How Binary, Decimal, and Hexadecimal Number Systems Actually Work in Computing

Introduction: Positional Radix Systems

All number systems used in modern mathematics and computing are positional radix systems, where the value of each digit depends upon its position relative to the base:

$$\text{Value} = \sum_{i=0}^{n-1} d_i \times \text{Base}^i$$

The 4 Primary Computing Bases

1. Decimal (Base 10): Uses digits 0–9. Built on human anatomy (10 fingers).
2. Binary (Base 2): Uses digits 0, 1. Powers physical CPU registers, memory buses, and logic gates.
3. Hexadecimal (Base 16): Uses digits 0–9 and letters A–F (where A=10, B=11, C=12, D=13, E=14, F=15). Widely used for memory addresses, CSS hex colors (#3B82F6), and MAC addresses.
4. Octal (Base 8): Uses digits 0–7. Commonly used in Unix file system permissions (e.g., chmod 755).

Conversion Examples

Converting Decimal 42 to Binary (Repeated Division by 2):

• 42 ÷ 2 = 21 remainder 0
• 21 ÷ 2 = 10 remainder 1
• 10 ÷ 2 = 5 remainder 0
• 5 ÷ 2 = 2 remainder 1
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Reading remainders bottom-to-top: `101010₂`

Converting Binary 101010₂ Back to Decimal:

$$\begin{aligned} & 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\ &= 32 + 0 + 8 + 0 + 2 + 0 = 42 \end{aligned}$$

Test synchronized 4-base conversions and toggle individual bits on our **Decimal & Binary Converter**.