Binary Number System

Binary Number System

Definition

• A binary number system is a base-2 numeral system that uses two symbols: 0 and 1.

• It is the foundation of all binary code used in computer systems and digital electronics.

Characteristics

• Base: 2

• Digits: 0, 1

• Place Value: Each digit represents a power of 2, starting from the rightmost digit (2^0).

Conversion

Binary to Decimal

• To convert binary to decimal, multiply each bit by 2 raised to the power of its position (starting from 0).

  Example:Binary: 1011Decimal: (1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11)

Decimal to Binary

• To convert decimal to binary, divide the number by 2 and record the remainder. Repeat with the quotient until it reaches 0. The binary number is the remainders read in reverse order.

  Example:Decimal: 11Steps:

  • 11 ÷ 2 = 5, remainder 1

  • 5 ÷ 2 = 2, remainder 1

  • 2 ÷ 2 = 1, remainder 0

  • 1 ÷ 2 = 0, remainder 1Binary: 1011

Applications

• Computing: All data in computers is represented in binary (e.g., text, images).

• Digital Electronics: Used in circuits, logic gates, and microcontrollers.

• Networking: IP addresses and data transmission protocols often utilize binary.

Binary Arithmetic

• Addition: Similar to decimal, but carries occur when the sum exceeds 1.

  • Example:

      1  (carry)
      1011
    + 1101
    ------
     11000

• Subtraction: Uses borrowing similar to decimal subtraction.

• Multiplication: Similar to decimal, but involves shifting and adding.

• Division: Similar to decimal long division.

Advantages

• Simplicity: Only two states (on/off