Number Systems
Computers use binary (base-2) to store and process all data. You need to understand three number systems and how to convert between them.
Denary (Base-10)
The number system we use every day.
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Example: 156 = (1×100) + (5×10) + (6×1)
Binary (Base-2)
The language of computers - only 0s and 1s.
Digits: 0, 1
Example: 1010 = 8 + 0 + 2 + 0 = 10
Hexadecimal (Base-16)
Shorthand for binary - used for colours, memory addresses.
Digits: 0-9, A, B, C, D, E, F
Example: 2F = (2×16) + (15×1) = 47
Binary Place Values
Each position in a binary number represents a power of 2:
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | Denary |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 10 |
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 101 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 255 |
Data Units
Key Definitions
- Bit - A single binary digit (0 or 1)
- Nibble - 4 bits
- Byte - 8 bits (can store values 0-255)
- Kilobyte (KB) - 1,000 bytes (or 1,024 bytes)
- Megabyte (MB) - 1,000 KB
- Gigabyte (GB) - 1,000 MB
- Terabyte (TB) - 1,000 GB
Hexadecimal Conversion
Hexadecimal uses 16 digits. Letters A-F represent values 10-15:
| Hex | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Denary | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Why Use Hexadecimal?
- Shorter: One hex digit = 4 binary digits (1 nibble)
- Easier to read: FF is easier than 11111111
- Common uses: Colour codes (#FF0000), MAC addresses, memory addresses
Craig 'n' Dave: Binary
Understanding binary number system
Craig 'n' Dave: Hexadecimal
Hexadecimal and binary conversions explained
Convert the binary number 11001010 to:
a) Denary [2 marks]
b) Hexadecimal [1 mark]
Show Mark Scheme Answer
a) Denary:
128 + 64 + 0 + 0 + 8 + 0 + 2 + 0 = 202
b) Hexadecimal:
Split into nibbles: 1100 1010 = C A = CA
Binary Arithmetic
Binary Addition
Binary addition follows these rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10
(0, carry 1)
Worked Example: 00110101 + 00011011
00110101 (53)
+ 00011011 (27)
----------
01010000 (80)
1111 (carries)
Overflow
Overflow Error
Overflow occurs when the result of a calculation is too large to fit in the available number of bits.
Example: In 8-bit binary, the maximum value is 255. If 200 + 100 = 300, this causes overflow because 300 > 255.
Binary Shifts
Shifting bits left or right is a quick way to multiply or divide by powers of 2.
Left Shift
Multiplies by 2 for each shift
00001010 (10)
00010100 (20) ← shift left 1
00101000 (40) ← shift left 2
Right Shift
Divides by 2 for each shift (integer division)
00101000 (40)
00010100 (20) → shift right 1
00001010 (10) → shift right 2
Exam Tip
When shifting, bits that "fall off" the end are lost. New bits added are always 0. A left shift can cause overflow if a 1 is shifted out of the most significant bit position.
Craig 'n' Dave: Binary Addition
How to add binary numbers
Craig 'n' Dave: Binary Shifts
Left and right shifts explained
Craig 'n' Dave: Two's Complement
Representing negative numbers in binary
Representing Text
Computers store text as numbers using character sets - tables that map each character to a unique number.
ASCII
Definition
ASCII (American Standard Code for Information Interchange) uses 7 bits to represent 128 characters (0-127), including letters, numbers, punctuation, and control characters.
Common ASCII values to remember:
| Character | Denary | Binary |
|---|---|---|
| A | 65 | 01000001 |
| Z | 90 | 01011010 |
| a | 97 | 01100001 |
| z | 122 | 01111010 |
| 0 | 48 | 00110000 |
| Space | 32 | 00100000 |
Unicode
Definition
Unicode is a universal character set that can represent characters from all languages in the world, plus emojis and symbols. It uses up to 32 bits per character, allowing for over 1 million characters.
