1.1 Data Representation

Binary Magnitudes and Prefixes

Binary vs Decimal Prefixes

Understanding the difference between binary and decimal prefixes is crucial, especially when dealing with storage capacities.

Binary Prefix	Value	Decimal Prefix	Value
Kibi (Ki)	2¹⁰ = 1,024	Kilo (K)	10³ = 1,000
Mebi (Mi)	2²⁰ = 1,048,576	Mega (M)	10⁶ = 1,000,000
Gibi (Gi)	2³⁰ = 1,073,741,824	Giga (G)	10⁹ = 1,000,000,000
Tebi (Ti)	2⁴⁰ = 1,099,511,627,776	Tera (T)	10¹² = 1,000,000,000,000

Why this matters:

Operating systems often report storage using binary prefixes (e.g., Windows shows 1 GiB as “1 GB” even though it’s actually 1.074 GB)
Hard drive manufacturers typically use decimal prefixes to make capacities sound larger
A “500 GB” hard drive actually has 500 × 10⁹ bytes ≈ 465.66 GiB of actual storage

Number Systems

1. Denary (Base-10)

Uses digits 0-9
Each place value is a power of 10
Example: 345₁₀ = (3 × 10²) + (4 × 10¹) + (5 × 10⁰)

2. Binary (Base-2)

Uses digits 0 and 1 (bits)
Each place value is a power of 2
Fundamental to all computer operations
Example: 1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀

3. Hexadecimal (Base-16)

Uses digits 0-9 and letters A-F
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
Each place value is a power of 16
Example: 2F₁₆ = (2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47₁₀

Why use hexadecimal?

More compact representation than binary
Easier for humans to read and remember
Each hex digit represents exactly 4 bits (a nibble)
Commonly used in memory addresses, MAC addresses, colour codes

4. Binary Coded Decimal (BCD)

Each denary digit is represented by its 4-bit binary equivalent
Not an efficient use of space but useful for displays and precise decimal calculations

Denary	BCD
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001

Example: 39₁₀ in BCD = 0011 1001 (each digit encoded separately)

Note: 1010, 1011, 1100, 1101, 1110, 1111 are invalid in BCD

Number System Conversions

Binary to Denary

Multiply each bit by its place value (powers of 2) and sum.

Example: Convert 1101₂ to denary

1   1   0   1
2³  2²  2¹  2⁰
8 + 4 + 0 + 1 = 13₁₀

Denary to Binary

Divide repeatedly by 2, reading remainders from bottom to top.

Example: Convert 25₁₀ to binary

25 ÷ 2 = 12 remainder 1 ↑
12 ÷ 2 = 6  remainder 0 |
6 ÷ 2 = 3  remainder 0 |
3 ÷ 2 = 1  remainder 1 |
1 ÷ 2 = 0  remainder 1 |

Reading upwards: 11001₂

Binary to Hexadecimal

Group binary digits into groups of 4 (from right), convert each group.

Example: Convert 10110111₂ to hexadecimal

1011   0111
 ↓       ↓
 B       7

Therefore: 10110111₂ = B7₁₆

Hexadecimal to Binary

Convert each hex digit to its 4-bit binary equivalent.

Example: Convert 3F₁₆ to binary

3       F
 ↓       ↓
0011   1111

Therefore: 3F₁₆ = 00111111₂

Hexadecimal to Denary

Multiply each digit by its place value (powers of 16) and sum.

Example: Convert A3₁₆ to denary

A    3
16¹  16⁰
10 × 16¹ = 160
3  × 16⁰ =   3
Total   = 163₁₀

Denary to Hexadecimal

Divide repeatedly by 16, reading remainders from bottom to top.

