Floating Point Number Systems and Representation

Limitations of Integer and Fixed-Bit Representations

• Standard Bit-Width Capabilities: In a system using $n$ bits, the range of representable numbers depends on the encoding scheme used:

  • Unsigned Integers: These can represent values ranging from $0$ to $2^n$.
  • 2’s Complement Signed Integers: These can represent values ranging from $-2^{n-1}$ to $2^{n-1} - 1$.

• Challenges for Fixed-Width Representations: Fixed-point or standard integer systems fail to efficiently handle specific categories of numbers that are essential for scientific and engineering calculations:

  • Extremely Large Numbers: For example, a value such as $9,349,398,989,787,762,244,859,087,678$.
  • Extremely Small Numbers: For example, a decimal value such as $0.0000000000000000000000045691$.
  • Rational Numbers: Fractions such as $\frac{2}{3}$.
  • Irrational Numbers: Values that cannot be expressed as simple fractions, such as $\sqrt{2}$.
  • Transcendental Numbers: Complex mathematical constants such as $e$ and $\pi$.

The Concept of Floating-Point Representation

• Overview: Floating-point representation is an approach where a number is represented according to a fixed number of decimal places and a specific level of precision.
• Methodology: This system allows the decimal or binary point to "float" rather than being in a fixed position, enabling the representation of a much wider range of magnitudes within the same number of bits.

Components of Scientific Notation

• General Formula: Scientists and engineers represent numbers using scientific notation, expressed in the form $M \times 10^E$.
• Key Definitions:
  • Mantissa ($M$): The significand or the part of the number containing its significant digits.
  • Exponent ($E$): The power to which the base is raised, determining the magnitude and the position of the decimal point.
  • Base: In standard scientific notation, the base is $10$. In computer systems, the base is typically $2$.
  • Scaling Factor: The component represented by the base raised to the exponent (e.g., $10^E$).

Normalization Principles and Examples

• Definition of Normalized Form: A number is considered to be in normalized form when there is exactly one non-zero digit to the left of the decimal (or binary) point.
• Normalization in Base 10: The number $51$ can be expressed in various non-normalized formats, but only one is strictly normalized:
  • $.51 \times 10^2$
  • $0.051 \times 10^3$
  • $5100. \times 10^{-2}$
  • Normalized Form: $5.100 \times 10^1$
• Normalization in Base 2 (Binary): The same principle applies to binary strings to ensure consistent representation and maximize precision:
  • Example 1: The binary string $01001$ is normalized to $1.001 \times 2^3$.
  • Example 2: The binary value $00101101.101$ is normalized to $1.01101101 \times 2^5$.
  • Example 3: The fractional binary value $0.0001101001110$ is normalized to $1.101001110 \times 2^{-4}$.

IEEE Half-Precision Standard

• Bit Allocation: The IEEE Half-Precision format uses a total of $16$ bits to encode a number, divided into three distinct fields:
  • Sign: $1$ bit is dedicated to indicating whether the number is positive or negative.
  • Exponent: $5$ bits are used for the exponent.
  • Mantissa: $10$ bits are used for the significand.
• Biasing: The exponent in this format is biased, specifically using an "excess $15$" representation. This means the actual exponent value is stored as the true exponent plus a bias of $15$.