2. Number Systems: Decimal, Binary, and Hexadecimal
Digital systems rely heavily on different number systems to represent data. We must understand how to move between them.
The Decimal System (Base 10)
We use the decimal system daily. It has 10 unique digits (0-9).
$$N = d_n \times 10^n + \dots + d_1 \times 10^1 + d_0 \times 10^0$$
The Binary System (Base 2)
Computers use binary, consisting only of 0 and 1. Each position represents a power of 2.
| Position | Weight | Example (1011) |
|---|---|---|
| 3 | $2^3 = 8$ | 1 |
| 2 | $2^2 = 4$ | 0 |
| 1 | $2^1 = 2$ | 1 |
| 0 | $2^0 = 1$ | 1 |
$$1011_2 = (1 \times 8) + (0 \times 4) + (1 \times 2) + (1 \times 1) = 11_{10}$$
The Hexadecimal System (Base 16)
Hexadecimal (Hex) uses 16 symbols (0-9 and A-F). It is used as a shorthand for binary because four binary digits (bits) map directly to one hex digit.
| Decimal | Binary | Hexadecimal |
|---|---|---|
| 10 | 1010 | A |
| 15 | 1111 | F |
Conversion Exercise (Binary to Decimal)
Convert $11010_2$ to decimal:
$$(1 \times 16) + (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) = 16 + 8 + 2 = 26_{10}$$