Number base conversion: binary, decimal, hexadecimal

Common bases

Binary to decimal

Each position is worth a power of 2, starting from the right (2⁰, 2¹, 2²…).

1010 (binary)
= 1×2³ + 0×2² + 1×2¹ + 0×2⁰
= 8 + 0 + 2 + 0
= 10 (decimal)

Decimal to binary

Repeated division by 2, keeping the remainders, read bottom to top:

10 ÷ 2 = 5 remainder 0
 5 ÷ 2 = 2 remainder 1
 2 ÷ 2 = 1 remainder 0
 1 ÷ 2 = 0 remainder 1
→ 1010

Hexadecimal

Each hex digit maps exactly to 4 bits (a "nibble"), which makes binary ↔ hex conversion direct, 4 bits at a time:

1010 1111 (binary)
= A       F   (hexadecimal)

In practice

# In a terminal (bash):
echo $((2#1010))    # binary to decimal: 10
echo $((16#FF))     # hex to decimal: 255
printf '%x\n' 255   # decimal to hex: ff