Data Representation

Data Representation

What is Data Representation?

Data representation refers to the methods used to store, process, and interpret data in a computer system. It defines how numbers, characters, and other data types are encoded in binary (0s and 1s) for CPU processing and memory storage.

Why Data Representation is Important

• Foundation of Computing: Enables computers to understand and manipulate data (e.g., numbers, text, images).

• Efficiency: Proper representation optimizes storage and processing speed.

• Accuracy: Ensures correct interpretation of data (e.g., signed vs. unsigned numbers).

• Compatibility: Standard formats allow data sharing across systems.

Types of Data Representation

Below are the primary ways data is represented in computers, with explanations and examples.

1. Numeric Representation

• Unsigned Integers:

  • Description: Represent positive numbers (including 0) using straight binary.

  • Example (4-bit): 1010 = 10 (decimal).

  • Range (n-bit): 0 to 2^n - 1 (e.g., 0 to 15 for 4-bit).

  • Use: Counters, addresses, non-negative values.

• Signed Integers (2’s Complement):

  • Description: Represent positive and negative numbers using 2’s complement for negatives.

  • Example (4-bit): 0011 = +3, 1101 = -3 (2’s complement of 0011).

  • Range (n-bit): -2^(n-1) to 2^(n-1) - 1 (e.g., -8 to +7 for 4-bit).

  • Use: Arithmetic operations, signed variables in programming.

• Floating-Point Numbers:

  • Description: Represent real numbers (e.g., decimals) using a sign, exponent, and mantissa (IEEE 754 standard).

  • Example (simplified): 1.5 = 1.1 x 2^0 (binary: sign = 0, mantissa = 1.1, exponent = 0).

  • Use: Scientific calculations, graphics, AI.

• BCD (Binary-Coded Decimal):

  • Description: Each decimal digit (0–9) is encoded in 4 bits.

  • Example: 59 = 0101 1001 (5 = 0101, 9 = 1001).

  • Use: Financial systems, calculators (avoids rounding errors).

2. Character Representation

• ASCII (American Standard Code for Information Interchange):

  • Description: Encodes characters (e.g., letters, digits) in 7 or 8 bits.

  • Example: A = 01000001 ( decimal 65), 5 = 00110101 (decimal 53).

  • Use: Text files, keyboards, basic communication.

• Unicode:

  • Description: Extends ASCII to support global characters (e.g., emojis, non-Latin scripts) using 16 or 32 bits.

  • Example: 😊 = U+1F60A (encoded as multiple bytes).

  • Use: Multilingual applications, web, modern OS.

3. Boolean Representation

• Description: Represents logical values (True/False) using 1 bit.

• Example: 1 = True, 0 = False.

• Use: Conditional statements, logic gates, flags.

4. Other Data Types

• Fixed-Point Numbers:

  • Description: Represents decimals with a fixed number of digits after the decimal point.

  • Example: 12.34 stored as binary with fixed scaling.

  • Use: Embedded systems, where floating-point is too complex.

• Images:

  • Description: Pixels encoded as RGB values (e.g., 24 bits for 8 bits each of Red, Green, Blue).

  • Example: White pixel = 11111111 11111111 11111111.

  • Use: Graphics, displays, image processing.

• Audio/Video:

  • Description: Sampled data encoded as binary (e.g., PCM for audio, MPEG for video).

  • Example: Audio sample at 44.1 kHz stored as 16-bit values.

  • Use: Streaming, multimedia applications.

Example Breakdown

• Signed Integer (2’s Complement):

  • Store -5 in 4-bit:

    • +5 = 0101.

    • 2’s complement: 1010 + 1 = 1011.

    • Result: -5 = 1011.

• ASCII Text:

  • Store “HI”:

    • H = 01001000, I = 01001001.

    • Result: 01001000 01001001.

• Floating-Point:

  • Store 3.5 (simplified IEEE 754):

    • Binary: 11.1 x 2^0 → sign = 0, mantissa = 1.1, exponent = 0.

    • Encoded in 32-bit format (details depend on standard).

How Data is Represented

• Binary Encoding: All data is ultimately stored as 0s and 1s in memory or registers.

• Bit Length: Determines range/precision (e.g., 8-bit vs. 32-bit integers).

• Endianness:

  • Big Endian: Most significant byte first (e.g., 1234 = 12 34).

  • Little Endian: Least significant byte first (e.g., 1234 = 34 12).

  • Use: Affects multi-byte data storage (e.g., integers, Unicode).

• Standards: IEEE 754 (floating-point), ASCII/Unicode (text) ensure compatibility.

Where Data Representation is Used

• CPU Operations: ALU processes numbers (e.g., addition, subtraction).

• Memory Storage: Data stored in RAM, cache, or disks.

• Programming: Languages like C, Java use specific data types (int, float, char).

• Communication: Data encoded for network transmission (e.g., ASCII, Unicode).

• Multimedia: Images, audio, and video rely on specialized formats.

Why Data Representation Matters in COA

• Performance: Efficient encoding (e.g., 2’s complement) speeds up arithmetic.

• Storage: Compact representations (e.g., BCD vs. binary) save space.

• Interoperability: Standard formats ensure data exchange across systems.

• Precision: Proper representation avoids errors (e.g., floating-point rounding).

Additional Insights

• Trade-Offs:

  • Precision vs. Storage: Floating-point uses more bits but handles decimals; integers are compact but limited.

  • Speed vs. Complexity: BCD is slower but avoids decimal errors; binary is faster.

• Errors:

  • Overflow: Exceeding bit range (e.g., 7 + 7 in 4-bit = -2).

  • Rounding: Floating-point numbers may lose precision (e.g., 0.1 + 0.2 ≠ 0.3).

• Modern Trends:

  • 64-bit Systems: Larger integers, more precise floating-point.

  • Unicode Dominance: Replaces ASCII for global applications.

  • Compression: Reduces storage for multimedia (e.g., JPEG, MP3).

• Limitations:

  • Fixed bit lengths limit number range (e.g., 32-bit max = 2^32 - 1).

  • Some formats (e.g., BCD) are less efficient for general computing.

Summary Table

Data Type	Representation	Example
Unsigned Integer	Straight binary.	1010 = 10 (4-bit).
Signed Integer	2’s complement.	1101 = -3 (4-bit).
Floating-Point	Sign, exponent, mantissa.	3.5 = 11.1 x 2^0.
BCD	4 bits per decimal digit.	59 = 0101 1001.
ASCII	7/8 bits per character.	A = 01000001.
Unicode	16/32 bits per character.	😊 = U+1F60A.

Example Breakdown: Multi-Type Data

Store “5 + 2.5”:

• Integer (5): 0101 (4-bit unsigned).

• Floating-Point (2.5): IEEE 754 (simplified: 1.01 x 2^1).

• Operation (+): ASCII + = 00101011.

• Result: Processed in ALU, stored as binary in memory.