Number System: Binary System Quiz

Number System: Binary System

• Definition: The binary system is a base-2 numeral system that uses only two symbols: 0 and 1.

• Base:

  • The binary system is called base-2 because it has two possible values for each digit.

• Position Value:

  • Each position in a binary number represents a power of 2, starting from the rightmost position (2^0).

• Binary to Decimal Conversion:

  • To convert a binary number to decimal, sum the products of each digit and its corresponding power of 2.
  • Example: Binary 1011 = (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0) = 8 + 0 + 2 + 1 = 11 in decimal.

• Decimal to Binary Conversion:

  • Divide the decimal number by 2, record the remainder, and repeat with the quotient until it reaches 0. The binary representation is the remainders read in reverse order.
  • Example: Decimal 13
    • 13 ÷ 2 = 6 (remainder 1)
    • 6 ÷ 2 = 3 (remainder 0)
    • 3 ÷ 2 = 1 (remainder 1)
    • 1 ÷ 2 = 0 (remainder 1)
    • Binary = 1101

• Binary Arithmetic:

  • Addition: Use the same rules as decimal addition, carrying over when the sum exceeds 1.
    • Example:
      • 0 + 0 = 0
      • 0 + 1 = 1
      • 1 + 0 = 1
      • 1 + 1 = 0 (carry 1)
      • 1 + 1 + 1 = 1 (carry 1)
  • Subtraction: Similar to decimal, but borrow from the next position when needed.

• Applications:

  • Used in computing and digital electronics.
  • Represents data in computers (bits).
  • Forms the basis for machine code and programming languages.

• Key Terms:

  • Bit: A binary digit (0 or 1).
  • Byte: A group of 8 bits.
  • Nibble: A group of 4 bits.

Binary System Overview

• The binary system is a base-2 numeral system featuring only two digits: 0 and 1.
• It is called base-2 due to its two possible values for each digit.

Position Values

• Each digit's position in a binary number signifies a power of 2, starting with 2^0 on the right.

Binary to Decimal Conversion

• To convert binary to decimal, multiply each digit by its corresponding power of 2 and sum the results.
• For instance, the binary number 1011 converts to decimal as follows:
  • 1 × 2^3 + 0 × 2^2 + 1 × 2^1 + 1 × 2^0 = 8 + 0 + 2 + 1 = 11.

Decimal to Binary Conversion

• Convert a decimal number to binary by dividing by 2, recording remainders, and continuing until the quotient reaches 0.
• Read the recorded remainders in reverse order for the binary representation.
• Example: Decimal 13 converts to binary as follows:
  • 13 ÷ 2 = 6 (remainder 1)
  • 6 ÷ 2 = 3 (remainder 0)
  • 3 ÷ 2 = 1 (remainder 1)
  • 1 ÷ 2 = 0 (remainder 1)
  • Resulting binary = 1101.

Binary Arithmetic

• Addition: Follows rules similar to decimal; carry over occurs when sums exceed 1.
  • Examples:
    • 0 + 0 = 0
    • 1 + 1 = 0 with a carry of 1.
• Subtraction: Mirrors decimal subtraction, with borrowing from the next position when needed.

Applications of Binary System

• The binary system is integral to computing and digital electronics.
• Data representation in computers is done using bits.
• Forms the fundamental basis for machine code and programming languages.

Key Terms

• Bit: The smallest unit in the binary system, representing 0 or 1.
• Byte: Comprised of 8 bits; commonly used as a standard unit of data storage.
• Nibble: A collection of 4 bits, often used in digital systems.