By Martin A Moskowitz
Although uncomplicated in nature, this e-book offers with basic matters in arithmetic вЂ” quantity, algebra, geometry (both Euclidean and non-Euclidean) and topology. those matters, on a sophisticated point, are a similar ones with which a lot of present mathematical examine is worried and have been themselves learn subject matters of previous classes. the cloth is especially appropriate either for complex highschool scholars and for students attracted to straightforward arithmetic from the next viewpoint. it is going to even be very helpful to highschool lecturers looking an outline in their material.
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That is, cannot be put in 1:1 correspondence with the positive integers, Z + . To do this it is clearly sufficient to show that the unit interval, [0,1], itself cannot be enumerated. Just as Cantor did, this will is done by use of his celebrated diagonal process . 8. M cannot be enumerated. Proof. This will be proved by contradiction. Suppose [0,1] can be enumerated. Let each real number between 0 and 1 be represented by a decimal. 0000... ) we select the terminating one. Let •%n,lZn,2%n,3 • • • %n,j • • • be the nth one.
Similarly, chose a number between 0 and 9 which is different from 22,2 a n d from 9 to be the second decimal. 5 The Integers Revisited 25 of the enumerated numbers since for each integer n it differs from the nth one in the nth spot. This contradiction proves the theorem. Of course since [0,1] is not enumerable neither is R. 5 The Integers Revisited We now use the Archimedean property (of R) to derive some facts about Z itself. First we turn to what is known as the Euclidean algorithm or division algorithm.
For a complex number z =• a + bi, we define its conjugate z — a — bi. Then it is a direct verification that ~z = z for all z € C. z — —z if and only if z is pure imaginary, z = z if and only if z is real and for all complex numbers z and w we have (zw) = zw. If \z\ is defined by \ / x 2 + y2, then zz = \z\2. Also \z\ = 0 if and only if x2 + y2 = 0. But since the square of a real number is always greater than or equal to zero, a sum of squares of real numbers can only be zero if each of the numbers is itself zero; thus in this case, if and only if both x and y = 0, that is, if and only if z — 0.
Adventures in Mathematics by Martin A Moskowitz