Dr. P. Sudhakara Rao, Dean UNIT1: NUMBER SYSTEMS & CODES • Philosophy of number systems • Complement representation of negative numbers • Binary arithmetic • Binary codes • Error detecting & error correcting codes • Hamming codes Switching Theory and Logic Design
HISTORY OF THE NUMERAL SYSTEMS:
A numeral system (or system of numeration) is a linguistic system and mathematical notation for representing numbers of a given set by symbols in a consistent manner. For example, It allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in different bases. Ideally, a numeral system will: • Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers) • Give every number represented a unique representation (or at least a standard representation) • Reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the experimental sciences, where greater precision is denoted by the trailing zero. The most commonly used system of numerals is known as Hindu-Arabic numerals. Great Indian mathematicians Aryabhatta of Kusumapura (5th Century) developed the place value notation. Brahmagupta (6th Century) introduced the symbol zero. Unary System: Every natural number is represented by a corresponding number of symbols, for example the number seven would be represented by ///////. Elias gamma coding which is commonly used in data compression expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral. With different symbols for certain new values, if / stands for one, - for ten and + for 100, then the number 123 as + - - /// without any need for zero. This is called signvalue notation. More elegant is a positional system, also known as place-value notation. Again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness
Vignan Institute of Technology and Science
Dr. P. Sudhakara Rao, Dean Switching Theory and Logic Design
caused by leading zeros. Bijective base-k numeration is also called k-adic notation, not to be confused with padic numbers. Bijective base-1 the same as unary.
Five A base-5 system (quinary), on the number of fingers, has been used in many cultures for counting. It may also be regarded as a sub-base of other bases, such as base 10 and base 60. Eight A base-8 system (octal), spaces between the fingers , was devised by the Yuki of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight Ten The base-10 system (decimal) is the one most commonly used today. It is assumed to have originated because humans...