1. What Is IEEE 754 Floating Point?

IEEE 754 is the technical standard for floating-point arithmetic established by the Institute of Electrical and Electronics Engineers. It defines formats for representing floating-point numbers and special values like NaN and infinity.

2. How Does The Conversion Work?

The conversion follows the IEEE 754 formula:

Where:

• \( sign \) — Determines positive (0) or negative (1)
• \( mantissa \) — Fractional part of the number
• \( exponent \) — Power of 2 scaling factor
• \( bias \) — Offset for exponent (127 for single, 1023 for double precision)

3. Single vs Double Precision

Single Precision (32-bit): 1 sign bit, 8 exponent bits, 23 mantissa bits
Double Precision (64-bit): 1 sign bit, 11 exponent bits, 52 mantissa bits
Double precision offers greater range and precision but uses more memory.

4. Using The Calculator

Instructions: Enter a decimal number and select the desired precision (single or double). The calculator will display the IEEE 754 binary representation, hexadecimal format, and individual components.

5. Frequently Asked Questions (FAQ)

Q1: What is the bias value in IEEE 754?
A: Bias is 127 for single precision and 1023 for double precision. It allows exponents to represent both positive and negative values.

Q2: How are special values represented?
A: NaN (Not a Number) has all exponent bits set to 1 and non-zero mantissa. Infinity has all exponent bits set to 1 and zero mantissa.

Q3: What is denormalized numbers?
A: Numbers with exponent field all zeros are denormalized, representing values very close to zero with reduced precision.

Q4: Why use hexadecimal representation?
A: Hexadecimal is more compact than binary and commonly used in programming and debugging floating-point values.

Q5: Are there rounding errors?
A: Yes, floating-point representation can have rounding errors due to finite precision, especially for numbers that cannot be exactly represented in binary.