By Schreier O., Sperner E.
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Extra resources for Projective geometry of N dimensions (of Intro. to modern algebra and matrix theory)
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.