ASCII
- 7 bits = 128 characters
- English only
- Smaller file sizes
- Faster to process
Unicode
- Up to 32 bits = 1+ million characters
- All world languages
- Includes emojis
- Larger file sizes
Craig 'n' Dave: Representing Text
ASCII and Unicode character sets explained
Representing Images
Key Definitions
Digital images are made up of pixels (picture elements). Each pixel's colour is stored as a binary number.
Key Terms
Resolution
The number of pixels in an image, usually expressed as width × height (e.g., 1920 × 1080).
Higher resolution = more detail but larger file size.
Colour Depth
The number of bits used to store each pixel's colour.
- 1-bit = 2 colours (black/white)
- 8-bit = 256 colours
- 24-bit = 16.7 million colours
Calculating Image File Size
Formula
File Size (bits) = Width × Height × Colour Depth
Example: A 800 × 600 image with 24-bit colour:
800 × 600 × 24 = 11,520,000 bits = 1,440,000 bytes = 1.44 MB
Metadata
Metadata is additional data stored with an image file, including:
- File format, dimensions, colour depth
- Date and time the photo was taken
- Camera settings (for photos)
- GPS location (for phones/cameras)
Craig 'n' Dave: Representing Images
Pixels, resolution, and colour depth
Calculate the file size in bytes of an image that is 640 × 480 pixels with a colour depth of 8 bits. Show your working.
Show Mark Scheme Answer
File size = width × height × colour depth
= 640 × 480 × 8 [1 mark]
= 2,457,600 bits [1 mark]
= 2,457,600 ÷ 8 = 307,200 bytes [1 mark]
Representing Sound
Sound is an analogue signal (continuous wave). To store it digitally, we must sample it at regular intervals.
Key Terms
Sample Rate
The number of samples taken per second, measured in Hertz (Hz).
CD quality: 44,100 Hz (44.1 kHz)
Higher sample rate = better quality, larger file
Bit Depth
The number of bits per sample, determining the range of values each sample can have.
CD quality: 16-bit
Higher bit depth = more accurate, larger file
Calculating Audio File Size
Formula
File Size (bits) = Sample Rate × Bit Depth × Duration (seconds)
Example: 30 seconds of audio at 44,100 Hz with 16-bit depth:
44,100 × 16 × 30 = 21,168,000 bits = 2,646,000 bytes ≈ 2.65 MB
Exam Tip
For stereo audio, multiply by 2 (left and right channels). The formula becomes:
File Size = Sample Rate × Bit Depth × Duration × Channels
Craig 'n' Dave: Representing Sound
Sample rate, bit depth, and file sizes
Compression
Definition
Compression reduces the size of a file so it takes up less storage space and can be transmitted faster. There are two types: lossy and lossless.
Lossy Compression
Permanently removes some data to reduce file size. Cannot be reversed.
Characteristics:
- Smaller file sizes
- Some quality is lost
- Original cannot be recovered
Examples:
- JPEG (images)
- MP3 (audio)
- MP4 (video)
Best for:
Streaming, web images, when quality loss is acceptable
Lossless Compression
Reduces file size without losing any data. Fully reversible.
Characteristics:
- Larger than lossy (but smaller than original)
- No quality loss
- Original can be perfectly reconstructed
Examples:
- PNG (images)
- FLAC (audio)
- ZIP (files)
Best for:
Text, code, medical images, when quality matters
Run Length Encoding (RLE)
A simple lossless compression technique that replaces repeated data with a count and value.
Example
Original: AAAAAABBBBCCCCCCCC
Compressed: 6A4B8C
18 characters reduced to 6 characters!
Craig 'n' Dave: Compression
Lossy vs lossless compression techniques
Explain the difference between lossy and lossless compression. Give an example of when each type would be appropriate.
Show Mark Scheme Answer
Lossy [2 marks]:
- Permanently removes data/reduces quality
- Appropriate for: streaming media, web images, where smaller size matters more than perfect quality
Lossless [2 marks]:
- No data is lost/original can be reconstructed
- Appropriate for: text files, program code, medical images, where accuracy is essential
Topic 2 Quiz
Test your knowledge of data representation with this quiz. Download your PDF certificate when complete!