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Company LOGO Edit your slogan here DKT 122/3 DIGITAL SYSTEM 1 WEEK #3 NUMBER SYSTEMS, OPERATION & CODES (PART 2)

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Company LOGO Add the 2 signed numbers and discard any final carry bit Example: 11111011 + 11110111 11111011 + 11110111 1 11110010 Discard carry bit Signed Numbers Arithmetic Operations - Addition

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Company LOGO take the 2’s complement of the subtrahend and add to the minuend discard any final carry bit Signed Numbers (Cont.) Arithmetic Operations - Subtraction Example: 00001000 – 00000011 minuend subtrahend 00001000 00001000 - 00000011 + 11111101 1 00000101 Discard carry 2’s complement of subtrahend

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Company LOGO Number System (Hexadecimal) Base 16 system Uses digits 0-9 & letters A,B,C,D,E,F Groups of four bits represent each base 16 digit

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Company LOGO Hexadecimal to Decimal Conversion Convert 3B4F 16 to its decimal equivalent: 3 B 4 F xx x 16 3 16 2 16 1 16 0 12288 +2816 + 64 +15 15,183 10 Positional Values Products x Hex digits Number System (Hex) (Cont.)

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Company LOGO Convert 830 10 to its hexadecimal equivalent: 830 / 16 = 51 R 14 51 / 16 = 3 R 3 3 / 16 = 0 R 3 33E 16 Number System (Hex) (Cont.) Decimal to Hexadecimal Conversion E in Hex

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Company LOGO Grouping the binary position in 4-bit groups, starting from the least significant position. Number System (Hex) (Cont.) Binary to Hexadecimal Conversion

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Company LOGO Example Convert 010101101010111001101010 2 to hex using the 4-bit substitution code : 0101 0110 1010 1110 0110 1010 5 6 A E 6 A Binary to Hexadecimal Conversion Number System (Hex) (Cont.)

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Company LOGO Solve this.. a)Convert hexadecimal number 23 16, 8D 16 and EB 16 to decimal b)Convert decimal number 52 10, 284 10 and 2890 10 to hexadecimal c)Convert hexadecimal number 38 16 and 5C8 16 to binary d)Convert binary number 1110 and 10111 to hexadecimal

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Company LOGO Numbering System (Octal) Also known as the Base 8 System Uses digits 0 - 7 Readily converts to binary Groups of three (binary) digits can be used to represent each octal digit Also uses multiplication and division algorithms for conversion to and from base 10

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Company LOGO Convert 653 8 to its decimal equivalent 6 5 3 xx x 8 2 8 1 8 0 384 + 40 + 3 427 10 Positional Values Products Octal Digits Numbering System (Octal) (Cont.) Octal to Decimal Conversion

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Company LOGO Use the repeated division-by-8 method Example Convert 427 10 to its octal equivalent 427 / 8 = 53 R 3LSB 53 / 8= 6 R 5 6 / 8 = 0 R 6MSB 653 8 Numbering System (Octal) (Cont.) Decimal to Octal Conversion

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Company LOGO Each octal number converts to 3 binary digits Example Convert 653 8 to binary 6 5 3 110 101 011 Octal to Binary Conversion Substitute numbers with 3-binary digits Numbering System (Octal) (Cont.)

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Company LOGO Grouping the binary position in groups of 3 starting at the least significant position. Example Convert the binary numbers 110101 to octal 1 1 0 1 0 1 6 5 65 8 Binary to Octal Conversion Numbering System (Octal) (Cont.)

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Company LOGO a)Convert octal number 64 8, 557 8 and 1024 8 to decimal b)Convert decimal number 46 10, 100 10 and 219 10 to octal c)Convert octal number 101 8 and 321 8 to binary d)Convert binary number 101010 and 110111 to octal Solve this..

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Company LOGO Digital Codes - BCD BCD (Binary Coded Decimal) Code Represent each of the 10 decimal digits (0~9) as a 4-bit binary code. Example Convert 15 to BCD. 1 5 0001 0101 BCD

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Company LOGO Step 1: Add the two BCD numbers, using the rules for binary addition Step 2: If a 4-bit sum is equal to or less than 9, it is a valid BCD number Step 3: If a 4-bit sum is greater than 9 / if a carry out is generated, it is an invalid BCD result. Add 6 (0110) to the 4-bit sum Digital Codes – BCD (Cont.) Addition

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Company LOGO Example: 0 0 1 1 + 0 1 0 0 0 1 1 1 (Valid BCD number) Digital Codes – BCD (Cont.) Addition – Valid number

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Company LOGO Example: 1 0 0 1 + 0 1 0 0 1 1 0 1 (Invalid BCD number coz >9) + 0 1 1 0 (Add 6) 0 0 0 1 0 0 1 1 (Valid BCD number) 1 Digital Codes – BCD (Cont.) Addition – Not valid number 3

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Company LOGO Digital Codes - ASCII ASCII (American Standard Code for Information Interchange) Code Used to translate from the keyboard characters to computer language

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Company LOGO ASCII Codes - Table

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Company LOGO Digital Codes - Gray Gray Code – Only 1 bit changes – Can’t be used in arithmetic circuits Exhibits only a single change from one code word to the next in sequence Need to know how to convert binary to Gray and vice versa. DecimalBinary Gray Code 00000 10001 200100011 3 0010 401000110 501010111 601100101

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Company LOGO Convert binary number 10110 to gray code. 10110 11101 Gray Codes ++++ Digital Codes – Gray (Cont.) Binary to Gray Conversion

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Company LOGO Convert gray code 10110 to binary. 10110 11011 Binary ++++ Gray to Binary Conversion Digital Codes – Gray (Cont.)

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Company LOGO Solve this.. (a) Convert binary number 11111 to gray code. (b) Convert gray code 11011 to binary. Answer : ? END

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Chapter 1: Binary Systems

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