Mathematica Tutorial: Notebooks And Documents - Wolfram Research

Notebooks and Documents 57Exposition in Mathematica NotebooksMathematica notebooks provide the basic technology that you need to be able to create a verywide range of sophisticated interactive documents. But to get the best out of this technologyyou need to develop an appropriate style of exposition.Many people at first tend to use Mathematica notebooks either as simple worksheets containinga sequence of input and output lines, or as onscreen versions of traditional books and otherprinted material. But the most effective and productive uses of Mathematica notebooks tend tolie at neither one of these extremes, and instead typically involve a fine-grained mixing ofMathematica input and output with explanatory text. In most cases the single most importantfactor in obtaining such fine-grained mixing is uniform use of the Mathematica language.One might think that there would tend to be four kinds of material in a Mathematica notebook:plain text, mathematical formulas, computer code, and interactive interfaces. But one of thekey ideas of Mathematica is to provide a single language that offers the best of both traditionalmathematical formulas and computer code.In StandardForm, Mathematica expressions have the same kind of compactness and eleganceas traditional mathematical formulas. But unlike such formulas, Mathematica expressions areset up in a completely consistent and uniform way. As a result, if you use Mathematica expressions,then regardless of your subject matter, you never have to go back and reexplain yourbasic notation: it is always just the notation of the Mathematica language. In addition, if youset up your explanations in terms of Mathematica expressions, then a reader of your notebookcan immediately take what you have given, and actually execute it as Mathematica input.If one has spent many years working with traditional mathematical notation, then it takes alittle time to get used to seeing mathematical facts presented as StandardForm Mathematicaexpressions. Indeed, at first one often has a tendency to try to use TraditionalForm wheneverpossible, perhaps with hidden tags to indicate its interpretation. But quite soon one tends toevolve to a mixture of StandardForm and TraditionalForm. And in the end it becomes clearthat StandardForm alone is for most purposes the most effective form of presentation.In traditional mathematical exposition, there are many tricks for replacing chunks of text byfragments of formulas. In StandardForm many of these same tricks can be used. But the factthat Mathematica expressions can represent not only mathematical objects but also procedures,algorithms, graphics, and interfaces increases greatly the extent to which chunks of text can bereplaced by shorter and more precise material.