Example: Convert 200₁₀ to hexadecimal

200 ÷ 16 = 12 remainder 8 ↑
12 ÷ 16 = 0  remainder 12 (C) |

Reading upwards: C8₁₆

Binary Addition and Subtraction

Binary Addition Rules

A	B	Sum	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Example: Add 1011₂ (11₁₀) and 1101₂ (13₁₀)

   1 0 1 1
+  1 1 0 1
-----------
 1 1 0 0 0

Check: 11 + 13 = 24, and 11000₂ = 24₁₀ ✓

Binary Subtraction

Can be performed using two’s complement (see below) or direct borrowing:

Example: Subtract 0110₂ (6₁₀) from 1010₂ (10₁₀)

   1 0 1 0
-  0 1 1 0
-----------
   0 1 0 0

Check: 10 – 6 = 4, and 0100₂ = 4₁₀ ✓

Representing Negative Binary Numbers

Sign and Magnitude

The leftmost bit (MSB) represents the sign: 0 = positive, 1 = negative
Remaining bits represent the magnitude

Example with 4 bits:

+5 = 0101
-5 = 1101

Problems:

Two representations for zero: 0000 (+0) and 1000 (-0)
Arithmetic is complicated

One’s Complement

Positive numbers are represented normally
Negative numbers are obtained by inverting all bits (0→1, 1→0)

Example with 4 bits:

+5 = 0101
-5 = 1010 (invert all bits)

Problems:

Still has two zeros: 0000 and 1111
End-around carry needed in arithmetic

Two’s Complement (Most Important!)

Positive numbers are represented normally
Negative numbers are obtained by:

Write the positive number in binary
Invert all bits (one’s complement)
Add 1 to the result

Example: Find -5 in two’s complement (4 bits)

Step 1: +5 = 0101
Step 2: Invert = 1010
Step 3: Add 1 = 1011

Therefore: -5 = 1011₂

Converting back: 1011₂

Step 1: Subtract 1 = 1010
Step 2: Invert = 0101 = +5
Therefore: 1011₂ = -5

Range for n-bit two’s complement:

Minimum: -2ⁿ⁻¹
Maximum: 2ⁿ⁻¹ – 1

For 4 bits: -8 to +7
For 8 bits: -128 to +127
For 16 bits: -32,768 to +32,767

Advantages of two’s complement:

Only one representation for zero (0000)
Addition works the same for positive and negative numbers
No special hardware needed for subtraction

Binary Addition with Two’s Complement

Example: Add +5 (0101) and -3 (1101)

First, find -3:
+3 = 0011
Invert = 1100
Add 1 = 1101

Now add:
  0101  (+5)
+ 1101  (-3)
-----------
 10010
  ↑
  Discard carry beyond 4 bits
-----------
Result: 0010 = +2 ✓

Overflow

What is overflow?
Overflow occurs when the result of an arithmetic operation is too large to fit in the available number of bits.

When does it happen?

When adding two positive numbers and the result becomes negative
When adding two negative numbers and the result becomes positive
When the carry into the sign bit differs from the carry out of the sign bit

Example with 4 bits (range -8 to +7):

Adding +5 (0101) and +4 (0100):

  0101 (+5)
+ 0100 (+4)
-----------
  1001

Result 1001 = -7 in two’s complement (incorrect!)
Overflow has occurred because 5 + 4 = 9, which exceeds +7.

Detecting overflow:
If carry into MSB ≠ carry out of MSB, overflow has occurred.

Practical Applications

Binary Coded Decimal (BCD) Applications

Digital Displays

Calculators, digital clocks, and measuring instruments
Each digit can be displayed independently

Financial Systems

Avoids rounding errors in decimal calculations
Used in banking and accounting software

Real-time Clocks

Time is naturally represented in decimal (hours, minutes, seconds)

Hexadecimal Applications

Memory Addresses

Debuggers show memory addresses in hex
Example: 0x7FFF is easier to read than 0111111111111111

MAC Addresses

48-bit addresses shown as 12 hex digits
Example: 00:1A:2B:3C:4D:5E

Colour Codes in Web Design

RGB colours in hex: #FF0000 = red
#FFFFFF = white, #000000 = black

Assembly Language and Machine Code

More readable than binary
Each hex digit represents exactly 4 bits

Error Codes

Blue Screen of Death shows error codes in hex
Memory dumps use hexadecimal

Character Representation

ASCII (American Standard Code for Information Interchange)

Standard ASCII (7-bit)

Uses 7 bits to represent 128 characters
Includes:
Uppercase letters A-Z (65-90)
Lowercase letters a-z (97-122)
Digits 0-9 (48-57)
Punctuation marks
Control characters (0-31)

Example ASCII codes:

