Digital Logic Designs

LECTURE # 9

BINARY ARITHMETIC AND ARITHMETIC CIRCUITS

�Introduction to Binary Arithmetic

Binary arithmetic is the branch of arithmetic that deals with binary numbers, using only two digits: 0 and 1. It is the foundation of all computer operations and digital systems. Understanding binary arithmetic is essential for any study or work involving digital logic design.

Binary Number System�

Binary Digits: The two digits in the binary system are known as bits.

Base-2: The binary system is base-2, meaning each position represents a power of 2.

- Example: (1011)2 = 1 times 2^3 + 0 times 2^2 + 1 times 2^1 + 1 times 2^0 = 8 + 0 + 2 + 1 = (11)10

�

In Digital Logic Design (DLD), arithmetic operations are fundamental for performing calculations in computer systems. Binary arithmetic forms the cornerstone of these operations since digital systems utilize binary digits (0s and 1s) for processing data.

Key Topics�

1. Overview of Number Systems

2. Principles of Binary Arithmetic

3. Design of Arithmetic Circuits

• Half Adders and Full Adders
• Ripple Carry Adders
• Carry Lookahead Adders

4. Binary Subtraction and Its Circuits

5. Multiplication and Division Circuits

6. Applications in Digital Systems

�Overview of Number Systems

Understanding number systems is critical for arithmetic in digital form.

• Decimal (Base 10): Uses digits 0-9. Common in everyday calculations.
• Binary (Base 2): Uses digits 0 and 1. Fundamental to digital electronics due to its compatibility with two-state devices (transistors).

 Conversion between Decimal and Binary is essential, allowing us to perform arithmetic in binary while interpreting results in decimal.

�Principles of Binary Arithmetic

Binary Addition

 Binary addition follows straightforward rules akin to decimal but with only 0 and 1:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10 (the output is 0 with a carry of 1)

 Binary Subtraction

 Binary subtraction involves borrowing:

Example: To subtract 1 from 0, we borrow 1 from the next column, converting the 0 to 10 (which represents 2 in binary).

Binary Multiplication

Binary multiplication mimics decimal, using AND operations:

1 * 1 = 1,

1 * 0 = 0,

0 * 0 = 0

Binary Division

Binary division is conceptually similar to decimal long division, often needing a process of shift and subtract.

�Design of Arithmetic Circuits

Understanding how to implement binary arithmetic operations through circuits is a key aspect of DLD.

Half Adder

A half adder adds two single binary digits.

Inputs: A, B

Outputs: Sum (S), Carry (C)

Logic Equations:

• S = A ⊕ B (XOR gate)
• C = A . B (AND gate)

Full Adder�

A full adder adds three bits: two significant bits and a carry-in bit.

Inputs: A, B, Cin

Outputs: Sum (S), Cout (Carry Out)

Logic Equations:

S= A ⊕ B ⊕ Cin

Cout = (A ⋅ B) + (Cin ⋅ (A⊕B)

​

�

Applications in Digital Systems

Arithmetic and binary circuits are vital components of:

​

ALUs (Arithmetic Logic Units): These perform both arithmetic operations and logical operations within CPUs and computer architecture.

​

Digital Signal Processors (DSPs): Used for complex arithmetic processes in signals.

​

Embedded Systems: Where calculators, controllers, and devices perform essential computations

The End