1. What is Floating Point Representation?

Floating-point representation is a way to store and manipulate real numbers in computers and mathematics. It represents numbers in scientific notation using a sign, mantissa (significand), base, and exponent.

2. How Does the Calculator Work?

The calculator uses the floating point formula:

Where:

• \( sign \) — 1 for positive, -1 for negative numbers
• \( mantissa \) — Significant digits (also called significand)
• \( base \) — Number base (typically 2 for binary, 10 for decimal)
• \( exponent \) — Power to which the base is raised

Explanation: The calculator normalizes the number so that the mantissa is in the range [1, base) for non-zero numbers.

3. Importance of Floating Point Representation

Details: Floating-point representation is crucial in computer science for storing real numbers efficiently and in mathematics for scientific notation. It allows representation of very large and very small numbers with consistent relative precision.

4. Using the Calculator

Tips: Enter any real number as the value and an integer ≥2 as the base. The calculator will decompose the number into its floating point components.

5. Frequently Asked Questions (FAQ)

Q1: What's the most common base used in computers?
A: Base 2 (binary) is used in most computer systems for floating-point arithmetic.

Q2: How is zero represented?
A: Zero is typically represented with a mantissa of 0 and exponent of 0 (with any sign).

Q3: What's the difference between floating-point and fixed-point?
A: Fixed-point has a constant number of digits after the decimal point, while floating-point can represent numbers of vastly different magnitudes.

Q4: What are the limitations of floating-point?
A: Floating-point cannot represent all real numbers exactly, leading to rounding errors in calculations.

Q5: What is IEEE 754?
A: It's the most common standard for floating-point arithmetic in computers, defining binary formats for single and double precision.