1 / 34

Week 2: Number Systems and Codes

Week 2: Number Systems and Codes A. Berrached 1 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

salena
Télécharger la présentation

Week 2: Number Systems and Codes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related