Decimal Number to IEEE 754 Calculator

Formula used

For a normal value, the stored number follows this expression:

value = (-1)^sign × (1 + fraction / 2^p) × 2^(stored exponent - bias)

Here, p is the number of fraction bits. Binary16 uses p = 10 and bias = 15. Binary32 uses p = 23 and bias = 127. Binary64 uses p = 52 and bias = 1023.

For subnormal values, the leading one is removed: value = (-1)^sign × (fraction / 2^p) × 2^(1 - bias).

How to use this calculator

1. Enter a decimal number, including scientific notation when needed.
2. Select Binary16, Binary32, or Binary64 for the target format.
3. Choose bit grouping and decoded display precision.
4. Select Convert decimal number.
5. Review the binary, hexadecimal, sign, exponent, fraction, and stored value.
6. Download CSV for records, or use the print control for PDF saving.

Example data

Understanding decimal conversion

IEEE 754 foundations

IEEE 754 is the common standard for storing floating-point numbers. It converts a decimal input into a binary pattern. That pattern contains a sign bit, an exponent field, and a fraction field. The pattern represents many ordinary values efficiently. It cannot represent every decimal fraction exactly. This is normal. For example, 0.1 repeats forever in binary. The stored result is the closest permitted binary value. A calculator exposes that stored pattern. It helps developers, students, testers, and engineers check numeric behavior before it creates confusion.

Stored binary fields

A sign bit records whether the number is positive or negative. An exponent shifts the binary point. A fraction, also called a significand or mantissa field, stores precision bits. Normal values use an implied leading one. This saves one bit of storage. Subnormal values handle tiny magnitudes near zero. They do not use the implied leading one. This design creates a gradual transition toward zero. Special exponent patterns represent positive infinity, negative infinity, and NaN. NaN means that a calculation has no defined numerical result.

Choosing precision

Half precision uses sixteen total bits. It is useful where memory matters greatly. Single precision uses thirty-two bits. It is common in graphics, sensors, and many data formats. Double precision uses sixty-four bits. It provides more significant digits and a wider exponent range. More bits improve accuracy and range. They also increase memory use. Select the format required by the system receiving the value. Matching the target format avoids misleading comparisons.

Rounding and limits

The conversion follows a consistent structure. First, the sign is separated from the magnitude. Next, the magnitude is written in binary. A normal binary value is normalized as one point fraction times two raised to an exponent. The stored exponent adds a fixed bias. The fraction stores digits after the binary point. When required digits exceed the fraction capacity, the value is rounded. IEEE 754 normally uses round to nearest, ties to even. This reduces long-term rounding bias across many calculations.

Practical checks

Use the output as a verification tool, not only as a conversion tool. Compare the original input with the decoded stored value. Review the hexadecimal encoding when debugging files, APIs, memory dumps, or network packets. Check the exponent and fraction when a value seems unusual. Very large values may overflow in smaller formats. Very small values can become subnormal or zero. A displayed bit pattern makes these limits visible. It also explains why two apparently similar decimals can produce different results after storage.

Reliable records

Input checks improve reliable results. Enter decimal values using ordinary or scientific notation. Keep the original text when auditing a conversion. Choose grouping for easier bit reading. Export a comma-separated record for documentation or regression tests. The print control creates a clean report that browsers can save as a PDF. These options make repeated checks faster. They keep essential values together. Review them later when needed. It supports clear communication across teams and versions.