number system and codes

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Digital logic and design

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Lecture No. 6:

Lecture No. 6 Number Systems and codes

Hexadecimal Number System:

Hexadecimal Number System Base 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Representing Binary in compact form 1101100000110 2 = 1B06 H

Counting in Hexadecimal:

Counting in Hexadecimal Decimal Binary Hexadecimal Decimal Binary Hexadecimal 0 0000 0 8 1000 8 1 0001 1 9 1001 9 2 0010 2 10 1010 A 3 0011 3 11 1011 B 4 0100 4 12 1100 C 5 0101 5 13 1101 D 6 0110 6 14 1110 E 7 0111 7 15 1111 F

Counting in Hexadecimal:

Counting in Hexadecimal Decimal Hexa- Decimal Decimal Hexa- Decimal Decimal Hexa- Decimal 16 10 24 18 32 20 17 11 25 19 33 21 18 12 26 1A 34 22 19 13 27 1B 35 23 20 14 28 1C 36 24 21 15 29 1D 37 25 22 16 30 1E 38 26 23 17 31 1F 39 27

Binary-Hexadecimal Conversion :

Binary-Hexadecimal Conversion Binary to Hexadecimal Conversion 11010110101110010110 1101 0110 1011 1001 0110 D 6 B 9 6 Hexadecimal to Binary Conversion FD13 1111 1101 0001 0011

Decimal-Hexadecimal Conversion :

Decimal-Hexadecimal Conversion Decimal to Hexadecimal Conversion Indirect Method Decimal →Binary → Hexadecimal Repeated Division by 16

Repeated Division by 16 :

Repeated Division by 16

Decimal-Hexadecimal Conversion :

Decimal-Hexadecimal Conversion Hexadecimal to Decimal Conversion Indirect Method Hexadecimal →Binary → Decimal Sum-of-Weights

Sum-of-Weights :

Sum-of-Weights

Hexadecimal Addition & Subtraction:

Hexadecimal Addition & Subtraction Hexadecimal Addition Carry generated Hexadecimal Subtraction Borrow weight 16

Hexadecimal Addition:

Hexadecimal Addition Carry 1 2AC6 6+5=11d Bh + 92B5 C+B=23d 17h BD7B A+2+1=13d Dh 2+9=11d Bh

Hexadecimal Subtraction:

Hexadecimal Subtraction Borrow 111 92B5 21-6=15d Fh - 2AC6 26-C=14d Eh 67EF 17-A=7d 7h 8-2=6d 6h

Octal Number System:

Octal Number System Base 8 0, 1, 2, 3, 4, 5, 6, 7 Representing Binary in compact form 1101100000110 2 = 15406 8

Counting in Octal:

Counting in Octal Decimal Binary Octal 0 000 0 1 001 1 2 010 2 3 011 3 4 100 4 5 101 5 6 110 6 7 111 7

Counting in Octal:

Counting in Octal Decimal Octal Decimal Octal Decimal Octal 8 10 16 20 24 30 9 11 17 21 25 31 10 12 18 22 26 32 11 13 19 23 27 33 12 14 20 24 28 34 13 15 21 25 29 35 14 16 22 26 30 36 15 17 23 27 31 37

Binary-Octal Conversion :

Binary-Octal Conversion Binary to Octal Conversion 11010110101110010110 011 010 110 101 110 010 110 3 2 6 5 6 2 6 Octal to Binary Conversion 1726 001 111 010 110

Decimal-Octal Conversion :

Decimal-Octal Conversion Decimal to Octal Conversion Indirect Method Decimal →Binary → Octal Repeated Division by 8

Repeated Division by 8:

Repeated Division by 8 Number Quotient Remainder 2075 259 3 (O 0 ) 259 32 3 (O 1 ) 8 4 0 (O 2 ) 4 0 4 (O 3 )

Decimal-Octal Conversion :

