Out[200]= {5020, 5564} Out[201]= {5564, 5020} Out[202]= {6232, 6368} Out[203]= {6368, 6232} To compute this with lists and cases, we ﬁrst compute the list containing all candidate pairs of n, m where m is the sum of the proper divisors of m, and then apply a pattern to the Cases function which matches a list of 2 elements, with the elements named n and m, and choose those pairs which match our criteria, above.