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Week 2: Number Systems and Codes

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  1. Week 2:Number Systems and Codes A. Berrached 1

  2. Numbers Systems and Codes How is information represented at the gate-level? • Only two symbols: 0 and 1 • Information is coded using string combinations of 0s and 1s. • A code is a standard set of rules for representing and interpreting information • Information: • Numerical: unsigned and signed numbers • Binary Coded Decimal codes • Characters (textual information)

  3. Positional Number Systems Positional Notation (polynomial notation) • a number is represented with a string (sequence) of digits • the position of each digit in the sequence carries a weight Example: decimal system • base 10 => 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • 4175.86 = 5*100 + 7* 101 + 1* 102 +4* 103 + 8* 10-1 +6* 10-2

  4. Positional Number Systems • In base 10: If a number N =dn-1dn-2 ….d1d0 .d-1… d-m then N = dn-1*10n-1 + dn-2*10n-2 +….+ d1*101 + d0*100 + d-1*10-1 + ... + d-m*10-m • In any base r: If a number (N)r = dn-1dn-2 ….d1d0 .d-1… d-m then (N)r = dn-1*rn-1 + dn-2*rn-2 +….+ d1*r1 + d0*r0 + d-1*r-1 + ...+ d-m*r-m

  5. Binary Number System N = 1101101.1012 N = (1*26) + (1*25) + (0*24) + (1*23) + (1*22) + (0*21) + (1*20) + (0*2-1) + (1*2-2) + (1*2-3) N = 64 + 32 + 0 + 8 + 4 + 0 + 1 + 1/2 + 0/4 + 1/8 in base 10 N = 109.625 in base 10 => N = (109.625)10 To convert a number N from base b to base 10use method as shown above (series substitution method)

  6. Base Conversion • From base b to base 10 • use series substitution method • do arithmetic in base 10. • From base 10 to base b • Integer part • use successive divisions by b, until quotient is 0 • collect remainders: first remainder is the least significant digit • fraction part • use successive multiplication by b • collect integer part of each product

  7. Base Conversion • Conversion from base A to base B • convert number from base A to base 10 • covert resulting number from base 10 to base B. • Special cases • Conversion frombinary to octal • Conversion from octal to binary • Conversion frombinary to hexadecimal • Conversion from hexadecimal to binary

  8. Binary Number Systems • Unsigned Binary Code • Signed Binary Codes • 2’s Complement System • BCD Code • Excess Codes • Floating-Point System

  9. Unsigned Binary Code • Given a number N in Unsigned Binary code, find the value of N in the decimal system • Use series substitution method • Given a number N in the decimal system, find the value of N in the Unsigned Binary Code. • Use successive division method (for integer part) • Use successive multiplication method (for fraction part)

  10. Unsigned Binary Code • Example 1: Represent (26)10 in Unsigned Binary Code 2610 = 11010 • Example 2: Represent (26)10 in Unsigned Binary Code using 8 bits. 2610 = 00011010 • Example 3:Represent (26)10 in Unsigned Binary Code using 4 bits. Can’t do. Not enough bits.

  11. Unsigned Binary Code ( 4 bits)

  12. Unsigned Binary Code • The Unsigned Binary Code is used to represent positive integer numbers. • What is the range of values that can be represented with n bits in the Unsigned Binary Code? [0, 2n-1] • How many bits are required to represent a given number N? number of bits = smallest integer greater than or equal to log(N)

  13. Unsigned Binary Code: Arithmetic & Logic Operations • Arithmetic Operations: • Addition • Subtraction • Multiplication • Division • Logic Operations • AND CONJUNCTION • OR DISJUNCTION • NOT NEGATION • XOR EXCLUSIVE OR

  14. Signed Binary Codes These are codes used to represent positive and negative numbers. • Sign and Magnitude Code • 1’s Complement Code • 2’s Complement Code

  15. Sign & Magnitude Code • The leftmost bit is the sign bit • 0 for positive numbers • 1 for negative numbers • The remaining bits represent the magnitude of the number Example: Sign & Mag. CodeDecimal 01101 +13 11101 -13 00101 +5 10101 -5

  16. Sign &Magnitude (4 bits) What is the range of values that can be represented in S&M code with n bits?

  17. Sign&Magnitude • Example 1: Represent (26)10 in Sign & Magnitude Code. 2610 = 011010 • Example 2: Represent (26)10 in Sign & Magnitude Code using 8 bits 2610 = 0001 1010 • Example 3: Represent (26)10 in Sign & Magnitude Code using 5 bits. • Need at least 6 bits.

