Binary to Octal and Number Operations

Conversion from Binary to Octal

Overview of Octal System

  • The octal number system includes digits from 0 to 7.
  • In binary, seven is represented as 111.

Converting Binary to Octal

  1. Identify the Binary Number: For example, consider the binary number 101101.010110.
  2. Grouping of Bits: Start grouping the binary digits into groups of three beginning from the radix (decimal point) to the left and right.
    • Right of the radix: 010 | 110
    • Left of the radix: 101 | 101
  3. Conversion to Decimal: Convert each group of three to its decimal equivalent:
    • Left Groups: 101 (binary) = 5 (decimal), 101 (binary) = 5 (decimal) → 55 (octal).
    • Right Groups: 010 (binary) = 2 (decimal), 110 (binary) = 6 (decimal) → 2.6 (octal)
  4. Final Result: The number in octal is 55.26.

Converting Hexadecimal to Binary and Vice-Versa

  1. Understanding base conversions involving hexadecimal (hex) and binary is essential.
  2. Each hex digit corresponds to four binary digits:
    • Example: Hex 8 = Binary 1000, Hex 9 = Binary 1001.
  3. **Converting from Decimal to Hexadecimal: **
    • Starting from decimal number 89:
    • Break down: 64 (16^1) + 8 (16^0) + 0 (16^-1) + 9 (16^-2) → This leads to hex 59.

Binary Addition

Basic Addition Rules

  1. In binary addition:

    • 1 + 1 = 10 (write 0, carry 1).
    • 0 + 0 = 0.
    • 1 + 0 = 1.

  2. To illustrate:

    • Add 1 and 1: yields 0, carry 1.
    • Adding a carry can complicate the addition: 111 + 101:
      • Process: ` Carry: 1 111
        • 101
          --------
          0 (run this column)
          +1 (this carry) +1 (from the next column) gives 11
          Current sum: 10 (carry 1)
          `

Adding Multiple Binary Numbers

  • Consider adding 1 + 1 + 1:
    • This results in carrying over
    • Binary Representation of Three = 11 (i.e., write down 1 and carry over 1).

Representing Negative Binary Numbers

How to Handle Negative Numbers

  1. Using signed binary for negative values involves a method called two's complement.
  2. Consider a binary number of four bits, the maximum positive number is 0111 (7).
  3. Any binary representation with a sign bit reduces the range of representable numbers:
    • Range for four bits with sign bit: from -7 to +7.

Two's Complement Conversion Steps

  1. One's Complement: Flip each bit of the binary number:

    • Example with 0001 (1): becomes 1110 (in one’s complement).

  2. Add One: Add 1 to the one's complement to get the two's complement:

    • Start with 1110:
      • 1110+0001=11111110 + 0001 = 1111 (Example: represents -1).

  3. Positive Number Representation: For a positive binary number, write it directly (e.g., 00001111 (15)).

  4. Example of 2's complement for -1:

    • As a signed binary in 8 bits, use:
      • 00000001 (One) → 11111110 (Flipping bits) → +1 makes it 11111111 (Negative One).

Summary of Number Conversion Techniques

  • For any conversion between binary, octal, and hexadecimal, practice grouping of bits and know how each system represents values:
    • Binary is base 2:
    • Octal is base 8:
    • Hexadecimal is base 16.
  • Familiarity with these bases aids in smooth conversion and arithmetic operations in programming and computer science contexts.