By Jonathan M. Borwein

Thirty years in the past mathematical, in place of utilized numerical, computation used to be tricky to accomplish and so quite little used. 3 threads replaced that: the emergence of the private computing device; the invention of fiber-optics and the resultant improvement of the trendy web; and the development of the 3 “M’s” Maple, Mathematica and Matlab.

We intend to cajole that Mathematica and different related instruments are worthy realizing, assuming simply that one needs to be a mathematician, a arithmetic educator, a working laptop or computer scientist, an engineer or scientist, or somebody else who wishes/needs to take advantage of arithmetic greater. We additionally desire to provide an explanation for the right way to develop into an "experimental mathematician" whereas studying to be larger at proving issues. to complete this our fabric is split into 3 major chapters by means of 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 resources for An Introduction to Modern Mathematical Computing: With Mathematica®**

**Sample text**

To do this we have Mathematica make a decision using an If function. Now, we need Mathematica to recognize which of the calculations are fractions, and which are whole numbers, but unfortunately we currently have no idea how we might do this. One possible answer is to use the modular arithmetic calculations from above, remembering that if n/a ∈ N then n ≡ 0 mod a. This is something we already know how to express in Mathematica. In order to see if, say, 3 was a divisor of our 6 then we could issue the command In[169]:= If[6 ~Mod~ 3 == 0, 6 / 3] Out[169]= 2 The above code should be read as “If 6 is equal to 0 modulo 3 then calculate 6/3,” and because 6 is most certainly equivalent to 0 modulo 3, Mathematica has correctly gone on to calculate 6/3 = 2.

In[112]:= N[Infinity] Out[112]= π2 6 Ih either case, there we have it. It looks very much as if the series converges to 1/6 π 2 , provided we trust Mathematica’s limit computation. To verify this analytically we would need to use the integral test, but we shall not do so here. We will see how to perform calculus with Mathematica in Chapter 2, but performing the integral test for this series is left as an exercise to the reader. 2 Loops Until now if we wanted to perform something several times, we either typed it in multiple times at the command prompt, or we constructed a list.

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. 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).