Moodle WILLINGDON COLLEGE SANGLI. ELECTRONICS (B. Sc.-I) Introduction to Number System

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1 Moodle 1 WILLINGDON COLLEGE SANGLI ELECTRONICS (B. Sc.-I) Introduction to Number System

2 E L E C T R O N I C S Introduction to Number System and Codes Moodle developed By Dr. S. R. Kumbhar Department of Electronics Willingdon College Sangli For B. Sc. I Electronics, Physics & Computer Science Students

3 N U M B E R S Y S T E M - 2 CHAPTER - I NUMBER SYSTEM AND BOOLEAN ALGEBRA 1.1 INTRODUCTION The number system is a system in which an ordered set of digits is used to specify number. The system having digits 0 to 9 is called decimal system. In decimal system the base or radix is 10. The system having numbers 0 and 1 is called binary with radix 2. The system having numbers 0 to 7 is called octal system and its radix is 8. In the hexadecimal system the numbers are represented from 0 to 9 and A to F with hex as base. Digital system works on the binary system. In digital system BIT is abbreviated from Binary digit. Bit is the basic unit of memory with 0 and 1 digit. Some of the common number systems are binary, decimal, octal and hexadecimal out of which decimal number system is most familiar to us. These systems are mostly used in the digital circuits, computers and many logic circuits. Therefore digital systems are required to handle data in the form of numeric, alphabets or symbols. Such data must be converted in to binary format by using different types of codes such as BCD, ASCII, GRAY, etc. 1.2 Binary number system Binary number system is represented with radix or base of 2 with only 0 and 1 digits. This 0 and 1 digits are called binary digits. A binary number with group of four bits is called Nibble and a group of 8 bits is called Byte. Following chart shows the binary equivalent of decimal number. B I N A R Y D E C I M A L In binary number system left most bit is called Most Significant Bit (MSB) and right most bit is called least significant bit (LSB). 2

4 N U M B E R S Y S T E M - 3 Example: A. MSB LSB a) Sign of numbers: In binary number system negative numbers are represented by placing a signed bit in the left most position of number and is separated by a comma. A 0 in this position means positive while 1 means negative number. Thus means 0, 111 and means 1, Decimal number system: Decimal number system has 10 digits such as 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are required to express a number. The base or radix of this number system is the total number of digits used in the system. Decimal number system has base 10 (ten). In decimal number system individual digits of the number represents the coefficients of some power of (10). Example : (2186) 10 is expanded in powers as follows 2186 = 2 x x x x Octal number system Octal number system has a base or radix of 8. In the octal system there are eight symbols are used such as 0, 1, 2, 3, 4, 5, 6 and 7. Example : (56) 8, (21) 8, (77) 8, etc. The octal number system does not have the number larger than decimal Hexadecimal number system The hexadecimal number system (Hex) uses a base or radix of 16. This system uses a set of 16 unique symbols i.e. 0 to 9 decimal digits having the same significance in decimal number system and six symbols A, B, C, D, E and F which represents decimal 10, 11, 12, 13, 14 and 15 respectively. Example: ( A7) Hex, (10) Hex, (FA) Hex, (1B) Hex s and 2 s complement 1 s Complement: 1 s complement of binary number is usually expressed by changing each 0 to 1 and each 1 to 0. Thus 1 s complement of is and 1 s complement of is s complement: 2 s complement of binary number is usually obtained by adding 1 to the least significant bit of the 1 s complement of binary number. 2 s complement of 0110 = = s complement of = =

5 N U M B E R S Y S T E M - 4 Example: Perform the subtraction by 1 s complement a) b) = = Add carry If the carry is not generated then the answer is negative and is in 1 s complement form. Perform the subtraction using 2 s complement a) b) In the subtraction using 2 s complement method first take the 2 s complement of the number to be subtracted by 1 s complement and adding 1 to the LSB of subtrahend. Then binary addition of two numbers is performed with dropping a carry, if generated. a) (2 s complement of 0111) carry dropped 0101 = Answer a) ( 2 s complement of 0111) carry dropped = Answer If the carry is not generated then the answer is negative and is in 2 s complement form. i.e. = 0 with carry 1 Rules for binary subtraction 0-0 = = 1 with borrow = = Binary addition Rules for binary addition = = = = 10 Perform the binary addition of the following numbers a) b) Perform the binary subtraction of the following numbers a) b)

