Converting Between Any Bases

To convert from base $b$ to decimal, multiply each digit by its position value ($b$ raised to the position) and sum. To convert from decimal to base $b$, repeatedly divide by $b$ and collect remainders.

Example: Convert $123_5$ (base-5) to decimal. $123_5 = 1 \times 5^2 + 2 \times 5^1 + 3 \times 5^0 = 25 + 10 + 3 = 38_{10}$. Now convert 38 to base-5: $38 \div 5 = 7$ rem $3$, $7 \div 5 = 1$ rem $2$, $1 \div 5 = 0$ rem $1$. Reading up: $123_5$.

The algorithm works for any base. The pattern is always the same: expand to decimal using powers, or divide repeatedly to get digits. Most programming languages have built-in functions for common bases.