Data Representation and Operations

Base Systems (Radix)

Radix-$r$ System ($r$-base) $K_{n} K_{n-1} K_{n-2} \dots K_{0} K_{-1} \dots K_{-m}$
Numerical expansion value: $K_{n} r^{n} + K_{n-1} r^{n-1} + \dots + K_{0} r^{0} + K_{-1} r^{-1} + \dots + K_{-m} r^{-m} = \sum_{i=n}^{-m} K_{i} r^{i}$
Binary System $\text{Digits: } 0, 1 \quad (r=2)$
Octal System $\text{Digits: } 0, 1, 2, 3, 4, 5, 6, 7 \quad (r=8=2^3)$
Hexadecimal System $\text{Digits: } 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F \quad (r=16=2^4)$

A 4-bit binary sequence directly corresponds to a single 1-digit hexadecimal character.
A 3-bit binary sequence directly corresponds to a single 1-digit octal character.

Definition: The highest-order bit represents the sign ($0$ for positive numbers, $1$ for negative numbers), while the remaining bits represent the absolute magnitude of the numerical value.
Characteristics: Has two distinct representations for zero ($+0$ and $-0$). It is intuitive for humans but complicates arithmetic hardware implementation.

Definition: * For positive numbers: Identical to its sign-and-magnitude representation.
- For negative numbers: The sign bit is maintained as $1$, while all magnitude bits are inverted ($0 \to 1$ and $1 \to 0$).
Characteristics: Also retains twin representations for zero ($+0$ and $-0$). Historically used as an intermediate state to simplify subtraction into addition.

Definition: * For positive numbers: Identical to its sign-and-magnitude representation.
- For negative numbers: Formed by taking its ones’ complement and adding $1$ to the least significant bit (LSB).
Characteristics: Features a unique representation for zero ($0000\dots00$). It unifies subtraction and addition into unified addition operations, and expands the representation range for negative values by one unit (e.g., matching $-128$ in an 8-bit integer system).

Definition: Shifting the true value by adding a fixed constant displacement bias value (typically $2^{n-1}$ or $2^{n-1}-1$, where $n$ represents the total tracking bit width count).
Characteristics: Preserves natural lexicographical ordering for values, making comparisons straightforward. Commonly used to represent the exponent field (characteristic) in floating-point notations (such as IEEE 754 formats).