Base Numbers Problems

A number in base-$b$ is written as a sequence of digits $d_n d_{n-1} \ldots d_1 d_0$, where each $d_i$ is an integer from $0$ to $b-1$. The value of the number is:

$d_n \times b^n + d_{n-1} \times b^{n-1} + \cdots + d_1 \times b^1 + d_0 \times b^0$

Base number problems often involve converting numbers between bases, performing arithmetic in different bases, or interpreting numbers written in unfamiliar bases.

Important properties

• Each digit in base-$b$ must be less than $b$.

• The rightmost digit is the 'ones' place ($b^0$), the next is $b^1$, and so on.

• To convert from base-$b$ to base-10, expand using powers of $b$.

• To convert from base-10 to base-$b$, repeatedly divide by $b$ and record remainders.