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Week 2: Number Systems and CodesPowerPoint Presentation

Week 2: Number Systems and Codes

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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

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 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 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

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

- 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

- Integer part

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

- Unsigned Binary Code
- Signed Binary Codes
- 2’s Complement System
- BCD Code
- Excess Codes

- Floating-Point System

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 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 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

- Arithmetic Operations:
- Addition
- Subtraction
- Multiplication
- Division

- Logic Operations
- AND CONJUNCTION
- OR DISJUNCTION
- NOT NEGATION
- XOR EXCLUSIVE OR

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

- 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)

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

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&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

- 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 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 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)

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

- 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 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 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 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 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

- 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 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 Arithmetic

- Addition
- Subtraction
- Overflow

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 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 each decimal digit is represented with 4 bits in base 2 N = 856.3710 N = 1000 0101 0110 . 0011 0111BCD

- 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)

- BCD codes allow easy conversion to/from decimal system

e.g. 0 = 0000 3 = 0011 5= 0101 9 = 1001

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