Comprehensive Number Base Conversions

Mastering Decimal, Binary, Octal, and Hexadecimal Systems

Key Takeaways

Understanding Base Systems: Grasp the fundamentals of decimal, binary, octal, and hexadecimal systems for effective number conversions.
Step-by-Step Conversion: Employ systematic division and multiplication methods to accurately convert numbers between different bases.
Error Recognition: Identify and rectify invalid digits in base-specific conversions to ensure accuracy.

1. Decimal to Octal (₁₀ → ₈)

Conversion Steps

To convert a decimal number to octal, repeatedly divide the number by 8 and record the remainders. The octal number is obtained by reading the remainders from bottom to top.

Examples

7772₁₀ → ₈
- 7772 ÷ 8 = 971 R4
- 971 ÷ 8 = 121 R3
- 121 ÷ 8 = 15 R1
- 15 ÷ 8 = 1 R7
- 1 ÷ 8 = 0 R1
Result: 17134₈
982₁₀ → ₈
- 982 ÷ 8 = 122 R6
- 122 ÷ 8 = 15 R2
- 15 ÷ 8 = 1 R7
- 1 ÷ 8 = 0 R1
Result: 1726₈
5327₁₀ → ₈
- 5327 ÷ 8 = 665 R7
- 665 ÷ 8 = 83 R1
- 83 ÷ 8 = 10 R3
- 10 ÷ 8 = 1 R2
- 1 ÷ 8 = 0 R1
Result: 12317₈

2. Decimal to Hexadecimal (₁₀ → ₁₆)

Conversion Steps

To convert a decimal number to hexadecimal, repeatedly divide the number by 16 and record the remainders. Replace remainders above 9 with their hexadecimal equivalents (A-F). The hexadecimal number is obtained by reading the remainders from bottom to top.

Examples

7772₁₀ → ₁₆
- 7772 ÷ 16 = 485 R12 (C)
- 485 ÷ 16 = 30 R5
- 30 ÷ 16 = 1 R14 (E)
- 1 ÷ 16 = 0 R1
Result: 1E5C₁₆
982₁₀ → ₁₆
- 982 ÷ 16 = 61 R6
- 61 ÷ 16 = 3 R13 (D)
- 3 ÷ 16 = 0 R3
Result: 3D6₁₆
5327₁₀ → ₁₆
- 5327 ÷ 16 = 332 R15 (F)
- 332 ÷ 16 = 20 R12 (C)
- 20 ÷ 16 = 1 R4
- 1 ÷ 16 = 0 R1
Result: 14C7₁₆

3. Decimal to Binary (₁₀ → ₂)

Conversion Steps

To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. The binary number is obtained by reading the remainders from bottom to top.

Examples

781₁₀ → ₂
- 781 ÷ 2 = 390 R1
- 390 ÷ 2 = 195 R0
- 195 ÷ 2 = 97 R1
- 97 ÷ 2 = 48 R1
- 48 ÷ 2 = 24 R0
- 24 ÷ 2 = 12 R0
- 12 ÷ 2 = 6 R0
- 6 ÷ 2 = 3 R0
- 3 ÷ 2 = 1 R1
- 1 ÷ 2 = 0 R1
Result: 1100001101₂
235₁₀ → ₂
- 235 ÷ 2 = 117 R1
- 117 ÷ 2 = 58 R1
- 58 ÷ 2 = 29 R0
- 29 ÷ 2 = 14 R1
- 14 ÷ 2 = 7 R0
- 7 ÷ 2 = 3 R1
- 3 ÷ 2 = 1 R1
- 1 ÷ 2 = 0 R1
Result: 11101011₂
9821₁₀ → ₂
- 9821 ÷ 2 = 4910 R1
- 4910 ÷ 2 = 2455 R0
- 2455 ÷ 2 = 1227 R1
- 1227 ÷ 2 = 613 R1
- 613 ÷ 2 = 306 R1
- 306 ÷ 2 = 153 R0
- 153 ÷ 2 = 76 R1
- 76 ÷ 2 = 38 R0
- 38 ÷ 2 = 19 R0
- 19 ÷ 2 = 9 R1
- 9 ÷ 2 = 4 R1
- 4 ÷ 2 = 2 R0
- 2 ÷ 2 = 1 R0
- 1 ÷ 2 = 0 R1
Result: 10011001011101₂
5327₁₀ → ₂
- 5327 ÷ 2 = 2663 R1
- 2663 ÷ 2 = 1331 R1
- 1331 ÷ 2 = 665 R1
- 665 ÷ 2 = 332 R1
- 332 ÷ 2 = 166 R0
- 166 ÷ 2 = 83 R0
- 83 ÷ 2 = 41 R1
- 41 ÷ 2 = 20 R1
- 20 ÷ 2 = 10 R0
- 10 ÷ 2 = 5 R0
- 5 ÷ 2 = 2 R1
- 2 ÷ 2 = 1 R0
- 1 ÷ 2 = 0 R1
Result: 1010011001111₂

