An Introduction to Modern Mathematical Computing: With by Jonathan M. Borwein

By Jonathan M. Borwein

Thirty years in the past mathematical, rather than utilized numerical, computation was once tricky to accomplish and so particularly little used. 3 threads replaced that: the emergence of the private laptop; 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 simply 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 exploit arithmetic greater. We additionally wish to provide an explanation for how you can turn 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 hide effortless quantity thought, calculus of 1 and a number of other variables, introductory linear algebra, and visualization and interactive geometric computation.

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If we want this new function to apply to the elements of a list we need to use either the Map function, or the map operator /@. In[213]:= Map[PerfectQ, {6, 10, 28}] PerfectQ /@ {6, 10, 28} 34 1 Number Theory Out[213]= {True, False, True} Out[214]= {True, False, True} Unfortunately, the pattern in our PerfectQ function causes another small problem. Our function is supposed to report whether its argument is a perfect number, but in many cases it does nothing at all. Any argument which is not a positive integer cannot possibly be a perfect number, and so our function should return False in these cases.

Furthermore, if a|n then n = ka for some k ∈ N and so, recalling modular arithmetic, n ≡ 0 mod a. The problem we now try to solve now with Mathematica is to find all the divisors of a number. To begin with, it is helpful to know that Mathematica can perform modular 24 1 Number Theory arithmetic using the Mod function. Simply put, entering Mod[a, b] will calculate the modulus of a (modulo b). 6, we may use it to perhaps make the input a little more intuitive. In this case a ~Mod~ b will compute the modulus of a (modulo b).

This is a form of pattern matching inside of Mathematica which we will later use to restrict the arguments of our functions to particular types of mathematical objects or Mathematica expressions. For the moment, however, we content ourselves with allowing any valid expression for our arguments. Fortunately, our function behaves quite sensibly with a variety of different arguments. In[96]:= p[2] p[4] p[A] p[{2, 4}] Out[96]= 18 Out[97]= 62 16 1 Number Theory Out[98]= − 2 + 4A + 3A2 Out[99]= {18, 62} It is interesting to see that in the last example above that the function was applied to each element of the list we used as our argument.

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