IEEE 754 Converter

Convert decimal numbers to IEEE 754 floating point binary representation. View sign, exponent, and mantissa bits.

Decimal Number

Precision

Breakdown

Actual value:3.140000104904175Hex:

0x4048F5C3

Exponent (8-bit):128 (bias 127) → 2^1

Binary Representation

Sign bit (positive)

Exponent (8 bits)

10000000

Mantissa / Significand (23 bits)

10010001111010111000011

Full binary:

01000000010010001111010111000011

How IEEE 754 floating point works

IEEE 754 is the standard for floating-point arithmetic. A 32-bit float has 1 sign bit, 8 exponent bits (biased by 127), and 23 mantissa bits. A 64-bit double has 1 sign bit, 11 exponent bits (biased by 1023), and 52 mantissa bits. The value is calculated as:(-1)^sign × 2^(exponent-bias) × 1.mantissa.

Built and maintained by Meet Shah · Last updated 1 August 2026

What this tool is used for

Seeing exactly which bits represent a float that is not behaving as expected.
Explaining why 0.1 plus 0.2 does not equal 0.3 by showing the stored values.
Checking the precision available at a given magnitude before choosing single or double.
Reading a float out of a binary protocol field or a hex dump.
Confirming a special value — an infinity or a NaN — from its bit pattern.

Frequently Asked Questions

What are the three parts of a float?: Sign, exponent and mantissa. A 32-bit single is 1 + 8 + 23 bits; a 64-bit double is 1 + 11 + 52. The mantissa carries an implied leading 1 that is never stored, which is where a double's famous "53 bits of precision" comes from — 52 stored plus the free one.
Why is 0.1 not exactly 0.1?: Because a binary fraction can only represent sums of powers of two, and 0.1 is a repeating fraction in binary just as a third is in decimal. The stored double is 0.1000000000000000055511151231257827…, which is why 0.1 + 0.2 famously misses 0.3.
What is the exponent bias?: 127 for single, 1023 for double. The stored exponent is the real one plus the bias, so the field is always unsigned — which lets two floats be compared as integers, a trick used in sorting and in hardware.
How are infinity and NaN encoded?: With an all-ones exponent. A zero mantissa alongside it means infinity; any non-zero mantissa is NaN — so there are millions of distinct NaN bit patterns. That is also why NaN ≠ NaN: the comparison is defined on the value, not the bits.
What is a subnormal number?: A value with an all-zero exponent, where the implied leading 1 becomes a leading 0 instead. That fills the gap between zero and the smallest normal float, at the cost of precision — and on some hardware it is dramatically slower to compute with.

Common errors and gotchas

Expecting an exact representation for a decimal fraction, which most cannot have in binary.
Confusing 32-bit and 64-bit layouts, which have different exponent and mantissa widths.
Overlooking byte order, so a hex pattern read from memory appears reversed.
Treating NaN as comparable. It is not equal to anything, including itself.
Assuming precision is uniform. The gap between representable values grows with magnitude.