Basic of digital logic- Number system,Codes,Arithmetic Operations,Signed Number Representation.

May 21, 2025May 22, 2025 Lui Kreyszig 0 Comments

Contents

1 💡 Basics of Digital Logic

Basics of Digital Logic

Digital logic forms the foundation of computer systems and electronic devices. It is based on binary operations and logical computations using number systems, codes, arithmetic operations, and signed number representation. Let’s break down these concepts:

1. Number Systems in Digital Logic

A number system defines how numbers are represented and used in digital circuits. The four main number systems are:

Number System	Base (Radix)	Digits Used	Example
Binary (Base 2)	2	0, 1	1011₂
Octal (Base 8)	8	0-7	57₈
Decimal (Base 10)	10	0-9	45₁₀
Hexadecimal (Base 16)	16	0-9, A-F	1A3₁₆

Conversions Between Number Systems:

Binary to Decimal: Multiply each binary digit by 2^position and sum them.
Decimal to Binary: Repeatedly divide by 2 and record the remainders.
Binary to Hexadecimal: Group 4 bits together and convert them into a hexadecimal digit.
Binary to Octal: Group 3 bits together and convert them into an octal digit.

Example: Convert 1011₂ to Decimal

$2^3 + 0 × 2^2 + 1 × 2^1 + 1 × 2^0 = 8 + 0 + 2 + 1 = 11₁₀$

2. Digital Codes in Logic Circuits

Binary codes are used to represent data in digital systems. Some important coding schemes includea) BCD (Binary-Coded Decimal)

Each decimal digit (0-9) is represented by 4-bit binary.
Example: 9₁₀ = 1001₂ in BCD.

b) Gray Code

A binary code where two successive values differ by only one bit.
Used to reduce errors in digital circuits (e.g., rotary encoders).
Example: Binary 110 → Gray 101.

c) ASCII (American Standard Code for Information Interchange)

7-bit or 8-bit codes used to represent characters.
Example: A = 65₁₀ = 1000001₂.

d) Excess-3 Code

A self-complementary code used in error detection.
Example: Decimal 5 → Binary 0101 → Excess-3 Code 1000.

3. Arithmetic Operations in Digital Logic

a) Binary Addition

Similar to decimal addition but uses only 0 and 1.

A	B	Sum (A+B)	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Example:

b) Binary Subtraction (Using 2’s Complement)

To subtract A – B, take 2’s complement of B and add it to A.
2’s Complement: Invert the bits and add 1.

Example:
Find 5 – 3 in Binary:

5 → 0101
3 → 0011
2’s Complement of 3 → 1101
Add:

c) Binary Multiplication

Follows the same rules as decimal multiplication but only involves 0s and 1s.

4. Signed Number Representation

In digital systems, signed numbers are used to represent both positive and negative values.

a) Sign-Magnitude Representation

The MSB (Most Significant Bit) is used as the sign bit (0 = Positive, 1 = Negative).
Example: +5 = 0101, -5 = 1101 (in 4-bit representation).

b) 1’s Complement

Invert all bits to get the negative value.
Example: +5 (0101) → -5 (1010).

c) 2’s Complement (Most Common)

Invert the bits and add 1.
Example: +5 (0101) → -5 (1011).
Used in most computers for easy arithmetic operations.

Why Use 2’s Complement?
No separate subtraction circuit needed—addition handles both positive and negative numbers.
Only one representation for 0 (unlike 1’s complement).

Applications of Digital Logic

Microprocessors & Computers – Use binary logic for computations.
Embedded Systems – Digital circuits in automobiles, IoT devices, and robotics.
Digital Communication – Used in networking, encoding, and data transmission.
Cryptography & Security – Binary operations in encryption algorithms.

Conclusion

Digital logic is the foundation of modern computing. Understanding number systems, coding techniques, arithmetic operations, and signed number representation is crucial for designing efficient circuits.

Would you like additional practice problems or real-world applications on any topic?

Here is a clear and concise explanation of the Basics of Digital Logic, covering the key topics: Number Systems, Codes, Arithmetic Operations, and Signed Number Representation — suitable for students, beginners, or revision material.

💡 Basics of Digital Logic

Digital logic is the foundation of computer systems, electronics, and digital devices. It uses binary signals (0 and 1) to represent and process data. Let’s explore the core building blocks:

🔢 1. Number Systems

a. Binary (Base 2)

Digits: 0, 1
Used in computers because hardware circuits understand two states: ON (1) and OFF (0).

b. Decimal (Base 10)

Digits: 0 to 9
Common human number system.

c. Octal (Base 8) and Hexadecimal (Base 16)

Octal digits: 0 to 7
Hex digits: 0 to 9 and A to F
Used to simplify binary representation.

✅ Conversions:

Binary → Decimal: Multiply each bit by 2ⁿ and sum.
Decimal → Binary: Divide by 2 repeatedly and collect remainders.

🧾 2. Codes in Digital Logic

a. BCD (Binary Coded Decimal)

Each decimal digit is represented by 4-bit binary (e.g., 9 = 1001).
Used in calculators and digital clocks.

b. Gray Code

Only one bit changes at a time.
Used in rotary encoders to avoid errors.

c. ASCII (American Standard Code for Information Interchange)

Represents characters using 7 or 8 bits (e.g., A = 65 = 1000001).

d. Excess-3 and Parity Codes

Error detection, data integrity in communication systems.

➕ 3. Arithmetic Operations

a. Binary Addition Rules

b. Binary Subtraction Rules

c. Binary Multiplication and Division

Follows same logic as decimal but with 0s and 1s.

➖ 4. Signed Number Representation

Digital systems must handle positive and negative numbers.

a. Sign-Magnitude

MSB (Most Significant Bit) is sign:
- 0 = positive
- 1 = negative
Value stored in remaining bits.

b. 1’s Complement

Negative number = Invert all bits of the positive version.

c. 2’s Complement (most common)

Negative = 1’s complement + 1
Makes subtraction easier in binary.
Range:
- For 4-bit: –8 to +7
- For 8-bit: –128 to +127

📌 Example: 2’s Complement (4-bit)

+5 = 0101
–5 = 1011 (invert 0101 → 1010, add 1 → 1011)

🧠 Quick Summary Table

Concept	Key Idea
Number System	Binary, Octal, Decimal, Hex
Codes	BCD, Gray, ASCII
Arithmetic	Binary addition/subtraction/multiplication
Signed Representation	Sign-magnitude, 1’s, and 2’s complement

🎯 Conclusion

Understanding the basics of digital logic is essential for:

Digital electronics
Computer architecture
Embedded systems
Programming logic

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