एन्कोडिंग योजनाएं एवं संख्या प्रणाली

In a positional number system, the value of any digit depends on three parameters: (1) the digit itself, (2) the position/place of the digit within the number, and (3) the base or radix ($r$) of the number system.

A general number with integer and fractional parts is formally expressed as:

A. Decimal to Any Base ($r$) Conversion

• Integer Part (Successive Division): Divide the decimal integer repeatedly by base $r$, recording remainders at each step until the quotient becomes 0. The result is read from bottom to top (Most Significant Bit to Least Significant Bit).
• Fractional Part (Successive Multiplication): Multiply the fractional component by base $r$. The integer part of the product forms the next fractional digit. Repeat with the remaining fractional part until the fraction reaches 0 or desired precision. Read top to bottom.

B. Binary ↔ Octal & Hexadecimal Direct Grouping

Because $8 = 2^3$ and $16 = 2^4$, conversions between Binary and Octal/Hexadecimal require no arithmetic division—only bit grouping:

• Binary to Octal: Group bits into sets of 3 starting from the binary point moving left for integers (pad left with 0s if needed) and moving right for fractions (pad right with 0s). Convert each 3-bit group to its octal digit.
• Binary to Hexadecimal: Group bits into sets of 4 (nibbles) starting from the binary point. Convert each 4-bit group to its hexadecimal digit ($0-\text{F}$).
• Example: Convert $(1101011011.1011)_2$ to Hexadecimal:
  • Integer: `0011 0101 1011` → `3 5 B`
  • Fraction: `1011` → `B`
  • Result: $(35\text{B}.\text{B})_{16}$

In digital electronics, negative numbers cannot be represented by a literal minus sign. Three standard binary conventions exist for $n$-bit signed integers:

1. Sign-Magnitude Representation: The most significant bit (MSB) acts as the sign bit ($0 = \text{positive}, 1 = \text{negative}$), while the remaining $n-1$ bits store the magnitude.
   • Flaw: Produces dual representations of zero ($+0 = 0000_2$ and $-0 = 1000_2$), complicating hardware ALU design.
2. 1's Complement Representation: Positive numbers are written in standard binary. Negative numbers are formed by inverting every bit ($0 \to 1, 1 \to 0$).
   • Flaw: Still possesses dual zero representations ($0000_2$ and $1111_2$).
3. 2's Complement Representation (Universal Standard): Positive numbers are written in standard binary. A negative number is formed by finding the 1's complement and adding 1 to the least significant bit: $$\text{2's Complement}(X) = \text{1's Complement}(X) + 1$$
   • Advantage: Possesses a single unique zero ($00000000_2$).
   • Range for $n$ bits: $[-2^{n-1} \text{ to } +2^{n-1} - 1]$. For an 8-bit byte: $-128 \text{ to } +127$.
   • Unified Hardware: Subtraction $A - B$ is performed as addition: $A + (\text{2's comp of } B)$, allowing the ALU to use identical adder circuits for both addition and subtraction!

To store human language characters, symbols, and punctuation in computer memory, each character must be assigned a unique numeric code point:

Unicode Transformation Formats (UTF):

• UTF-8: Variable-length encoding using 1 to 4 bytes. 100% backward compatible with 7-bit ASCII (uses only 1 byte for ASCII chars). Dominates the World Wide Web (used by >98% of websites).
• UTF-16: Uses 2 or 4 bytes per character. Default internal string representation in Java, Python 3 (variable width internally), and Windows APIs.
• UTF-32: Fixed-width 4 bytes (32 bits) for every character. Fast memory indexing, but highly memory-inefficient for Latin text.