B-Parts Of Real Numbers And Their Generalization - Hooshmand, Mohamad Hadi (2010)
Tiga aspek iaitu peringkat permulaan, analisis dan kaedah algebra dikaji dengan mempertimbangkan ciri-ciri asas dan utama bagi bahagian-b nombor nyata. Bagi aspek pertama, terdapat beberapa subjek dan aplikasi seperti: (a) Penjelasan teori nombor bagi bahagian-b dan ciri-ciri asasnya. (b) Aplikasi bahagian-b untuk perwakilan unik nombor nyata: terhingga dan tidak terhingga, penggunaannya dan algoritma pembahagian itlak; beberapa rumus langsung untuk digit bagi pengembangan unik nombor' nyata tidak terhingga terhadap asas integer diperkenalkan serta perwakilan unik nombor nyata terhingga terhadap asas sebarang (nombor nyata 0, ±1) dibuktikan. (c) Aplikasi bahagian-b untuk menentukan bentuk am subset nyata bagi kala-b dan fungsinya. Considering the basic and main properties of the b-parts of real numbers, we have studied them in three aspects: elementary, analytic and algebraic methods. As the first aspect we have several subjects and their applications such as: (a) Number theoretic explanations of b-parts and their elementary properties. (b) Application of b-parts for unique finite and infinite representation of real numbers, applying them and the generalized division algorithm, we not only introduce some direct formulas for digits of the unique infinite expansion of real numbers to the base an integer but also prove a (new) unique finite representation of real numbers to the base an arbitrary real number (not equal 0, ±1). (c) Application of b-parts for determining the general form of b-periodic real subsets and functions.