IEEE 754 Floating Point

Sign, exponent, and significand

IEEE 754 is the standard almost every modern CPU uses for floating-point numbers. A value is split into a sign bit, a biased exponent field, and a fraction (mantissa / significand) field. This page shows binary32 (single precision, 32 bits) and binary64 (double precision, 64 bits) for a decimal you enter, including hex patterns and field breakdowns. Bytes are shown in big-endian (network) order for readability.

Field widths

• binary32: 1 sign + 8 exponent + 23 fraction bits
• binary64: 1 sign + 11 exponent + 52 fraction bits

Normal numbers use an implicit leading 1 in the significand. Special patterns encode subnormals, ±infinity, and NaN (not a number).

Worked example

The decimal 0.1 cannot be represented exactly in binary floating point. binary64 stores the closest value; the hex pattern is the well-known repeating approximation that causes classic 0.1 + 0.2 ≠ 0.3 surprises in many languages. Inspecting the bits shows why pocket-decimal fractions misbehave.

Why endianness matters

The logical fields are defined left-to-right in the standard diagrams, but memory order on a machine may be little-endian. This tool displays big-endian byte order so hex dumps match common textbook diagrams and network traces. Use Byte Storage Order when comparing CPU memory dumps.

Common mistakes

• Comparing floats with == after decimal arithmetic.
• Assuming more decimal digits always increase accuracy — binary64 still has a finite significand.
• Forgetting NaN ≠ NaN under IEEE comparison rules.

FAQs

Is this the same as decimal128?

No. Decimal floating formats exist in IEEE 754 but this page focuses on the ubiquitous binary32/binary64 pair.

Can I edit hex to change fields?

Enter a decimal; the layout updates from the rounded binary encoding.

Related: Byte Storage Order, Signed & Unsigned, Bitwise Operations.