Decimal-Octal Conversion Octal to Decimal Conversion Indirect Method Octal →Binary → Decimal Sum-of-Weights

Sum-of-Weights:

Sum-of-Weights 4033 (4 x 8 3 ) + (0 x 8 2 ) + (3 x 8 1 ) + (3 x 8 0 ) (4 x 512) + (0 x 64) + (3 x 8) + (3 x 1) 2048 + 0 + 24 + 3 2075

Octal Addition & Subtraction:

Octal Addition & Subtraction Octal Addition Carry generated Octal Subtraction Borrow weight 8

Octal Addition:

Octal Addition Carry 1 7602 2+1=3d 3O + 5771 0+7=7d 7O 15573 6+7=13d 15O 1+7+5=13d 15O

Octal Subtraction:

Octal Subtraction Borrow 11 7602 2-1=1d 1O - 5771 8-7=1d 1O 1611 13-7=6d 6O 6-5=1d 1O

Alternate Representations:

BCD Code BCD Addition Gray Code Alternate Representations

Alternate Representations:

Alternate Representations BCD (Binary Coded Decimal) Code Decimal BCD Decimal BCD 0 0000 5 0101 1 0001 6 0110 2 0010 7 0111 3 0011 8 1000 4 0100 9 1001

BCD Addition:

BCD Addition Multi-digit BCD numbers can be added together 23 0010 0011 45 0100 0101 68 0110 1000 23 0010 0011 48 0100 1000 71 0110 1011 1011 is illegal BCD number

BCD Addition:

BCD Addition Add a 0110 (6) to an invalid BCD number Carry added to the most significant BCD digit 23 0010 0011 48 0100 1000 71 0110 1011 0110 0111 0001

Slide 28:

32 0011 0010 84 1000 0100 71 1011 0110 0110 10001 0110

Gray Code:

Gray Code Binary Code more than 1 bit change Electromechanical applications of digital systems restrict bit change to 1 Shaft encoders Un-Weighted Code

Gray Code:

Gray Code Decimal Gray Binary 0 0000 0000 1 0001 0001 2 0011 0010 3 0010 0011 4 0110 0100 5 0111 0101 6 0101 0110 7 0100 0111

Gray Code Application:

Gray Code Application

Alphanumeric Code:

Alphanumeric Code Numbers, Characters, Symbols ASCII 7-bit Code American Standard Code for Information Interchange 10 Numbers (0-9) 26 Lower Case Characters (a-z) 26 Upper Case Characters (A-Z) Punctuation and Symbols 32 Control Characters

ASCII Code:

ASCII Code Numbers 0 to 9 ASCII 0110000 (30h) to 0111001 (39h) Alphabets a to z ASCII 1100001 (61h) to 1111010 (7Ah) Alphabets A to Z ASCII 1000001 (41h) to 1011010 (5Ah) Control Characters ASCII 0000000 (0h) to 0011111 (1Fh)

Alphanumeric Code:

Alphanumeric Code Extended ASCII 8-bit Code Additional 128 Graphic characters Unicode 16-bit Code

Error Detection:

Error Detection Digital Systems are very Reliable Errors during storage or transmission Parity Bit Even Parity Odd Parity

Odd Parity Error Detection:

Odd Parity Error Detection Original data 10011010 With Odd Parity 1 10011010 1-bit error 110 1 11010 Number of 1s even indicates 1-bit error 2-bit error 110 1 1 0 010 Number of 1s odd no error indicated 3-bit error 1 0 0 1 1 0 010 Number of 1s even indicates error

Summary:

Summary 2’s Complement Range and Overflow Floating Point representation

Summary:

Summary Hexadecimal Number System Binary-Hexadecimal Conversion Decimal-Hexadecimal Conversion Octal Number System Binary-Octal Conversion Decimal-Octal Conversion

Summary:

Summary Alternate Representations BCD Code Gray Code Alphanumeric Codes ASCII Error Detection Parity Bit