9.2 Binary, Octal, Decimal, Hexadecimal Conversions & File Formats

Binary, Octal, Decimal, Hexadecimal Conversions & File Formats

Quick Answer: Digital computing systems utilize positional number systems with distinct radix bases: Binary (base 2), Octal (base 8), Decimal (base 10), and Hexadecimal (base 16, where $A=10$ through $F=15$). Conversions rely on successive division/multiplication for decimal mappings and direct 3-bit (octal) or 4-bit (hexadecimal) binary grouping. Storage capacity scales in binary factors of $2^{10} = 1024$ (Byte, KB, MB, GB, TB, PB, EB, ZB, YB), while multimedia file extensions are governed by lossy or lossless compression standards.

1. Positional Number Systems in Digital Computing

In a positional number system, the value of any digit depends on its face value, its position relative to the radix point, and the base (radix, denoted $r$) of the system. The value of a general number $N$ with integer digits $d_i$ and fractional digits $d_{-j}$ is expressed as:

$N = \sum_{i=0}^{n-1} d_i \cdot r^i + \sum_{j=1}^{m} d_{-j} \cdot r^{-j}$

[!NOTE] In Hexadecimal, alphabetic letters represent values from 10 to 15:
$\text{A} = 10, \quad \text{B} = 11, \quad \text{C} = 12, \quad \text{D} = 13, \quad \text{E} = 14, \quad \text{F} = 15$

2. Step-by-Step Conversion Algorithms and Numerical Methods

A. Decimal to Any Base (Successive Division / Multiplication Method)

1. Integer Part: Continuously divide the decimal integer by the target base $r$. Record the integer remainder at each stage until the quotient becomes 0. Read remainders from bottom to top (Most Significant Digit [MSD] to Least Significant Digit [LSD]).
2. Fractional Part: Continuously multiply the fractional part by the target base $r$. Record the generated integer digit at each step. Multiply only the remaining fractional component until it reaches 0 or achieves desired precision. Read generated integers from top to bottom.

Worked Example 1: Convert $(75.625){10}$ to Binary $(?){2}$

• Integer Conversion ($75 \div 2$):
  • $75 \div 2 = 37$, remainder 1 (LSD)
  • $37 \div 2 = 18$, remainder 1
  • $18 \div 2 = 9$, remainder 0
  • $9 \div 2 = 4$, remainder 1
  • $4 \div 2 = 2$, remainder 0
  • $2 \div 2 = 1$, remainder 0
  • $1 \div 2 = 0$, remainder 1 (MSD)
  • Reading bottom-up: $(75)_{10} = (1001011)_2$
• Fractional Conversion ($0.625 \times 2$):
  • $0.625 \times 2 = \mathbf{1}.25 \rightarrow$ Integer 1
  • $0.25 \times 2 = \mathbf{0}.50 \rightarrow$ Integer 0
  • $0.50 \times 2 = \mathbf{1}.00 \rightarrow$ Integer 1
  • Reading top-down: $(0.625)_{10} = (0.101)_2$
• Combined Result: $(75.625)_{10} = (1001011.101)_2$

B. Direct Grouping Methods (Binary $\leftrightarrow$ Octal $\leftrightarrow$ Hexadecimal)

Because $8 = 2^3$ and $16 = 2^4$, conversions between Binary, Octal, and Hexadecimal bypass decimal arithmetic through direct bit-grouping:

• Binary to Octal: Partition the binary bitstream into groups of 3 bits starting from the radix point (leftward for integers, rightward for fractions, padding with leading/trailing zeros as needed). Replace each 3-bit cluster with its octal equivalent.
• Binary to Hexadecimal: Partition the binary bitstream into groups of 4 bits starting from the radix point. Replace each 4-bit cluster with its hexadecimal digit ($0\text{--}9, \text{A--F}$).
• Octal / Hexadecimal to Binary: Expand each octal digit into its precise 3-bit binary equivalent, or each hexadecimal digit into its 4-bit binary equivalent.

Direct Binary Equivalence Reference Table:

Worked Example 2: Convert Binary $(1101011110.1011)_2$ to Hexadecimal and Octal

1. Hexadecimal Grouping (4 bits from radix point):
   • Integer part: 0011 0101 1110 (padded two leading zeros to complete left cluster)
     • 0011 = $3$, 0101 = $5$, 1110 = $\text{E} (14)$
   • Fractional part: 1011 = $\text{B} (11)$
   • Hexadecimal result: $(35\text{E}.\text{B})_{16}$
2. Octal Grouping (3 bits from radix point):
   • Integer part: 001 101 011 110 (padded two leading zeros)
     • 001 = $1$, 101 = $5$, 011 = $3$, 110 = $6$
   • Fractional part: 101 100 (padded two trailing zeros)
     • 101 = $5$, 100 = $4$
   • Octal result: $(1536.54)_8$

Worked Example 3: Convert Hexadecimal $(2\text{C}7)_{16}$ to Decimal

3. Hierarchy of Digital Data Storage Units

Digital data is quantified in discrete binary units. UGC NET questions frequently test hierarchical sequencing and conversions across both binary (IEC) and decimal (SI) notations:

Elementary Storage Units

• Bit (b): Binary Digit; the fundamental atomic unit of digital information, representing either a 0 (low voltage / off) or a 1 (high voltage / on).
• Nibble: A cluster of 4 contiguous bits (or half a byte). One hexadecimal digit precisely maps to one nibble.
• Byte (B): A collection of 8 bits. The universal basic addressable unit of computer memory, capable of representing $2^8 = 256$ distinct values (e.g., one standard ASCII character).
• Word: The natural data width processed by a computer's CPU architecture in a single clock cycle (e.g., 16-bit word = 2 bytes; 32-bit word = 4 bytes; 64-bit word = 8 bytes).

Storage Capacity Hierarchy (Powers of 1024 vs. Powers of 1000)

In computer architecture, data storage scales in powers of 2 ($2^{10} = 1024$), whereas telecommunication bandwidth and commercial drive manufacturers often use decimal SI powers of 10 ($10^3 = 1000$).

[!TIP] Memory Scale Mnemonic (Ascending Order):
Bit $\rightarrow$ Nibble $\rightarrow$ Byte $\rightarrow$ Kilo $\rightarrow$ Mega $\rightarrow$ Giga $\rightarrow$ Tera $\rightarrow$ Peta $\rightarrow$ Exa $\rightarrow$ Zetta $\rightarrow$ Yotta
(Remember: Kids Make Great Teachers, Polishing Every Zesty Yarn)

4. File Formats, MIME Types & Compression Paradigms

Digital files are encoded using specialized formatting structures and compression algorithms tailored to their respective media categories.

Lossy vs. Lossless Compression

• Lossy Compression: Permanently discards mathematically redundant or perceptually imperceptible information (e.g., frequencies outside human hearing range, subtle color variations). Yields very high compression ratios but prevents bit-for-bit reconstruction of the original file (ideal for web streaming and consumer multimedia).
• Lossless Compression: Reduces file size by identifying and eliminating statistical redundancy using encoding algorithms (e.g., Run-Length Encoding, Huffman Coding, LZW). Allows 100% bit-for-bit reconstruction of original source data upon decompression (mandatory for text, executables, medical imaging, and archival records).

Comprehensive Media Format Classification