History of the Numeral Systems
www.jntuworld.com
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
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