What is a Number Base?

A number base (or radix) defines how many unique digits a positional number system uses. Each digit position represents a power of the base.

Positional Notation

In a base-n system, each digit position has a value of n raised to a power:

Binary: 1011₂
= 1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 8  +  0  +  2  +  1  = 11₁₀

Hexadecimal: 2A₁₆
= 2×16¹ + A×16⁰
= 32  + 10  = 42₁₀

Binary (Base 2)

Binary is the foundation of all computing — transistors have two states (on/off) mapping to 1 and 0.

Converting decimal to binary

Divide repeatedly by 2, recording remainders from bottom to top:

Convert 43₁₀ to binary:
43 ÷ 2 = 21 remainder 1
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

Read remainders upward: 43₁₀ = 101011₂

Converting binary to decimal

101011₂ = 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 1×1
        = 32 + 0 + 8 + 0 + 2 + 1 = 43₁₀

Hexadecimal (Base 16)

Hexadecimal is a compact way to represent binary — each hex digit represents exactly 4 bits (a nibble).

Binary ↔ Hexadecimal conversion

Binary to Hex — group bits in 4s from the right:
10110101₂ → 1011 | 0101 → B | 5 → B5₁₆

Hex to Binary — expand each digit to 4 bits:
3F₁₆ → 0011 | 1111 → 00111111₂

Hex to decimal

B5₁₆ = 11×16 + 5×1 = 176 + 5 = 181₁₀

Decimal to hex

Convert 255₁₀ to hex:
255 ÷ 16 = 15 remainder 15 → F
 15 ÷ 16 =  0 remainder 15 → F
Read upward: 255₁₀ = FF₁₆

Why Use Hexadecimal?

• Much more compact than binary — 1 hex digit = 4 binary bits
• Easier for humans to read and write than long binary strings
• Used for: memory addresses (e.g. 0xFF3C10), colour codes (#FF6600), MAC addresses, machine code output