IEEE 754 is the international standard for floating-point arithmetic used in virtually all modern computers and programming languages. It defines how floating-point numbers are stored and calculated, ensuring consistent behavior across different systems. The standard specifies formats for single precision (32-bit), double precision (64-bit), and extended precision, along with rules for rounding, special values (NaN, Infinity), and exception handling.

Single precision (32-bit): 1 sign bit + 8 exponent bits + 23 mantissa bits. It provides about 7 significant decimal digits of precision and can represent values from roughly ±1.18×10⁻³⁸ to ±3.4×10³⁸.

Double precision (64-bit): 1 sign bit + 11 exponent bits + 52 mantissa bits. It provides about 15-17 significant decimal digits and ranges from ±2.23×10⁻³⁰⁸ to ±1.80×10³⁰⁸. Double is the default in JavaScript, Python, and most scientific computing.

Endianness refers to the byte order in memory. Big Endian stores the most significant byte at the lowest memory address (used in network protocols and some embedded systems). Little Endian stores the least significant byte first (used by x86/x64 processors). When reading raw hex dumps from a debugger or binary file, knowing the endianness is crucial for correct interpretation of floating-point values.

NaN (Not a Number) represents undefined results like 0/0 or √(-1). It has all exponent bits set to 1 and a non-zero mantissa. There are two types: quiet NaN (most significant mantissa bit = 1) and signaling NaN (MSB of mantissa = 0).

Infinity represents overflow (e.g., 1/0). It has all exponent bits = 1 and mantissa = 0. The sign bit determines positive or negative infinity.

Many decimal fractions (like 0.1) become repeating binary fractions, similar to how 1/3 = 0.333... in decimal. In binary, 0.1₁₀ = 0.0001100110011...₂ which is infinite. IEEE 754 must truncate this to fit the available mantissa bits, causing a tiny rounding error. This is why 0.1 + 0.2 !== 0.3 in most programming languages.

Subnormal numbers fill the gap between zero and the smallest normalized number. When the exponent bits are all zero and the mantissa is non-zero, the implicit leading bit becomes 0 (instead of 1), and the actual exponent is fixed at 1−bias (e.g., −126 for single, −1022 for double). This allows gradual underflow, representing extremely small values at reduced precision, avoiding a sudden jump to zero.

The bias is a fixed offset added to the actual exponent to allow storing both positive and negative exponents as unsigned integers. For single precision, the bias is 127; for double precision, it's 1023. An exponent field of 0 represents the minimum (actual exponent = 1−bias for subnormals, or zero values), while the maximum (all 1s) is reserved for Infinity and NaN.