‘A’ = 65₁₀ = 1000001₂ = 41₁₆
‘a’ = 97₁₀ = 1100001₂ = 61₁₆
‘0’ = 48₁₀ = 0110000₂ = 30₁₆
Space = 32₁₀ = 0100000₂ = 20₁₆

Extended ASCII (8-bit)

Uses 8 bits to represent 256 characters
First 128 characters same as standard ASCII
Additional 128 characters include:
Accented letters (é, ñ, ü)
Special symbols (£, ©, ®)
Line drawing characters
Mathematical symbols

Problem: Different manufacturers used different extended character sets → incompatibility issues

Unicode

Why Unicode was needed:

ASCII only supports Western languages
Need to represent all world’s writing systems
Single unified standard

Key features:

Can use 16, 24, or 32 bits per character
Over 140,000 characters supported
Includes emojis and ancient scripts
First 128 characters match ASCII for backward compatibility

Common encodings:

UTF-8: Variable length (1-4 bytes), backward compatible with ASCII
UTF-16: Uses 2 or 4 bytes
UTF-32: Fixed 4 bytes per character

Examples:

‘A’ = U+0041 (same as ASCII)
‘€’ (euro symbol) = U+20AC
‘中’ (Chinese character) = U+4E2D
‘😊’ (emoji) = U+1F60A

1.2 Multimedia

Graphics

Bitmap Images

Key Terminology:

Term	Definition
Pixel	Picture element; the smallest addressable element in a display device
File Header	Contains metadata about the image (dimensions, colour depth, file type)
Image Resolution	The number of pixels in an image (width × height)
Screen Resolution	The number of pixels a monitor can display
Colour Depth / Bit Depth	Number of bits used to represent each pixel’s colour

How Bitmap Data is Encoded:

Image divided into a grid of pixels
Each pixel stores colour information as binary data
Number of bits per pixel = colour depth
Header contains dimensions and other metadata
Pixel data stored sequentially (row by row)

File Size Calculation for Bitmap Images

Formula:

File Size = Image Resolution × Colour Depth
          = (Width × Height) × Colour Depth

Example 1: Calculate file size for 1024 × 768 image with 24-bit colour

Resolution = 1024 × 768 = 786,432 pixels
Colour depth = 24 bits = 3 bytes
File size = 786,432 × 3 = 2,359,296 bytes
          = 2,359,296 ÷ 1024 = 2,304 KiB
          = 2,304 ÷ 1024 = 2.25 MiB

Example 2: 1920 × 1080 (Full HD) with 32-bit colour

Resolution = 1920 × 1080 = 2,073,600 pixels
Colour depth = 32 bits = 4 bytes
File size = 2,073,600 × 4 = 8,294,400 bytes
          = 8,294,400 ÷ 1024 = 8,100 KiB
          = 8,100 ÷ 1024 = 7.91 MiB

Effects of Changing Elements

Change	Effect on Quality	Effect on File Size
Increase image resolution	Higher quality, more detail	Increases significantly
Decrease image resolution	Lower quality, pixelation	Decreases significantly
Increase colour depth	More colours, smoother gradients	Increases
Decrease colour depth	Fewer colours, banding	Decreases

Colour Depth Examples:

1-bit: 2 colours (black and white)
4-bit: 16 colours
8-bit: 256 colours
16-bit: 65,536 colours (High Colour)
24-bit: 16.7 million colours (True Colour)
32-bit: 16.7 million colours + alpha channel (transparency)

Vector Graphics

How Vector Graphics are Encoded:

Images are stored as mathematical formulas and geometric objects
Each object has properties and is stored in a drawing list

Key Terminology:

Term	Definition
Drawing Object	A geometric shape (line, circle, rectangle, polygon)
Property	Attributes of an object (colour, line thickness, fill, position)
Drawing List	Collection of all objects that make up the image

Common Vector Objects and Their Properties:

Circle:
- Centre coordinates (x, y)
- Radius
- Fill colour
- Outline colour
- Outline thickness

Rectangle:
- Top-left corner (x, y)
- Width, Height
- Fill colour
- Outline colour
- Rotation angle

Line:
- Start point (x1, y1)
- End point (x2, y2)
- Colour
- Thickness

Vector File Formats:

