Lecture Notes: Binary to Decimal and Decimal to Binary Conversion

Key Concepts

Binary to Decimal Conversion
- Example: Convert 101 in base 2 to base 10.
  - Number positions: 0, 1, 2
  - Calculation: 1 * 2^2 + 0 * 2^1 + 1 * 2^0
  - 4 (from 2^2) + 0 (from 0 * 2^1) + 1 (from 2^0)
  - Result: 5 in base 10
- Example: Convert 1010101 in base 2 to decimal.
  - Number positions: 3, 2, 1, 0, -1, -2, -3
  - Whole number calculation: 8 + 0 + 2 + 0 + 0.5 + 0 + 0.125
  - Result: 10.625 in base 10
Decimal to Binary Conversion
- Convert 10.625 in base 10 to binary
  - Whole number part: 10
  - Fractional part: 0.625
  - Divide whole number by 2, track remainders for binary:
    - 10 / 2 = 5 R0
    - 5 / 2 = 2 R1
    - 2 / 2 = 1 R0
    - 1 / 2 = 0 R1
  - Binary whole: 1010
  - Multiply fractional by 2, track whole parts:
    - 0.625 * 2 = 1.25 (Whole 1)
    - 0.25 * 2 = 0.5 (Whole 0)
    - 0.5 * 2 = 1.0 (Whole 1)
  - Binary fractional: .101
  - Result: 1010.101 base 2
Iterative Technique for Base Conversion
- Convert decimal to base 3 example:
  - Convert 15 base 10 to base 3
  - Divide by 3, note remainders:
    - 15 / 3 = 5 R0
    - 5 / 3 = 1 R2
    - 1 / 3 = 0 R1
  - Result: 120 base 3
Special Cases for Base Conversion
- Binary to Octal (base 8): Group binary digits in sets of three.
- Binary to Hexadecimal (base 16): Group binary digits in sets of four.
- Example: Convert binary 1010110 to octal and hexadecimal.
- Hexadecimal digits: 0-9 and A-F (A=10, B=11,..., F=15).

Convert Decimal to Binary
- Problem: Convert 13.25 base 10 to binary.
- Options:
  - 1101.01
  - 1100.01
  - 1011.11
  - 1101.101
Convert Decimal to Hexadecimal
- Problem: Convert 77 base 10 to hexadecimal.
- Options:
  - 3F
  - 4D
  - 21
  - 37