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Binary Converter Calculator

Convert numbers between decimal, binary, octal, and hexadecimal number systems with ease.

Enter a decimal number (base 10)
Enter a binary number (base 2)
Enter an octal number (base 8)
Enter a hexadecimal number (base 16)

How to Use This Calculator

  1. Enter a number in any of the four number systems (decimal, binary, octal, or hexadecimal)
  2. Adjust the number of bits if needed (8, 16, 24, 32, 40, 48, 56, or 64)
  3. Enable Two's Complement if you're working with negative numbers
  4. Click Convert to see the number in all four number systems

Conversion Methods

Decimal to Binary: Divide by 2 and record remainders
Binary to Decimal: Sum of (bit × 2^position)
Decimal to Hex: Divide by 16 and record remainders
Hex to Decimal: Sum of (digit × 16^position)

Where:

  • Position starts from 0 at the rightmost digit
  • Binary digits (bits) can only be 0 or 1
  • Octal digits range from 0 to 7
  • Hexadecimal digits range from 0 to 9 and A to F

Example Calculation

Real-World Scenario:

A programmer needs to convert the decimal number 255 to binary and hexadecimal for memory addressing.

Given:

  • Decimal = 255

Decimal to Binary:

255 ÷ 2 = 127 remainder 1
127 ÷ 2 = 63 remainder 1
63 ÷ 2 = 31 remainder 1
31 ÷ 2 = 15 remainder 1
15 ÷ 2 = 7 remainder 1
7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading remainders from bottom to top: 11111111

Decimal to Hexadecimal:

255 ÷ 16 = 15 remainder 15
15 in hexadecimal is F
Reading remainders from bottom to top: FF

Result: Decimal 255 = Binary 11111111 = Hexadecimal FF

Why This Calculation Matters

Practical Applications

  • Computer programming and debugging
  • Digital electronics and circuit design
  • Network configuration (IP addresses)
  • Cryptography and data encoding

Key Benefits

  • Understanding computer data representation
  • Efficient memory management
  • Low-level programming optimization
  • Better grasp of digital systems

Common Mistakes & Tips

Remember that binary numbers only use 0 and 1, octal numbers use 0-7, and hexadecimal numbers use 0-9 and A-F. Using digits outside these ranges will result in errors.

When working with fixed bit lengths, leading zeros are important. For example, in 8-bit binary, the number 5 should be represented as 00000101, not just 101.

Two's complement is used to represent negative numbers in binary. To convert a negative number to two's complement, invert all bits and add 1. For example, -5 in 8-bit two's complement is 11111011.

Frequently Asked Questions

Binary is base-2 and uses only 0 and 1, while hexadecimal is base-16 and uses digits 0-9 and letters A-F. Hexadecimal is often used in programming because it's more compact than binary and easier to read than decimal for certain applications.

Computers use binary because their electronic components (transistors) have two states: on and off, which naturally correspond to 1 and 0. This binary system is simpler to implement in hardware than a decimal system would be.

Two's complement is a method for representing negative numbers in binary. It's used because it simplifies arithmetic operations - addition, subtraction, and multiplication work the same for both positive and negative numbers without needing special circuitry for sign handling.

References & Disclaimer

Mathematical Disclaimer

This calculator provides accurate conversions between number systems. However, for critical applications, always verify results independently. The calculator assumes valid input in the correct format for each number system.

References

Accuracy Notice

This calculator accurately converts numbers between decimal, binary, octal, and hexadecimal systems. For very large numbers beyond standard JavaScript integer limits, precision may be affected. For scientific or engineering applications requiring high precision, consider using specialized software.

About the Author

Kumaravel Madhavan

Web developer and data researcher creating accurate, easy-to-use calculators across health, finance, education, and construction and more. Works with subject-matter experts to ensure formulas meet trusted standards like WHO, NIH, and ISO.

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