By George E. Martin (auth.)

This ebook is a textual content for junior, senior, or first-year graduate classes commonly titled Foundations of Geometry and/or Non Euclidean Geometry. the 1st 29 chapters are for a semester or yr path at the foundations of geometry. the remainder chap ters may perhaps then be used for both a standard direction or self sustaining examine classes. one other hazard, that's additionally in particular suited to in-service lecturers of highschool geometry, is to survey the the basics of absolute geometry (Chapters 1 -20) in a short time and start earnest learn with the speculation of parallels and isometries (Chapters 21 -30). The textual content is self-contained, other than that the ordinary calculus is thought for a few elements of the cloth on complex hyperbolic geometry (Chapters 31 -34). There are over 650 workouts, 30 of that are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic improvement of the Euclidean and hyperbolic planes, together with the category of the isometries of those planes, is balanced via the dialogue approximately this improvement. types, corresponding to Taxicab Geometry, are used exten sively to demonstrate concept. old elements and choices to the chosen axioms are fashionable. The classical axiom platforms of Euclid and Hilbert are mentioned, as are axiom structures for 3 and 4-dimensional absolute geometry and Pieri's process in response to inflexible motions. The textual content is split into 3 components. The advent (Chapters 1 -4) is to be learn as quick as attainable after which used for ref erence if necessary.

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**Example text**

C) + (b . c) for all a, b, c in Q. The number system (Q, +, ',0,1) is called the field of rationals. The very early Greeks thought that all numbers had to be rational numbers. The whole of religion and philosophy of the early Pythagorean school was based on this supposed fact. It came as quite a shock to find that the diagonal of a square with sides of length 1 could not be expressed as a quotient of in~ers. In other words, there do not exist integers a and b such that V 2 = al b. To prove this fact, one begins by assuming Y2=alb where a and b are integers and alb has already been reduced to its lowest terms.

The Pythagorean idea that all (real) numbers eventually depend on the integers for their definition was vindicated by the work of Richard Dedekind in 1872. Dedekind (1831-1916), following in the footsteps of Eudoxus, was among those who first gave a rigorous definition of the real numbers. A thorough understanding of the real numbers is only a hundred years old! Dedekind defined an infinite set to be any set such that there is a one-to-one correspondence between the set and some proper subset of the set.

Example 4 For real numbers a and b, let a':' b = a2 + b2 • Then ", is an example of a binary operation on R that is commutative but not associative. Example 5 Define binary operation ", on R by a'" b= lalb. Then ", is a binary operation that is associative but not commutative. ) Recall that Ial = a if a ~ 0 but Ial = -a if a < O. So Iai, called the absolute value of ai is always nonnegative. Example 6 Another example of an associative binary operation that is not commutative is composition of permutations on a set A where A has at least three elements.