1. What is IEEE 754 Floating Point?

The IEEE 754 standard is the most widely used standard for floating-point computation. It defines formats for representing floating-point numbers, special values (infinity and NaN), and rounding rules.

2. How Does the Calculator Work?

The calculator uses the IEEE 754 single-precision (32-bit) formula:

Where:

• \( sign \) — 1 bit (0 for positive, 1 for negative)
• \( exponent \) — 8 bits (stored with bias of 127)
• \( mantissa \) — 23 bits (fractional part)
• \( bias \) — 127 for single-precision

3. Components of IEEE 754

Details: The 32-bit format consists of:

• 1 sign bit
• 8 exponent bits (range -126 to +127 after unbias)
• 23 mantissa bits (implied leading 1)

Special values:

• Exponent all 1s, mantissa 0: ±Infinity
• Exponent all 1s, mantissa ≠ 0: NaN
• Exponent all 0s: subnormal numbers

4. Using the Calculator

Tips: Enter a 32-bit binary string representing an IEEE 754 single-precision floating-point number. The calculator will decode it into its decimal equivalent.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between single and double precision?
A: Single precision uses 32 bits (1-8-23), double uses 64 bits (1-11-52) with larger range and precision.

Q2: Why is there a bias in the exponent?
A: The bias allows the exponent to be stored as an unsigned number while still representing negative exponents.

Q3: What is the "hidden bit" in the mantissa?
A: For normalized numbers, there's an implied leading 1 that isn't stored, giving 24 bits of precision.

Q4: How are subnormal numbers represented?
A: When exponent is 0, the implied leading bit is 0 instead of 1, allowing representation of very small numbers.

Q5: What are some common pitfalls with floating-point?
A: Rounding errors, non-associativity of operations, and inability to precisely represent many decimal fractions.