Data Representation

Data Representation — CIE A Level Computer Science 9618

1. Number Systems

Binary (base 2): Uses digits 0 and 1. Each position represents a power of 2.

Converting binary to denary:

1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀

Converting denary to binary (repeated division by 2):

45 ÷ 2 = 22 R 1

22 ÷ 2 = 11 R 0

11 ÷ 2 = 5 R 1

5 ÷ 2 = 2 R 1

2 ÷ 2 = 1 R 0

1 ÷ 2 = 0 R 1

Read remainders upward: 45₁₀ = 101101₂

Hexadecimal (base 16): Digits 0–9, A–F (A=10, B=11, C=12, D=13, E=14, F=15).

Used because 1 hex digit = 4 binary bits (a nibble).

Binary to hex: Group bits in 4s from right, convert each group.

10110111₂ = 1011 0111 = B 7 = B7₁₆

Hex to denary:

3F₁₆ = 3×16 + 15 = 48 + 15 = 63₁₀

Binary arithmetic:

• Addition: 0+0=0; 0+1=1; 1+1=10 (carry 1); 1+1+1=11 (carry 1)
• Example: 1101 + 0110 = 10011 (13 + 6 = 19 ✓)
• Overflow: Result requires more bits than available.

2. Signed Integers: Two's Complement

Two's complement represents negative numbers in binary.

For an n-bit system: Range is −2ⁿ⁻¹ to 2ⁿ⁻¹−1. For 8-bit: −128 to +127.

Finding negative of a number:

1. Write positive binary representation
2. Invert all bits (flip 0s and 1s)
3. Add 1

Example: Find −45 in 8-bit two's complement.

+45 = 00101101

Invert: 11010010

Add 1: 11010011 = −45 in two's complement

Recognising negative numbers: MSB (most significant bit) = 1 means negative.

Converting two's complement to denary:

11010011: MSB=1, so negative. Invert: 00101100; add 1: 00101101 = 45. Answer: −45.

3. Real Numbers: Floating Point

Floating point represents very large or very small numbers.

Format: Value = mantissa × 2^exponent (similar to scientific notation)

In binary: A normalised floating point number has mantissa beginning 0.1... (positive) or 1.0... (negative).

Trade-offs:

• More bits in mantissa → greater precision (more significant figures)
• More bits in exponent → greater range (larger/smaller numbers)
• Fixed total bits: increasing one reduces the other

Example issue: 0.1 cannot be represented exactly in binary floating point — causes rounding errors in financial software.

4. Text Representation

ASCII (American Standard Code for Information Interchange):

• 7-bit code: 128 characters (0–127)
• Extended ASCII: 8-bit, 256 characters
• 'A' = 65 (01000001₂); 'a' = 97; '0' = 48
• Covers: uppercase, lowercase, digits, punctuation, control characters

Unicode:

• Designed to represent every character in every language
• UTF-8: Variable length (1–4 bytes); backward compatible with ASCII
• UTF-16: 2 or 4 bytes; efficient for languages like Urdu, Arabic, Chinese
• UTF-32: Fixed 4 bytes per character; simple but wasteful

Why Unicode? ASCII cannot represent Urdu, Arabic, Chinese, Devanagari, etc. Unicode enables global software and multilingual text.

5. Image Representation

A bitmap image stores colour value of each pixel.

Key terms:

• Pixel: Smallest addressable element of an image
• Resolution: Width × Height in pixels (e.g., 1920×1080)
• Colour depth (bit depth): Number of bits per pixel
• 1 bit: monochrome (black/white only)
• 8 bits: 256 colours
• 24 bits: 16,777,216 colours ("true colour") — 8 bits each for R, G, B

File size (uncompressed):

File size = width × height × colour depth (in bits) / 8 bytes

Example: A 1920×1080, 24-bit image:

Size = 1920 × 1080 × 24 / 8 = 6,220,800 bytes ≈ 5.93 MB

Metadata: Stored with image file — includes width, height, colour depth, file format.

Vector graphics: Store geometric objects (lines, curves, shapes) as mathematical descriptions. Advantages: scale without quality loss; smaller file size for simple images. Disadvantage: complex for photographic images.

6. Sound Representation

Analogue to Digital Conversion (ADC):

1. Sampling: Measure amplitude of analogue signal at regular intervals
2. Quantisation: Round each sample to nearest available level

Key parameters:

• Sampling rate: Number of samples per second (Hz). CD quality: 44,100 Hz.
• Bit depth (resolution): Number of bits per sample. CD: 16 bits (65,536 levels).
• Higher sampling rate + higher bit depth = better quality BUT larger file size

File size:

Sound file size (bits) = sampling rate × bit depth × duration (s) × number of channels

Example: 44,100 Hz, 16-bit, 2 channels (stereo), 3 minutes:

= 44100 × 16 × 180 × 2 = 254,016,000 bits = 31.75 MB uncompressed

7. Lossless vs Lossy Compression

Lossless compression: Original data fully recoverable after decompression.

• Run-length encoding (RLE): replace repeating values with count + value
• Huffman coding: frequent characters get shorter codes
• Used for: text, program files, PNG images

Lossy compression: Some data permanently discarded; smaller files but quality reduced.

• JPEG (images): removes high-frequency detail human eye is less sensitive to
• MP3 (audio): removes frequencies outside human hearing range
• Cannot recover original; not suitable for text or program files

When to use lossy: Images/audio/video for web distribution where some quality loss acceptable.

When to use lossless: Text, code, archival quality images.