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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

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numbers systems and codes
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
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
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
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
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
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
Binary Number Systems
  • Unsigned Binary Code
  • Signed Binary Codes
    • 2’s Complement System
    • BCD Code
    • Excess Codes
  • Floating-Point System
unsigned binary code
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
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
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
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
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
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
Sign &Magnitude (4 bits)

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

sign magnitude
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
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
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
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
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
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
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
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
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
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
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
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
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
2’s Complement Code (4 bits)

Range of values with n bits:

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

2 s complement arithmetic
2’s Complement Arithmetic
  • Addition
  • Subtraction
  • Overflow
hexadecimal notation
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
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
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