SVG (Scalable Vector Graphics)
EPS (Encapsulated PostScript)
AI (Adobe Illustrator)
CDR (CorelDRAW)

Bitmap vs Vector Graphics: Comparison

Feature	Bitmap	Vector
Composition	Grid of pixels	Mathematical formulas
Scaling	Loses quality when enlarged (pixelates)	Can be scaled infinitely without quality loss
File Size	Large, depends on resolution	Small, depends on complexity
Editing	Pixel-level editing possible	Edit objects and properties
Realism	Excellent for photographs	Better for illustrations, logos
Conversion	Can be converted to vector (tracing)	Can be converted to bitmap (rasterisation)
Common Uses	Photos, scanned images	Logos, diagrams, fonts, illustrations

Justifying Image Type for a Given Task

Choose Bitmap when:

Working with photographs or realistic images
Need complex colour variations and gradients
Image source is from a camera or scanner
Need pixel-level editing
Web photos and digital art

Choose Vector when:

Creating logos that need to scale to different sizes
Designing diagrams, flowcharts, or technical drawings
Creating fonts and text
Image will be used at multiple resolutions
File size needs to be minimised
Need precise, clean lines

Sound

How Sound is Represented and Encoded

Key Terminology:

Term	Definition
Analogue Data	Continuous signal that varies over time
Digital Data	Discrete samples representing the analogue signal
Sampling	Process of converting analogue to digital by taking measurements at intervals
Sampling Rate	Number of samples taken per second (measured in Hz or kHz)
Sampling Resolution	Number of bits used to store each sample (bit depth)

The Analogue-to-Digital Conversion Process

Sound waves (analogue) are picked up by a microphone
Sampling: Measurements taken at regular intervals (sampling rate)
Quantisation: Each sample assigned a digital value (sampling resolution)
Encoding: Digital values stored as binary data

Example:

CD quality: 44.1 kHz sampling rate, 16-bit resolution
44,100 samples per second, each stored in 16 bits

File Size Calculation for Sound

Formula:

File Size = Sampling Rate × Sampling Resolution × Duration × Channels

Example: Calculate file size for 3-minute stereo song at CD quality

Sampling rate = 44,100 Hz
Resolution = 16 bits = 2 bytes
Duration = 3 minutes = 180 seconds
Channels = 2 (stereo)

File size = 44,100 × 2 × 180 × 2
          = 44,100 × 720
          = 31,752,000 bytes
          = 31,752,000 ÷ 1024 = 31,008 KiB
          = 31,008 ÷ 1024 = 30.28 MiB

Impact of Changing Sampling Parameters

Increasing Sampling Rate:

More accurate representation of high frequencies
Higher quality recording
Larger file size

Increasing Sampling Resolution:

More precise amplitude values
Better dynamic range (quieter and louder sounds)
Reduced quantisation noise
Larger file size

Nyquist Theorem:
To accurately represent a frequency, you must sample at at least twice that frequency. This is why CD quality (44.1 kHz) can represent up to 22.05 kHz (just above human hearing range of 20 kHz).

Quality	Sampling Rate	Resolution	Use Case
Telephone	8 kHz	8-bit	Voice calls
Radio	22.05 kHz	8-bit	AM-quality audio
CD	44.1 kHz	16-bit	Music albums
DVD	48 kHz	16/24-bit	Movies
Studio	96 kHz	24/32-bit	Professional recording

1.3 Compression

Need for Compression

Why compress files?

Reduce storage space requirements
Faster file transfer over networks
Reduce bandwidth usage when streaming
Make more efficient use of limited resources
Enable storage of more files on devices

Examples where compression is essential:

Streaming video (Netflix, YouTube)
Downloading software
Sending email attachments
Storing photos on phones
Music streaming services
Video conferencing

Lossy vs Lossless Compression

Lossless Compression

How it works:

Reduces file size without losing any original data
Original file can be perfectly reconstructed
Works by identifying and eliminating redundancy

Common methods:

Run-Length Encoding (RLE)
Huffman coding
LZW compression

Advantages:

No quality loss
Original data preserved exactly
Essential for text, programs, critical data

Disadvantages:

Compression ratio limited (typically 2:1 to 4:1)
Less effective than lossy for some media types

Lossy Compression

How it works:

