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Computer Science Notes: Information
Representation
Chapter 1: General Introduction
Introduction to information representation and base numbering systems.
Chapter 2: Numbering Systems
1. Introduction
Every numbering system is based on a base (b) where:
Base 10 (Decimal): Uses 0-9 digits.
Base 2 (Binary): Uses 0-1 digits.
Each number in base (b) is made by combining digits from the set of symbols allowed in that
base.
Note: Always specify the base to avoid confusion (101) 2
, except for base 10 which
is the useful base for humans.
2. Representation of a Positive Integer in Base (B)
Principle of Euclidean Division: X = q.B + r
Divide the number by the base repeatedly.
Record the remainders (until q < B).
Write remainders from last to first.
Example: (19) 10
Base 16 Digits (Hexadecimal)
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
Basic Operations in Binary System
Binary Addition
Follows 4 simple rules:
Operation Carry Result
Ex: 1011 + 1101 = 11000
(When 1 + 1, we write 0 and carry 1 to the next column, just like 9 +
1 in decimal).
Binary Subtraction
Also has 4 rules:
Operation Carry (Borrow) Result
Ex: 1001 - 1101 = 0100 (borrowing 1 turns the 1 to 0 and adds 2 in
binary to the current column).
Chapter 03: Information Representation
This chapter explains how diverse information (numbers, text, images, sound) is represented
in a computer using binary code (0s and 1s).
1. Binary Coding
Computers use bits (binary digits) to represent information. n bits can represent 2
n unique
pieces of information.
Explanation:
1 bit → 2 informations (
2 bits → 4 informations (
3 bits → 8 informations (
1.2 Gray Code
The procedure:
G
n
= B
n
G
i
= B
i+
⊕ B
i
(for i = n-1 to 0)
⊕ is called XOR: a ⊕ b = 1 if a ≠ b (different), else 0.
Example: (1011) 2
to Gray Code
n=4 digits (B 3
B
B
B
G
= B
G
= B
⊕ B
G
= B
⊕ B
G
= B
⊕ B
Result: Gray Code = 1110
2.3 BCD (Binary-Coded Decimal) Code
In Base 10, digits 0-9 are replaced by the BCD code (4-bit binary equivalent).
Example: X = 687
BCD code: 0110 1000 0111
2.4 Excess-3 (XS-3) Code
Used in digital systems. Represents each decimal digit by adding 3 (0011) 2
to its BCD code.
Principle: Xs-3 = BCD code + (0011)
Example: X = 9
BCD(9) = 1001
XS-3 = (1001) + (0011) = 1100
4. Number Representation
4.1 Integers
4.3.1 Representation of Negative Integers in SAV (Sign and Absolute Value)
The Most Significant Bit (MSB) is the sign:
0 indicates a positive number
1 indicates a negative number
Range for n bits: -(
n-
n-
4.3.2 Representation of Negative Integers in 1's Complement (CP1)
Principle: A negative number is the bitwise logical NOT of its positive counterpart.
SAV is positive: Range same as SAV.
SAV is negative: Change 0 by 1 and vice versa (flip bits), keeping the sign bit.
Example: +30 = (11110) 2
in 5 bits + sign bit
SAV (+30) = 0 | 11110
CP1 (+30) = 0 | 11110
SAV (-30) = 1 | 11110
CP1 (-30) = 1 | 00001 (flip number bits, keep sign)
