Number systems
1. The Concept of Base
A number system is defined by its base (or radix), which dictates the number of unique digits available and the positional value of each digit.
The Rule: For any system with base $n$, the digits range from $0$ to $n-1$.
For example, Base 2 (Binary) uses only $0$ and $1$.
2. Key Systems Comparison
| System | Base | Digits/Symbols | Primary Purpose |
|---|---|---|---|
| Denary | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | Human-centric calculations. |
| Binary | 2 | 0, 1 | Internal processing (Logic gates/Transistors). |
| Hexadecimal | 16 | 0-9 and A-F | Memory addresses, Color codes (HTML), MAC addresses. |
3. Binary (Base 2)
Computers use binary because they are made of electronic circuits that can only exist in two states: High Voltage (1) or Low Voltage (0).
Place Values
Binary values are calculated using powers of 2. For a standard 8-bit byte:
| 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
4. Hexadecimal (Base 16)
Hexadecimal uses 16 symbols. After 9, we use letters to represent values from 10 to 15:
A=10 | B=11 | C=12 | D=13 | E=14 | F=15
Why do we use Hex?
Many students mistakenly think computers "understand" Hex. They do not. We use it for the following human-centric reasons:
- Readability: It is much easier to read
3Fthan00111111. - Efficiency: It is faster to type and takes up less screen space.
- Error Reduction: Humans are less likely to make mistakes when copying Hex compared to long strings of 1s and 0s.
- Easy Mapping: One Hex digit perfectly represents exactly one nibble (4 bits).
5. Data Storage Units
The IGCSE syllabus requires knowledge of the difference between Base 10 (Metric) and Base 2 (IEC) units:
- Bit: The smallest unit of data.
- Byte: 8 bits.
- Kibibyte (KiB): 1024 bytes ($2^{10}$).
- Mebibyte (MiB): 1024 KiB ($2^{20}$).
- Gibibyte (GiB): 1024 MiB ($2^{30}$).