Digital Logic Fundamentals – Number Systems & Codes

DIGITAL
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Part – I (N
um
ber System
&
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NUMBER SYSTEMS
 A number system is a way of representing numbers using symbols (digits)
and a base (radix).
 Types of Number Systems
System Base Digits Used Example
Binary 2 0, 1 (1011) = 11
₂
Octal 8 0 – 7 (745) = 485
₈
Decimal 10 0 – 9 (253) = 253
₁₀
Hex 16 0 – 9, A – F (2F) = 47
₁₆

REASON FOR USING BINARY IN DIGITAL COMPUTERS
 Two-State Devices: Digital components like transistors, switches, and magnetic storage naturally operate
in only two stable states (e.g., ON/OFF, High/Low, Saturated/Cutoff).
 Noise Immunity: Two distinct states (0 and 1) make signals less prone to error due to noise or signal
degradation.
 Simplicity in Circuit Design: Using only two states simplifies logic circuit design, enabling the use of
logic gates (AND, OR, NOT).
 Reliable Storage: Data storage (e.g., in magnetic tape, punched cards, memory cells) easily maps to two
states (hole/no hole, magnetized/non-magnetized).
 Universal Representation: Any numeric, character, or symbolic information can be encoded into binary
format for processing and storage.
CONCLUSION
 Binary is chosen because it aligns perfectly with the physical behavior of electronic components,
ensuring simplicity, reliability, and cost-effectiveness in computing systems.

BASE CONVERSIONS
A. Decimal → Any Base
Divide repeatedly by base, note remainders, reverse order.
Example: 43 → Binary → (101011)₂

ANY BASE DECIMAL
→
 Multiply each digit by positional weight.
 Example: (1101)₂
= 1×2³ + 1×2² + 0×2¹ + 1×2 = 13
⁰

BINARY OCTAL
↔
 Group bits in 3 from right.
 Example: (101011) = (53)
₂ ₈

BINARY HEXADECIMAL
↔
 Group bits in 4 from right.
 Example: (10101111) = (AF)
₂ ₁₆

HEXADECIMAL TO DECIMAL CONVERSION

BINARY CODES
A. BCD (Binary Coded Decimal)
Each decimal digit represented using 4 bits.
Example: 59 → (0101 1001)
B. ASCII
American Standard Code for Information Interchange
7-bit code representing English letters and symbols.
Example: 'A' → 65 → (1000001)₂
C. EBCDIC
Extended Binary Coded Decimal Interchange Code (8-bit, used in IBM mainframes).
D. Unicode
Supports multiple languages (8 to 32 bits).
E. Gray Code
Successive values differ in only one bit (reduces errors).
Decimal Binary Gray
0 000 000
1 001 001
2 010 011
3 011 010
Example Table

CODE CONVERSION SHORTCUTS
A. Binary → Gray Code
 First bit same, next bits = XOR of consecutive binary bits.
 Example: Binary 1011 → Gray 1110
B. Gray → Binary
 First bit same, next bits = XOR(previous binary bit, current Gray bit).
 Example: Gray 1110 → Binary 1011
C. Binary ↔ Hex
 Group 4 bits for hex conversion.
D. Binary ↔ Octal
 Group 3 bits for octal conversion.

COMPLEMENTS
 1’s Complement: Invert all bits.
 Example: 1010 → 0101
 2’s Complement: 1’s complement + 1.
 Example: 1010 → 0101+1 = 0110
 Use: Subtraction: A – B = A + (2’s complement of B)

QUESTIONS
Short Answer Questions
1. What are the different types of number systems?
2. What is the advantage of using binary number system?
3. How do you perform addition and subtraction using 1’s and
2’s complement? Illustrate with examples.
4. Convert the following numbers to their binary equivalents:
(a) 37 (b) 14 (c) 167 (d) 72.45
5. Convert the following decimal numbers to equivalent binary
numbers:
(a) 43 (b) 64 (c) 4096 (d) 0.375
Long Answer Questions
1. Convert the following from binary to octal:
(a) 101101 (b) 101101110 (c) 10110111 (d) 11010.011
2. Convert the following from decimal to octal and then to binary:
(a) 59 (b) 0.58 (c) 64.2 (d) 199.3
3. Convert each octal number to binary:
(a) 15 (b) 24 (c) 167 (d) 234
₈ ₈ ₈ ₈
(e) 173 (f) 157 (g) 4653 (h) 1723 (i) 2645
₈ ₈ ₈ ₈ ₈
4. Convert each hexadecimal number to binary:
(a) 49 (b) 324 (c) 649 (d) ABC
₁₆ ₁₆ ₁₆ ₁₆
5. Convert each binary number to hexadecimal:
(a) 1100110 (b) 11101111 (c) 1011110101.0111
₂ ₂ ₂
6. Convert each decimal number to hexadecimal:
(a) 16 (b) 19 (c) 35 (d) 439
₁₀ ₁₀ ₁₀ ₁₀
7. Convert the following binary numbers to octal and then to
hexadecimal:
(a) 101100110011 (b) 1011101.1011
8. Convert each hexadecimal number to decimal:
(a) 49 (b) 632 (c) 54 (d) AB0
₁₆ ₁₆ ₁₆ ₁₆

Digital Logic Fundamentals – Number Systems & Codes

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