Permanently removes some data considered less important
Original cannot be perfectly reconstructed
Explores limitations of human perception

Advantages:

Much higher compression ratios (10:1 to 100:1)
Smaller file sizes
Good for media where perfect reproduction isn’t needed

Disadvantages:

Permanent quality loss
Cannot recover original data
Visible/audible artefacts at high compression

Run-Length Encoding (RLE)

How RLE works:

Replaces repeated consecutive characters with a count and the character
Works well for images with large areas of same colour
Simple lossless compression method

Text Example:

Original:    "AAAABBBCCDAA"
Encoded:     "4A3B2C1D2A"

Compression: 12 characters → 10 characters

Bitmap Image Example:

Original pixels: 5 white, 3 black, 4 white
Without RLE:     WWWWWBBBWWWW
With RLE:        5W3B4W

Advantages of RLE:

Simple to implement
Fast compression and decompression
Effective for simple images (icons, line art, text)

Disadvantages:

Inefficient for complex images with frequent changes
Can actually increase file size for “noisy” data

Compression by File Type

Text File Compression

Methods used:

Run-Length Encoding (RLE)
Dictionary-based compression (LZW)
Huffman coding (variable-length codes)

How it works:

Find repeated patterns or words
Replace with shorter references
Build dictionary of common sequences

Example: “the cat sat on the mat”

“the” appears twice → store once, reference twice
Common in ZIP, gzip formats

Bitmap Image Compression

Lossless methods:

RLE: Effective for images with large uniform areas
LZW: Used in GIF and TIFF formats
PNG: Uses DEFLATE algorithm (combination of LZ77 and Huffman)

Lossy methods:

JPEG:
Divides image into 8×8 blocks
Applies Discrete Cosine Transform (DCT)
Quantisation (removes less visible detail)
Can achieve 10:1 to 20:1 compression

Vector Graphic Compression

How vectors compress well:

Already stored as compact mathematical formulas
File size depends on complexity, not dimensions
SVG files can be compressed with gzip (SVGZ)

Compression methods:

Remove unnecessary whitespace
Simplify paths (reduce number of points)
Use relative instead of absolute coordinates
Reuse definitions (symbols, patterns)

Sound File Compression

Lossless audio:

FLAC (Free Lossless Audio Codec)
ALAC (Apple Lossless)
Typically 2:1 to 3:1 compression

Lossy audio:

MP3 (MPEG-1 Audio Layer 3)
AAC (Advanced Audio Coding)
Uses psychoacoustic models to remove inaudible sounds
Typical compression: 10:1 (CD quality → 1.4 Mbps → 128 kbps)

How lossy audio compression works:

Analysed in frequency domain
Masking effects applied (louder sounds hide quieter ones)
Less audible frequencies given fewer bits
Result encoded efficiently

Justifying Compression Methods

When to use Lossless Compression:

Text documents and source code
Executable programs
Spreadsheets with critical data
Medical images (X-rays, MRI scans)
Archival purposes
When any data loss is unacceptable

When to use Lossy Compression:

Streaming video (Netflix, YouTube)
Music streaming (Spotify, Apple Music)
Sharing photos on social media
Video calls (Zoom, Skype)
When bandwidth/storage is limited
When slight quality loss is acceptable

Summary Checklist for Assessment Objectives

AO1 (Knowledge) – You should be able to:

✓ Define binary prefixes (kibi, mebi, gibi, tebi)
✓ Describe different number systems
✓ Explain BCD and two’s complement
✓ Define pixel, resolution, colour depth
✓ Define sampling, sampling rate, resolution
✓ Explain need for compression
✓ Define lossy and lossless compression
✓ Describe RLE

AO2 (Application) – You should be able to:

✓ Convert between number bases
✓ Perform binary addition/subtraction
✓ Identify overflow
✓ Calculate bitmap file sizes
✓ Calculate sound file sizes
✓ Apply RLE to given data
✓ Justify compression method for given scenario

AO3 (Design/Evaluation) – You should be able to:

✓ Evaluate effects of changing image parameters
✓ Compare bitmap vs vector for tasks
✓ Evaluate impact of changing sound parameters
✓ Justify image/sound format choices
✓ Compare lossy vs lossless for situations