  18. Sign&Magnitude • Example 1: Represent (-26)10 in Sign & Magnitude Code. • 26 = 11010 • -2610 = 111010 • Example 2: Represent (-26)10 in Sign & Magnitude Code using 8 bits • 26 = 00011010 • - 2610 = 10011010 • Example 3: Represent (-26)10 in Sign & Magnitude Code using 5 bits. • Need at least 6 bits.

  19. 1’s Complement Code • Positive numbers: • same as in unsigned binary code • pad a 0 at the leftmost bit position • Negative numbers: 1. Represent the magnitude of the number in unsigned binary system 2. pad a 0 at the leftmost bit position 3. complement every bit

  20. 1’s Complement Code • Example: represent 2610 in 1’s complement code • 2610 = 11010 • Pad a 0: = 011010 • Example: Represent (-26)10 in 1’s complement code. 1. 26= 11010 2. Pad a 0: 011010 3. Complement: 100101(-26)10 = (100101)1’s comp

  21. 1’s Complement Code • Example: Represent (-26)10 in 1’s comp. code using 8 bits 1. Represent magnitude in unsigned binary using 8 bits 26 = 0001 1010 2. Complement every bit 11100101 -2610 = (1110 0101) 1’s comp

  22. 1’s Complement Code (4 bits) What is the range of values that can be represented in S&M code with n bits? [ -(2 (n-1) -1) , 2(n-1) -1]

  23. 2’s Complement Code • This is the code commonly used to represent integer numbers. • Positive Numbers: • same as in unsigned binary code • pad with a 0 leftmost bit position • Negative Numbers 1. represent magnitude in unsigned binary code 2. pad leftmost positions with 0s 3. complement every bit 4. add 1

  24. 2’s Complement Code • Example 1: Represent 26 in 2’s complement code. 26 = 011010 • Example 2: Represent 26 in 2’s complement code using 8 bits 26 = 00011010 • Example 3: Represent 26 in 2’s complement code using 5 bits • Need at least 6 bits.

  25. 2’s Complement Code • Example 3: Represent - 26 in 2’s comp. Code 1. +26 = 11010 2. Pad with a 0: 011010 3. Complement: 100101 4. Add 1: + 1 --------------- 100110

  26. 2’s Complement Code • Example 4: Represent - 26 in 2’s comp. Code using 8 bits 1. +26 = 11010 2. Pad 0s: 00011010 3. Complement: 11100101 4. Add 1: + 1 --------------- 11100110

  27. 2’s Complement Code • More example: represent 65 in 2’s comp. Code. • 65 = (0100 0001)2’s comp • Represent - 65 in 2’s comp • 65 = 0100 0001 • -65 = 1011 1111

  28. Conversion from 2’s comp code to decimal code • How to convert a number in 2’s Comp. Code into the decimal code. There 2 cases: Case 1: If leftmost bit of the number is 0 => number is positive => conversion is the same as in unsigned binary code

  29. Conversion from 2’s comp code to decimal code Case 2: If leftmost bit is 1 => the number is negative step1: complement every bit step2: add 1 step3: convert result to decimal code using same method as in unsigned binary code. Answer = the negative of the result of step 3.

  30. 2’s Complement Code (4 bits) Range of values with n bits: [ -2 (n-1), 2(n-1) -1]

  31. 2’s Complement Arithmetic • Addition • Subtraction • Overflow

  32. Hexadecimal Notation • Hexadecimal system: base 16 • There are 16 digits: • 0 1 2 3 4 5 6 7 8 9 A B C D E F • Each Hex digit represents a group of 4 bits (i.e. half of a byte) 0000 thru 1111 • Generally used as shorthand notation for binary numbers => easier to read • Binary: 0101 1010 1001 1110 • Decimal: 5 10 9 14 • Hex: 5 A 9 E

  33. Hexadecimal Notation • Examples: • Binary: 1111 0110 • Hex: F6 • Binary: 1001 1101 0000 1010 • Hex: 9D0A • Hex: F6E7 • Binary 1111 0110 1110 0111

  34. Binary Coded Decimal Codes • A number is a sequence of decimal digits • Each digit is represented with a 4-bit number. • BCD codes allow easy conversion to/from decimal system Natural Binary Coded Decimal Code (the BCD code) • each decimal digit is represented with 4 bits in base 2 e.g. 0 = 0000 3 = 0011 5= 0101 9 = 1001 • N = 856.3710 • N = 1000 0101 0110 . 0011 0111BCD