Week 2:
Download
1 / 34

Numbers Systems and Codes - PowerPoint PPT Presentation


  • 907 Views
  • Updated On :

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Numbers Systems and Codes' - salena


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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Slide1 l.jpg

Week 2:Number Systems and Codes

A. Berrached

1


Numbers systems and codes l.jpg
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)


Positional number systems l.jpg
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


Positional number systems4 l.jpg
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


Binary number system l.jpg
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)


Base conversion l.jpg
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


Base conversion7 l.jpg
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


Binary number systems l.jpg
Binary Number Systems

  • Unsigned Binary Code

  • Signed Binary Codes

    • 2’s Complement System

    • BCD Code

    • Excess Codes

  • Floating-Point System


Unsigned binary code l.jpg
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)


Unsigned binary code10 l.jpg
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.



Unsigned binary code12 l.jpg
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)


Unsigned binary code arithmetic logic operations l.jpg
Unsigned Binary Code: Arithmetic & Logic Operations

  • Arithmetic Operations:

    • Addition

    • Subtraction

    • Multiplication

    • Division

  • Logic Operations

    • AND CONJUNCTION

    • OR DISJUNCTION

    • NOT NEGATION

    • XOR EXCLUSIVE OR


Signed binary codes l.jpg
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


Sign magnitude code l.jpg
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


Sign magnitude 4 bits l.jpg
Sign &Magnitude (4 bits)

What is the range of values that can be represented in S&M code with n bits?


Sign magnitude l.jpg
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.


Sign magnitude18 l.jpg
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.


1 s complement code l.jpg
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


1 s complement code20 l.jpg
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


1 s complement code21 l.jpg
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


1 s complement code 4 bits l.jpg
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]


2 s complement code l.jpg
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


2 s complement code24 l.jpg
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.


2 s complement code25 l.jpg
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


2 s complement code26 l.jpg
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


2 s complement code27 l.jpg
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


Conversion from 2 s comp code to decimal code l.jpg
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


Conversion from 2 s comp code to decimal code29 l.jpg
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.


2 s complement code 4 bits l.jpg
2’s Complement Code (4 bits)

Range of values with n bits:

[ -2 (n-1), 2(n-1) -1]


2 s complement arithmetic l.jpg
2’s Complement Arithmetic

  • Addition

  • Subtraction

  • Overflow


Hexadecimal notation l.jpg
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


Hexadecimal notation33 l.jpg
Hexadecimal Notation

  • Examples:

    • Binary: 1111 0110

    • Hex: F6

    • Binary: 1001 1101 0000 1010

    • Hex: 9D0A

    • Hex: F6E7

    • Binary 1111 0110 1110 0111


Binary coded decimal codes l.jpg
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


  • ad