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**Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP**- Computer Science Engineering
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Digital Logic Design Number System Unit -I II B.Tech I Sem – CSE, IT

Number System Introduction to Digital Systems Digital systems have such a prominent role in everyday life that we refer to the present technological period as the digital age. Digital systems are used in communication, business transactions, traffic control, spacecraft guidance, medical treatment, weather monitoring, the Internet, and many other commercial, industrial, and scientific enterprises. We have digital telephones, digital televisions, digital versatile discs, digital cameras, handheld devices, and, of course, digital computers. One characteristic of digital systems is their ability to represent and manipulate discrete elements of information. Any set that is restricted to a finite number of elements contains discrete information. Examples of discrete sets are the 10 decimal digits, the 26 letters of the alphabet, the 52 playing cards, and the 64 squares of a chessboard. Discrete elements of information are represented in a digital system by physical quantities called signals. Electrical signals such as voltages and currents are the most common. The signals in most present-day electronic digital systems use just two discrete values and are therefore said to be binary. A binary digit, called a bit, has two values: 0 and 1. Discrete elements of information are represented with groups of bits called binary codes. Binary Numbers A decimal number such as 8,473 represents a quantity equal to 8 thousands, plus 4 hundreds, plus 7 tens, plus 3 units. The thousands, hundreds, etc., are powers of 10 implied by the position of the coefficients (symbols) in the number. To be more exact, 8,473 is a shorthand notation for what should be written as 8 X 103 + 4 X 102 + 7 X 101 + 3 X 100 However, the convention is to write only the numeric coefficients and, from their position, deduce the necessary powers of 10 with powers increasing from right to left. In general, a number with a decimal point is represented by a series of coefficients: a5a4a3a2a1a0 . a-1a-2a-3 The coefficients aj are any of the 10 digits (0, 1, 2, …,9), and the subscript value j gives the place value and, hence, the power of 10 by which the coefficient must be multiplied. Thus, the preceding decimal number can be expressed as 105a5 + 104a4 + 103a3 + 102a2 + 101a1 + 100a0 + 10-1a-1 + 10-2a-2 + 10-3a-3 With a3 = 8, a2 = 4, a3=7, a4=3 The decimal number system is said to be of base, or radix, 10 because it uses 10 digits and the coefficients are multiplied by powers of 10. The binary system is a different number system. The coefficients of the binary number system have only two possible values: 0 and 1. Each coefficient aj is multiplied by a power of the radix, e.g., 2j, and the results are added to obtain the decimal equivalent of CSE & IT Dept., PEC Page 2

Number System the number. The radix point (e.g., the decimal point when 10 is the radix) distinguishes positive powers of 10 from negative powers of 10. For example, the decimal equivalent of the binary number 10010.11 is 18.75, as shown from the multiplication of the coefficients by powers of 2: 1 X 24 + 0 X 23 + 0 X 22 + 1 X 21 + 0 X 20 + 1 X 2-1 + 1 X 2-2 = 18.75 There are many different number systems. In general, a number expressed in a base-r system has coefficients multiplied by powers of r: an . r n + an-1 . r n-1 + … + a2 . r 2 + a1 . r + a0 + a-1 . r-1+ a-2 . r-2 + … + a-m . r-m The coefficients aj range in value from 0 to r - 1. To distinguish between numbers of different bases, we enclose the coefficients in parentheses and write a subscript equal to the base used (except sometimes for decimal numbers, where the content makes it obvious that the base is decimal). The octal number system is a base-8 system that has eight digits: 0, 1, 2, 3, 4, 5, 6, 7. An example of an octal number is 127.4. To determine its equivalent decimal value, we expand the number in a power series with a base of 8: (127.4)8 = 1 X 82 + 2 X 81 + 7 X 80 + 4 X 8-1 = (87.5)10 Note that the digits 8 and 9 cannot appear in an octal number. It is customary to borrow the needed r digits for the coefficients from the decimal system when the base of the number is less than 10. The letters of the alphabet are used to supplement the 10 decimal digits when the base of the number is greater than 10. For example, in the hexadecimal (base-16) number system, the first 10 digits are borrowed from the decimal system. The letters A, B, C, D, E, and F are used for the digits 10, 11, 12, 13, 14, and 15, respectively. An example of a hexadecimal number is (A89)16 = 10 X 162 + 8 X 161 + 9 X 160 = (2697)10 The hexadecimal system is used commonly by designers to represent long strings of bits in the addresses, instructions, and data in digital systems. Binary to Decimal Conversion As noted before, the digits in a binary number are called bits. When a bit is equal to 0, it does not contribute to the sum during the conversion. Therefore, the conversion from binary to decimal can be obtained by adding only the numbers with powers of two corresponding to the bits that are equal to 1. For example, (101010)2 (101010)2 = 1 X 25 + 0 X 24 + 1 X 23 + 0 X 22 + 1 X 21 + 0 X 20 = (42)10 There are three 1’s in the binary number. The corresponding decimal number is the sum of the three powers of two. Zero and the first 24 numbers obtained from 2 to the power of n are listed in Table below. CSE & IT Dept., PEC Page 3

Number System In computer work, 210 is referred to as K (kilo), 220 as M (mega), 230 as G (giga), and 240 as T (tera). Thus, 4K = 212 = 4,096 and 16M = 224 = 16,777,216. Computer capacity is usually given in bytes. A byte is equal to eight bits and can accommodate (i.e., represent the code of) one keyboard character. A computer hard disk with four gigabytes of storage has a capacity of 4G = 232 bytes (approximately 4 billion bytes). A terabyte is 1024 gigabytes, approximately 1 trillion bytes. Arithmetic operations with numbers in base r follow the same rules as for decimal numbers. When a base other than the familiar base 10 is used, one must be careful to use only the r-allowable digits. Examples of addition, subtraction, and multiplication of two binary numbers are as follows: In binary addition, 0 + 0 = 0, 1 + 0 = 1, 0 + 1 = 1, 1 + 1 = 10, where sum is 0 and 1 will be a carry added to next digit same as in decimal addition. In binary subtraction, 0 – 0 = 0, 1 – 0 = 1, 1 – 1 =0, 0 – 1 = 1 with a borrow of 1. Multiplication is simple: The multiplier digits are always 1 or 0; therefore, the partial products are equal either to a shifted (left) copy of the multiplicand or to 0. Number-Base Conversions Decimal to Binary The conversion of a decimal integer to a number in base r is done by dividing the number and all successive quotients by r and accumulating the remainders. This procedure is best illustrated by example. Ex: Convert decimal 41 to binary. First, 41 is divided by 2 to give an integer quotient of 20 and a remainder of 1. Then the quotient is again divided by 2 to give a new quotient and remainder. The process is continued until the integer quotient becomes 0. The coefficients of the desired binary number are obtained from the remainders as follows: CSE & IT Dept., PEC Page 4

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