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

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

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:

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

1×23+0×22+1×21+1×20=8+0+2+1=11101 × 2^3 + 0 × 2^2 + 1 × 2^1 + 1 × 2^0 = 8 + 0 + 2 + 1 = 11₁₀1×23+0×22+1×21+1×20=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.

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.

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