Floating Point Converter — IEEE 754
Convert decimal numbers to IEEE 754 32-bit and 64-bit floating point binary representation.
What Is the Floating Point Converter — IEEE 754?
The Floating-Point Converter converts decimal numbers to IEEE 754 binary representation (32-bit single precision and 64-bit double precision), showing the sign bit, exponent bits, and mantissa bits. It also converts IEEE 754 bit patterns back to decimal values, revealing rounding errors and special values.
Formula
How to Use
Enter a decimal number to see its IEEE 754 binary representation split into sign, exponent, and mantissa fields. Or enter a 32-bit hex value to decode it. The converter shows the exact decimal value stored (which may differ from input due to rounding), the relative error, and any special values (NaN, Infinity, denormalized).
Example Calculation
0.1 in IEEE 754 (32-bit): Sign=0, Exponent=01111011 (123, biased: 123−127=−4), Mantissa=10011001100... Value = 1.10011001100...×2⁻⁴ = 0.100000001490... (not exactly 0.1 — this is why 0.1+0.2≠0.3 in floating point).
Understanding Floating Point Converter — IEEE 754
IEEE 754 is the technical standard for floating-point arithmetic used in virtually all modern computers, processors, and programming languages. It defines how real numbers are approximated as a sum of a sign bit, an exponent, and a significand (mantissa), allowing a wide range of magnitudes to be represented compactly in 32 or 64 bits.
The fundamental insight of floating-point is the trade-off between range and precision: the same number of bits can represent either very large or very small numbers, but not all real numbers exactly. Most decimal fractions (like 0.1) cannot be represented exactly in binary — they must be rounded to the nearest representable value. This is why floating-point arithmetic can produce unexpected results in programming.
Understanding IEEE 754 is essential for numerical programming, scientific computing, game development (where floating-point precision affects physics simulations), financial applications (where exact decimal arithmetic is critical — hence why financial software often uses integer arithmetic for currency), and low-level systems programming. The converter reveals the exact bits behind any floating-point number, making precision issues transparent.
Frequently Asked Questions
Why can't computers represent 0.1 exactly?
0.1 in binary is 0.000110011... (repeating). IEEE 754 truncates this at 23 bits (single) or 52 bits (double) of mantissa, introducing a small rounding error. This is why 0.1 + 0.2 = 0.30000000000000004 in most programming languages.
What is the precision of single vs double precision?
Single precision (32-bit) has about 7 significant decimal digits. Double precision (64-bit) has about 15-16 significant decimal digits. For most scientific computing, double precision is used.
What are special IEEE 754 values?
Special values include: +Infinity and −Infinity (overflow), NaN (Not a Number, from 0/0 or √(−1)), positive zero (+0) and negative zero (−0), and denormalized (subnormal) numbers near zero.
What is the exponent bias?
The bias converts the stored unsigned exponent to a signed value. Single precision uses bias 127: stored exponent 127 means actual exponent 0. Range: −126 to +127 for normal numbers.
Is this converter free?
Yes, completely free with no registration needed.
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