Floating Point Number Converter Decimal Binary and Hexadecimal

How to Use This Calculator

1. Enter a floating-point number in the input field (e.g., 123.456, -0.1, 3.14e8)
2. Select the precision (32-bit for single precision or 64-bit for double precision)
3. Check which output formats you want to see (binary and/or hexadecimal)
4. Click the "Calculate" button to see the conversion results
5. Click "Reset" to clear all inputs and results

Understanding IEEE 754 Representation

The IEEE 754 standard is a widely used floating-point representation that encodes real numbers in three parts:

• Sign bit (1 bit): Indicates if the number is positive (0) or negative (1)
• Exponent (8 bits for 32-bit, 11 bits for 64-bit): Represents the power of 2 by which the mantissa is multiplied. It uses a biased representation to handle both positive and negative exponents
• Mantissa (fraction) (23 bits for 32-bit, 52 bits for 64-bit): Contains the significant digits of the number. It's normalized to start with a 1 (implied for non-zero numbers)

Example

Let's take the number 123.456 in single precision (32-bit):

1. Decimal to binary: 123 = 1111011, 0.456 ≈ 0.0111001100110011001100110011001100110011
2. Combined binary: 1111011.0111001100110011001100110011001100110011
3. Normalized: 1.1110110111001100110011001100110011001100110011 × 2⁶
4. Exponent (6) + bias (127) = 133 = 1000 0101 in binary
5. Sign bit = 0 (positive)
6. Final 32-bit representation: 0 1000 0101 1110 1101 0111 0011 0011 0011
7. Hexadecimal: 4C8D733

Precision Considerations

Single precision (32-bit) provides less precision than double precision (64-bit). This means that:

• Small differences may not be distinguishable in single precision
• Very large or very small numbers may be rounded or lose precision
• For scientific calculations requiring high precision, use double precision (64-bit)