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Comprehensive Guide to Number System Conversions

Mastering Decimal, Binary, Octal, and Hexadecimal Conversions

number systems in computer programming

Key Takeaways

  • Understanding Base Systems: Grasping the fundamentals of decimal, binary, octal, and hexadecimal systems is crucial for seamless conversions.
  • Step-by-Step Conversion Processes: Employ repetitive division and remainder methods to accurately convert numbers between different bases.
  • Common Conversion Challenges: Recognizing and addressing potential errors, such as invalid digits in specific base systems, ensures accurate results.

1. Converting Decimal to Octal (₈) and Hexadecimal (₁₆)

1) 7772₁₀ → ₈

To convert the decimal number 7772 to octal:

  1. Divide 7772 by 8: 7772 ÷ 8 = 971 with a remainder of 4.
  2. Divide 971 by 8: 971 ÷ 8 = 121 with a remainder of 3.
  3. Divide 121 by 8: 121 ÷ 8 = 15 with a remainder of 1.
  4. Divide 15 by 8: 15 ÷ 8 = 1 with a remainder of 7.
  5. Divide 1 by 8: 1 ÷ 8 = 0 with a remainder of 1.

Reading the remainders from bottom to top gives the octal representation: 17134₈.

2) 7772₁₀ → ₁₆

To convert the decimal number 7772 to hexadecimal:

  1. Divide 7772 by 16: 7772 ÷ 16 = 485 with a remainder of 12 (C).
  2. Divide 485 by 16: 485 ÷ 16 = 30 with a remainder of 5.
  3. Divide 30 by 16: 30 ÷ 16 = 1 with a remainder of 14 (E).
  4. Divide 1 by 16: 1 ÷ 16 = 0 with a remainder of 1.

Reading the remainders from bottom to top gives the hexadecimal representation: 1E5C₁₆.

2. Converting Decimal to Binary (₂)

3) 781₁₀ → ₂

To convert the decimal number 781 to binary:

  1. 781 ÷ 2 = 390 remainder 1
  2. 390 ÷ 2 = 195 remainder 0
  3. 195 ÷ 2 = 97 remainder 1
  4. 97 ÷ 2 = 48 remainder 1
  5. 48 ÷ 2 = 24 remainder 0
  6. 24 ÷ 2 = 12 remainder 0
  7. 12 ÷ 2 = 6 remainder 0
  8. 6 ÷ 2 = 3 remainder 0
  9. 3 ÷ 2 = 1 remainder 1
  10. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top gives the binary representation: 1100001101₂.

4) 235₁₀ → ₂

To convert the decimal number 235 to binary:

  1. 235 ÷ 2 = 117 remainder 1
  2. 117 ÷ 2 = 58 remainder 1
  3. 58 ÷ 2 = 29 remainder 0
  4. 29 ÷ 2 = 14 remainder 1
  5. 14 ÷ 2 = 7 remainder 0
  6. 7 ÷ 2 = 3 remainder 1
  7. 3 ÷ 2 = 1 remainder 1
  8. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top gives the binary representation: 11101011₂.

5) 9821₁₀ → ₂

To convert the decimal number 9821 to binary:

  1. 9821 ÷ 2 = 4910 remainder 1
  2. 4910 ÷ 2 = 2455 remainder 0
  3. 2455 ÷ 2 = 1227 remainder 1
  4. 1227 ÷ 2 = 613 remainder 1
  5. 613 ÷ 2 = 306 remainder 1
  6. 306 ÷ 2 = 153 remainder 0
  7. 153 ÷ 2 = 76 remainder 1
  8. 76 ÷ 2 = 38 remainder 0
  9. 38 ÷ 2 = 19 remainder 0
  10. 19 ÷ 2 = 9 remainder 1
  11. 9 ÷ 2 = 4 remainder 1
  12. 4 ÷ 2 = 2 remainder 0
  13. 2 ÷ 2 = 1 remainder 0
  14. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top gives the binary representation: 10011001011101₂.

3. Converting Decimal to Octal (₈) and Hexadecimal (₁₆)

6) 982₁₀ → ₈

To convert the decimal number 982 to octal:

  1. 982 ÷ 8 = 122 remainder 6
  2. 122 ÷ 8 = 15 remainder 2
  3. 15 ÷ 8 = 1 remainder 7
  4. 1 ÷ 8 = 0 remainder 1

Reading the remainders from bottom to top gives the octal representation: 1726₈.

7) 982₁₀ → ₁₆

To convert the decimal number 982 to hexadecimal:

  1. 982 ÷ 16 = 61 remainder 6
  2. 61 ÷ 16 = 3 remainder 13 (D)
  3. 3 ÷ 16 = 0 remainder 3

Reading the remainders from bottom to top gives the hexadecimal representation: 3D6₁₆.

4. Converting Decimal to Binary (₂)

8) 5327₁₀ → ₂

To convert the decimal number 5327 to binary:

  1. 5327 ÷ 2 = 2663 remainder 1
  2. 2663 ÷ 2 = 1331 remainder 1
  3. 1331 ÷ 2 = 665 remainder 1
  4. 665 ÷ 2 = 332 remainder 1
  5. 332 ÷ 2 = 166 remainder 0
  6. 166 ÷ 2 = 83 remainder 0
  7. 83 ÷ 2 = 41 remainder 1
  8. 41 ÷ 2 = 20 remainder 1
  9. 20 ÷ 2 = 10 remainder 0
  10. 10 ÷ 2 = 5 remainder 0
  11. 5 ÷ 2 = 2 remainder 1
  12. 2 ÷ 2 = 1 remainder 0
  13. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top gives the binary representation: 1010011001111₂.