4. Octal to Decimal (₈ → ₁₀)

Conversion Steps

To convert an octal number to decimal, multiply each digit by the corresponding power of 8 and sum the results.

Examples

5327₈ → ₁₀
- 5 × 8³ = 5 × 512 = 2560
- 3 × 8² = 3 × 64 = 192
- 2 × 8¹ = 2 × 8 = 16
- 7 × 8⁰ = 7 × 1 = 7
Result: 2775₁₀

5. Binary to Decimal (₂ → ₁₀)

Conversion Steps

To convert a binary number to decimal, multiply each bit by the corresponding power of 2 and sum the results.

Examples

    
<table border="1">
  <tr>
    <th>Binary Digit</th>
    <th>Power of 2</th>
    <th>Value</th>
  </tr>
  <tr>
    <td>1</td>
    <td>2⁷ = 128</td>
    <td>128</td>
  </tr>
  <tr>
    <td>0</td>
    <td>2⁶ = 64</td>
    <td>0</td>
  </tr>
  <tr>
    <td>1</td>
    <td>2⁵ = 32</td>
    <td>32</td>
  </tr>
  <tr>
    <td>1</td>
    <td>2⁴ = 16</td>
    <td>16</td>
  </tr>
  <tr>
    <td>0</td>
    <td>2³ = 8</td>
    <td>0</td>
  </tr>
  <tr>
    <td>1</td>
    <td>2² = 4</td>
    <td>4</td>
  </tr>
  <tr>
    <td>1</td>
    <td>2¹ = 2</td>
    <td>2</td>
  </tr>
  <tr>
    <td>0</td>
    <td>2⁰ = 1</td>
    <td>0</td>
  </tr>
</table>

Example: 10110110₂

Calculation: 128 + 0 + 32 + 16 + 0 + 4 + 2 + 0 = 182₁₀

6. Summary of Conversions

Conversion	Input	Output
7772₁₀ → ₈	7772₁₀	17134₈
7772₁₀ → ₁₆	7772₁₀	1E5C₁₆
781₁₀ → ₂	781₁₀	1100001101₂
235₁₀ → ₂	235₁₀	11101011₂
9821₁₀ → ₂	9821₁₀	10011001011101₂
982₁₀ → ₈	982₁₀	1726₈
982₁₀ → ₁₆	982₁₀	3D6₁₆
5327₁₀ → ₂	5327₁₀	1010011001111₂
5327₁₀ → ₈	5327₁₀	12317₈
5327₁₀ → ₁₆	5327₁₀	14C7₁₆
5327₈ → ₁₀	5327₈	2775₁₀
7298₈ → ₁₀	7298₈	3792₁₀
10110110₂ → ₁₀	10110110₂	182₁₀
A13F₁₆ → ₁₀	A13F₁₆	41279₁₀
B897A₁₆ → ₁₀	B897A₁₆	756090₁₀

Conclusion

Mastering number base conversions is essential in computer programming and digital electronics. By understanding and applying systematic conversion methods, you can seamlessly navigate between decimal, binary, octal, and hexadecimal systems. Always ensure to follow the step-by-step processes to maintain accuracy and recognize any invalid inputs specific to each base.