6 N U M B E R S Y S T E M Conversion of numbers Binary to Octal conversion: Note: make a group of 3 bits from decimal point a) to Octal b) to Octal = (207) 8 = (13.54) Binary to Decimal conversion Binary number can be converted into its equivalent decimal number using weighted position method. a) Convert ( ) 2 to decimal equivalent = 1 x x x x x x x x 2-2 = = (27.75) 10 b) Convert to decimal equivalent = 1x x x x x 2-3 = = (3.375) Binary to hexadecimal conversion a) Convert binary to hexadecimal number Note: make a group of 4 bits from decimal point and convert binary number into its decimal equivalent. a) b) to Hex B = (265) Hex = (3.B) 8 C) Convert to Hexadecimal E. F 6 = (73E.F6) Hex Conversion of octal to binary 1) Convert (2176) 8 to Binary

7 N U M B E R S Y S T E M - 6 2) Convert (0.75) 8 to Binary. 3) Convert to binary = = ( ) Octal to decimal converter a) convert (315.1) to decimal b) Convert (724) to decimal x x x x x x 8 0 = = (210.15) 10 = (261) Octal to Hexadecimal converter 1) Convert (63.5) 8 to hexadecimal Convert it into group of four digits from the decimal point By making group of 4 digits Decimal to binary 3 3. C = (33. C) Hex For integers divide successively by 2, by writing quotient and its reminder till zero. The reminder is taken in reverse order. a) Convert decimal 15 to binary 15 2 = 7 1 LSB 7 2 = 3 1 = (1111) = = 0 1 MSB B) Convert decimal to binary Note : For fraction multiply given number by 2, record the carry till quotient becomes zero and then represent the binary number in downward direction = 14 1 LSB x 2 = = x 2 = = x 2 = = x 2 = = 0 1 MSB = ( ) Convert decimal to Octal While converting decimal to octal, divide the number by 8 successively and note down the reminder in reverse order. Similarly fractional numbers are multiplied by 8 and carry is recorded. 6

8 N U M B E R S Y S T E M - 7 a) Convert ( ) to octal = 67 7 LSB x 8 = = x 8 = = x 8 = = 0 1 MSB x 8 = x 8 = = ( ) 8 b) Convert ( ) to octal = 68 6 LSB 0.55 x 8 = = x 8 = = x 8 = = 0 1 MSB 0.6 x 8 = x 8 = Reoccurring = ( ) conversion of decimal to hexadecimal a) Convert ( ) to hexadecimal = 135 B LSB 0.1 x 16 = = x 16 = = 0 8 MSB 0. 6 x 16 = Reoccurring = (87B.199) Hex b) Convert ( ) to hexadecimal = LSB 0.1 x 16 = = 12 A 0.6 x 16 = = 0 C MSB 0. 6 x 16 = Reoccurring = (CA7.199) Hex conversion of hexadecimal to binary a) convert (D0C) Hex to binary and b) (BC.B3C) to binary a) D 0 C b) B C. B 3 C = ( ) 2 = ( ) conversion of hexadecimal to Octal First convert the given hexadecimal number into binary and then make the group of three bits from the decimal point and write the corresponding octal number a) Convert (CAB) Hex to Octal C A B Hexadecimal number Binary number ( ) 2 Binary 3 digit groups = (6253) 8 Octal number 7

9 N U M B E R S Y S T E M - 8 a) Convert (BA. 1B) Hex to Octal B A. 1 B Hexadecimal number Binary number ( ) 2 Binary 3 digit groups = (135013) 8 Octal number conversion of hexadecimal to Octal a) Convert (150) Hex to decimal = 1x x x 16 0 = = (336) Hex B) Convert (18B. 88) Hex to decimal 1 8 B = 1x x B x x x 16-2 = B = ( ) Hex 8