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NUMBER SYSTEMS AND CODES

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CS Digital LogicNumber Systems and Codes2 Outline Number systems –Number notations –Arithmetic –Base conversions –Signed number representation Codes –Decimal codes –Gray code –Error detection code –ASCII code

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CS Digital LogicNumber Systems and Codes3 Number Systems The decimal (real), binary, octal, hexadecimal number systems are used to represent information in digital systems. Any number system consists of a set of digits and a set of operators (+, , , ).

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CS Digital LogicNumber Systems and Codes4 Radix or Base Decimal (base 10) Binary (base 2) 0 1 Octal (base 8) Hexadecimal (base 16) A B C D E F The radix or base of the number system denotes the number of digits used in the system.

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CS Digital LogicNumber Systems and Codes5 DecimalBinaryOctalHexadecimal A B C D E F

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CS Digital LogicNumber Systems and Codes6 Positional Notation It is convenient to represent a number using positional notation. A positional notation is written as a sequence of digits with a radix point separating the integer and fractional part. where r is the radix, n is the number of digits of the integer part, and m is the number digits of the fractional part.

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CS Digital LogicNumber Systems and Codes7 Polynomial Notation A number can be explicitly represented in polynomial notation. where r p is a weighted position and p is the position of a digit.

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CS Digital LogicNumber Systems and Codes8 Examples In binary number system In octal number system In hexadecimal number system

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CS Digital LogicNumber Systems and Codes9 Arithmetic (101101) 2 +(11101) 2 : Addition: In binary number system,

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CS Digital LogicNumber Systems and Codes10 Addition (6254) 8 +(5173) 8 : In octal number system, (9F1B) 16 +(4A36) 16 : F1B 4A36 D951 In hexadecimal number system,

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CS Digital LogicNumber Systems and Codes11 Subtraction (101101) 2 -(11011) 2 : In binary number system,

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CS Digital LogicNumber Systems and Codes12 Subtraction In octal number system, In hexadecimal number system, (6254) 8 -(5173) 8 : (9F1B) 16 -(4A36) 16 : F1B 4A36 54E5

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CS Digital LogicNumber Systems and Codes13 Multiplication (1101) 2 (1001) 2 : In binary number system,

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CS Digital LogicNumber Systems and Codes14 Division ( ) 2 (1001) 2 : In binary number system,

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CS Digital LogicNumber Systems and Codes15 Base Conversions Convert ( ) 2 to base 8

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CS Digital LogicNumber Systems and Codes16 Base Conversion Convert ( ) 2 to base 10

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CS Digital LogicNumber Systems and Codes17 Base Conversion Convert ( ) 2 to base 16

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CS Digital LogicNumber Systems and Codes18 Base Conversion from base 8 Convert (372) 8 to base 2 Convert (372) 8 to base 10 Convert (372) 8 to base 16

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CS Digital LogicNumber Systems and Codes19 Base Conversion from base 16 Convert (9F2) 16 to base 2 Convert (9F2) 16 to base 8 Convert (9F2) 16 to base 10

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CS Digital LogicNumber Systems and Codes20 Binomial expansion (series substitution) To convert a number in base r to base p. –Represent the number in base p in binomial series. –Change the radix or base of each term to base p. –Simplify.

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CS Digital LogicNumber Systems and Codes21 Convert Base 10 to Base r Convert (174) 10 to base 8 Therefore (174) 10 = (256) LSB MSB 00

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CS Digital LogicNumber Systems and Codes22 Convert Base 10 to Base r Convert (0.275) 10 to base 8 Therefore (0.275) 10 = ( ) 8 8 2.200MSD 8 3.200LSD

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CS Digital LogicNumber Systems and Codes23 Convert Base 10 to Base r Convert ( ) 10 to base 2 Therefore ( ) 10 = ( ) 2 2 MSD 2 LSD

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CS Digital LogicNumber Systems and Codes24 Signed Number Representation There are 3 systems to represent signed numbers in binary number system: – Signed-magnitude –1's complement –2's complement

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CS Digital LogicNumber Systems and Codes25 Signed-magnitude system In signed-magnitude systems, the most significant bit represents the number's sign, while the remaining bits represent its absolute value as an unsigned binary magnitude. –If the sign bit is a 0, the number is positive. –If the sign bit is a 1, the number is negative.

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CS Digital LogicNumber Systems and Codes26 Signed-magnitude system

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CS Digital LogicNumber Systems and Codes27 1's Complement system A 1's complement system represents the positive numbers the same way as in the signed-magnitude system. The only difference is negative number representations. Let be N any positive integer number and be a negative 1's complement integer of N. If the number length is n bits, then

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CS Digital LogicNumber Systems and Codes28 Example of 1's Complement For example in a 4-bit system, 0101 represents +5 and 1010 represents 5

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CS Digital LogicNumber Systems and Codes29 1's Complement system

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CS Digital LogicNumber Systems and Codes30 2's Complement system A 2's complement system is similar to 1's complement system, except that there is only one representation for zero. Let be N any positive integer number and be a negative 2's complement integer of N. If the number length is n bits, then

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CS Digital LogicNumber Systems and Codes31 Example of 2's Complement For example in a 4-bit system, 0101 represents +5 and 1011 represents 5

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CS Digital LogicNumber Systems and Codes32 2's Complement system

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CS Digital LogicNumber Systems and Codes33 Addition and Subtraction in Signed and Magnitude

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CS Digital LogicNumber Systems and Codes34 Addition and Subtraction in 1’s Complement

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CS Digital LogicNumber Systems and Codes35 Addition and Subtraction in 2’s Complement

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CS Digital LogicNumber Systems and Codes36 Overflow Conditions Carry-in carry-out Carry-in = carry-out

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CS Digital LogicNumber Systems and Codes37 Addition and Subtraction in Hexadecimal System Addition Subtraction

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CS Digital LogicNumber Systems and Codes38 Codes Decimal codes Gray code Error detection code ASCII code

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CS Digital LogicNumber Systems and Codes39 Decimal codes Decimal DigitBCDExcess

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CS Digital LogicNumber Systems and Codes40 Gray Code Decimal EquivalentBinary CodeGray Code

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CS Digital LogicNumber Systems and Codes41 Error detection code Parity Bit (odd)Message

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CS Digital LogicNumber Systems and Codes42 Error detection code Parity Bit (even)Message

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CS Digital LogicNumber Systems and Codes43 ASCII Code ASCII: American Standard Code for Information Interchange. Used to represent characters and textual information Each character is represented with 1 byte –upper and lower case letters: a..z and A..Z –decimal digits -- 0,1,…,9 – punctuation characters -- ;,. : –special characters --$ / { –control characters -- carriage return (CR), line feed (LF), beep

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CS Digital LogicNumber Systems and Codes44 Assignment 1 Page 74 –1.1: Only A+B and A B (a), (c), (f), and (g) –1.2: Only A+B and A B (a), (c) –1.3: Only A+B and A B (a), (c) –1.4: (a), (c), (e) –1.5: (a), (c), (e) –1.6: (a), (e) –1.7: (a), (b) –1.8: (a), (b) –1.10: (a), (c) –1.11: (a), (c) –1.12: (a), (c) –1.13: (a), (b)

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