By Jonathan M. Borwein
Thirty years in the past mathematical, in place of utilized numerical, computation used to be tough to accomplish and so really little used. 3 threads replaced that: the emergence of the non-public laptop; the invention of fiber-optics and the resultant improvement of the fashionable net; and the construction of the 3 “M’s” Maple, Mathematica and Matlab.
We intend to cajole that Mathematica and different comparable instruments are worthy understanding, assuming purely that one needs to be a mathematician, a arithmetic educator, a working laptop or computer scientist, an engineer or scientist, or an individual else who wishes/needs to take advantage of arithmetic larger. We additionally wish to give an explanation for tips on how to turn into an "experimental mathematician" whereas studying to be larger at proving issues. to complete this our fabric is split into 3 major chapters via a postscript. those conceal easy quantity thought, calculus of 1 and a number of other variables, introductory linear algebra, and visualization and interactive geometric computation.
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Extra info for An Introduction to Modern Mathematical Computing: With Mathematica®
In this case, however, the action we are performing is not a computation, but is to print a message to the screen, as such the Print is key, and takes centre stage, so to speak, when we read the input. To be properly illustrative, we should see a case where the If function evaluates its third argument. That is, we should see a case where the criterion fails. To this end, we will try n = 7. Note that this also illustrates a weakness with the cheat we performed to write a message to the output; the english becomes broken due to Mathematica re-ordering the unknowns when it outputs the expression.
To illustrate this idea, and how we implement decisions in Mathematica we look at the following problem. Let us say we have a natural number n. Recall that if n can be divided by another natural number a evenly—that is, n/a is a natural number—we use the notation a|n and say that a divides n or that a is a divisor of n. 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.
Observe that the fraction is undeﬁned for N = −1, but this isn’t a concern for us as we’re only interested in natural values of N . However, we may need to be a little speciﬁc in what we ask Mathematica. In:= Abs[a[N + 1] / a[N]] < 1 Out= Abs N2 <1 2+N The less than operator (<) should have produced an output of either True or False. The fact that it simply re-iterated the input tells us that Mathematica does not know the answer. With a bit of thought this is hopefully not too surprising.