[PDF] Concrete Mathematics a Foundation for Computer Science

THIS BOOK IS BASED on a course of the identical identify that has been taught yearly at Stanford College since 1970. About fifty college students have taken it every year-juniors and seniors, however largely graduate students-and alumni of those courses have begun to spawn comparable programs elsewhere. Thus the time appears ripe to current the fabric to a wider viewers (together with sophomores). It was a darkish and stormy decade when Concrete Mathematics was born. Lengthy-held values had been continuously being questioned throughout these turbulent years; school campuses had been hotbeds of controversy. The faculty curriculum itself was challenged, and arithmetic didn’t escape scrutiny.

John Hammersley had simply written a thought-provoking article “On the enfeeblement of mathematical abilities by ‘Trendy Mathematics’ and by comparable gentle mental trash in faculties and universities” [145]; different apprehensive mathematicians [272] even requested, “Can arithmetic be saved?” One of many current authors had launched into a collection of books referred to as The Artwork of Computer Programming, and in writing the primary quantity he (DEK) had discovered that there have been mathematical instruments lacking from his repertoire; the arithmetic he wanted for a thorough, well-grounded understanding of laptop applications was fairly completely different from what he’d discovered as a arithmetic main in school. So he launched a new course, instructing what he wished anyone had taught him.

The course title “Concrete Mathematics” was initially meant as an antidote to “Summary Mathematics,” since concrete classical outcomes had been quickly being swept out of the trendy mathematical curriculum by a new wave of summary concepts popularly referred to as the “New Math!’ Summary arithmetic is a great topic, and there’s nothing improper with it: It’s lovely, normal, and helpful. However its adherents had change into deluded that the remainder of arithmetic was inferior and now not worthy of consideration. The objective of generalization had change into so trendy that a era of mathematicians had change into unable to relish magnificence within the specific, to benefit from the problem of fixing quantitative issues, or to understand the worth of method.

Summary arithmetic was changing into inbred and dropping contact with actuality; mathematical schooling wanted a concrete counterweight in an effort to restore a wholesome stability. When DEK taught Concrete Mathematics at Stanford for the primary time, he defined the considerably unusual title by saying that it was his try to show a math course that was exhausting as an alternative of sentimental. He introduced that, opposite to the expectations of a few of his colleagues, he was not going to show the Idea of Aggregates, nor Stone’s Embedding Theorem, nor even the Stone-Tech compactification. (A number of college students from the civil engineering division acquired up and quietly left the room.)

Though Concrete Mathematics started as a response towards different traits, the primary causes for its existence had been constructive as an alternative of detrimental. And because the course continued its well-liked place within the curriculum, its subject material “solidified” and proved to be invaluable in a number of new purposes. In the meantime, impartial affirmation for the appropriateness of the identify got here from one other course, when Z. A. Melzak printed two volumes entitled Companion to Concrete Mathematics [214]. The fabric of concrete arithmetic could seem at first to be a disparate bag of methods, however apply makes it into a disciplined set of instruments. Certainly, the methods have an underlying unity and a sturdy enchantment for many individuals.

When one other one of many authors (RLG) first taught the course in 1979, the scholars had such enjoyable that they determined to carry a class reunion a yr later. However what precisely is Concrete Mathematics? It’s a mix of steady and diSCRETE arithmetic. Extra concretely, it’s the managed manipulation of mathematical formulation, utilizing a assortment of methods for fixing issues. When you, the reader, have discovered the fabric on this guide, all you’ll need is a cool head, a giant sheet of paper, and pretty respectable handwriting in an effort to consider horrendous-looking sums, to unravel advanced recurrence relations, and to find refined patterns in information. You may be so fluent in algebraic methods that you’ll typically discover it simpler to acquire precise outcomes than to settle for approximate solutions which might be legitimate solely in a limiting sense.

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