5. Converting Decimal to Octal (₈) and Hexadecimal (₁₆)

9) 5327₁₀ → ₈

To convert the decimal number 5327 to octal:

  1. 5327 ÷ 8 = 665 remainder 7
  2. 665 ÷ 8 = 83 remainder 1
  3. 83 ÷ 8 = 10 remainder 3
  4. 10 ÷ 8 = 1 remainder 2
  5. 1 ÷ 8 = 0 remainder 1

Reading the remainders from bottom to top gives the octal representation: 12317₈.

10) 5327₁₀ → ₁₆

To convert the decimal number 5327 to hexadecimal:

  1. 5327 ÷ 16 = 333 remainder 15 (F)
  2. 333 ÷ 16 = 20 remainder 13 (D)
  3. 20 ÷ 16 = 1 remainder 4
  4. 1 ÷ 16 = 0 remainder 1

Reading the remainders from bottom to top gives the hexadecimal representation: 14CF₁₆.

6. Converting Octal and Binary to Decimal (₁₀)

11) 5327₈ → ₁₀

To convert the octal number 5327₈ to decimal:

  1. Identify each digit's place value based on powers of 8.
  2. Calculate: (5 × 8³) + (3 × 8²) + (2 × 8¹) + (7 × 8⁰)
  3. Compute each term:
    • 5 × 512 = 2560
    • 3 × 64 = 192
    • 2 × 8 = 16
    • 7 × 1 = 7
  4. Sum the results: 2560 + 192 + 16 + 7 = 2775₁₀

Thus, 5327₈ = 2775₁₀.

12) 7298₈ → ₁₀

Attempting to convert the octal number 7298₈ to decimal reveals an issue:

  • Octal digits range from 0 to 7.
  • The digit '9' in 7298₈ exceeds this range, making it an invalid octal number.

Therefore, the conversion cannot be performed due to the presence of invalid digits.

13) 10110110₂ → ₁₀

To convert the binary number 10110110₂ to decimal:

  1. Identify each digit's place value based on powers of 2.
  2. Calculate: (1 × 2⁷) + (0 × 2⁶) + (1 × 2⁵) + (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰)
  3. Compute each term:
    • 1 × 128 = 128
    • 0 × 64 = 0
    • 1 × 32 = 32
    • 1 × 16 = 16
    • 0 × 8 = 0
    • 1 × 4 = 4
    • 1 × 2 = 2
    • 0 × 1 = 0
  4. Sum the results: 128 + 0 + 32 + 16 + 0 + 4 + 2 + 0 = 182₁₀

Thus, 10110110₂ = 182₁₀.

14) A13F₁₆ → ₁₀

To convert the hexadecimal number A13F₁₆ to decimal:

  1. Identify each digit's place value based on powers of 16.
  2. Calculate: (A × 16³) + (1 × 16²) + (3 × 16¹) + (F × 16⁰)
  3. Convert hexadecimal letters to decimal:
    • A = 10
    • F = 15
  4. Compute each term:
    • 10 × 4096 = 40960
    • 1 × 256 = 256
    • 3 × 16 = 48
    • 15 × 1 = 15
  5. Sum the results: 40960 + 256 + 48 + 15 = 41279₁₀

Thus, A13F₁₆ = 41279₁₀.

15) B897A₁₆ → ₁₀

To convert the hexadecimal number B897A₁₆ to decimal:

  1. Identify each digit's place value based on powers of 16.
  2. Calculate: (B × 16⁴) + (8 × 16³) + (9 × 16²) + (7 × 16¹) + (A × 16⁰)
  3. Convert hexadecimal letters to decimal:
    • B = 11
    • A = 10
  4. Compute each term:
    • 11 × 65536 = 720896
    • 8 × 4096 = 32768
    • 9 × 256 = 2304
    • 7 × 16 = 112
    • 10 × 1 = 10
  5. Sum the results: 720896 + 32768 + 2304 + 112 + 10 = 755090₁₀

Thus, B897A₁₆ = 755090₁₀.

7. Summary of Number Conversions

Conversion Result
7772₁₀ → ₈ 17134₈
7772₁₀ → ₁₆ 1E5C₁₆
781₁₀ → ₂ 1100001101₂
235₁₀ → ₂ 11101011₂
9821₁₀ → ₂ 10011001011101₂
982₁₀ → ₈ 1726₈
982₁₀ → ₁₆ 3D6₁₆
5327₁₀ → ₂ 1010011001111₂
5327₁₀ → ₈ 12317₈
5327₁₀ → ₁₆ 14CF₁₆
5327₈ → ₁₀ 2775₁₀
7298₈ → ₁₀ Invalid Octal Number
10110110₂ → ₁₀ 182₁₀
A13F₁₆ → ₁₀ 41279₁₀
B897A₁₆ → ₁₀ 755090₁₀

Conclusion

Mastering number system conversions is fundamental in computer programming and digital electronics. By understanding the mechanisms of converting numbers between decimal, binary, octal, and hexadecimal systems, programmers can efficiently interpret and manipulate data in various formats. This guide provides a step-by-step approach to ensure accuracy and confidence in performing these essential conversions.

References


Last updated February 13, 2025
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