# 10 8 unit ii introduction to digital electronics 21

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UNIT IIINTRODUCTION TO DIGITAL ELECTRONICS2.1 Number systems and conversionsNumber systems provide the basis for all operations in information processingsystems. In a number system the information is divided into a group of symbols; forexample, 26 English letters, 10 decimal digits etc. In conventional arithmetic, anumber system based upon ten units (0 to 9) is used. However, arithmetic and logiccircuits used in computers and other digital systems operate with only 0's and 1'sbecause it is very difficult to design circuits that require ten distinct states. Thenumber system with the basic symbols 0 and 1 is called binary. ie. A binary systemuses just two discrete values. The binary digit (either 0 or 1) is called a bit.A group of bits which is used to represent the discrete elements of information is asymbol. The mapping of symbols to a binary value is known a binary code. Thismapping must be unique. For example, the decimal digits 0 through 9 are representedin a digital system with a code of four bits. Thus a digital system is a system thatmanipulates discrete elements of information that is represented internally in binaryform.Computers process data in binary form, or base 2.Number systems are classified on the basis of radix/base, as shown in Table 2.1.Table 2.1Classification of number systemsBinary0, 1Ternary0, 1, 2Quaternary0, 1, 2, 3Octal0, 1, 2, 3, 4, 5, 6, 7Decimal0, 1, 2, 3, 4, 5, 6, 7, 8, 9Hexadecimal0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,C, D, E, FDecimal Numbers109
The invention of decimal number system has been the most important factor in thedevelopment of science and technology. The decimal number system uses positionalnumber representation, which means that the value of each digit is determined by itsposition in a number.The base, also called the radix of a number system is the number of symbols that thesystem contains. The decimal system has ten symbols: 0,1,2,3,4,5,6,7,8,9. In otherwords, it has a base of 10. Each position in the decimal system is 10 times moresignificant than the previous position. The numeric value of a decimal number isdetermined by multiplying each digit of the number by the value of the position inwhich the digit appears and then adding the products. Thus the number 2734 isinterpreted asHere 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD).In general in a number system with a base or radix r, the digits used are from 0 to r-1and the number can be represented asEquation (1) is for all integers and for the fractions (numbers between 0 and 1), thefollowing equation holds.Thus for decimal fraction 0.7123110
Binary NumbersThe binary number has a radix of 2. As r = 2, only two digits are needed, and theseare 0 and 1. Like the decimal system, binary is a positional system, except that eachbit position corresponds to a power of 2 instead of a power of 10. In digital systems,the binary number system and other number systems closely related to it are usedalmost exclusively. Hence, digital systems often provide conversion between decimaland binary numbers. The decimal value of a binary number can be formed bymultiplying each power of 2 by either 1 or 0 followed by adding the values together.

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