# Number Systems | System | Base | Digits | | ------------ | ------- | :-------------------------------: | | Decimal | Base 10 | $0,1,2,3,4,5,6,7,8,9$ | | [[Binary]] | Base 2 | $0,1$ | | Octal | Base 8 | $0,1,2,3,4,5,6,7$ | | Hexa-decimal | Base 16 | $0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F$ | ## Base Expansions > [!info] **Base $b$ Definition** > **Theorem:** Let $b$ be a positive integer greater than $1$. Then if $n$ is a positive integer, it can be expressed uniquely in the form: > $n=a_k b^k + a_{k-1} b^{k-1}+...+a_1 b^1 + a_0$ > where $k$ is a nonnegative integer and $a_0,a_1,…,a_k$ are nonnegative integers less than $b$. This representation of $n$ is called the *“base $b$ expansion of $n$”* and can be denoted as: $(a_ka_{k-1}...a_1a_0)_b$ - The subscript 10 is usually omitted for base 10 expansions, but these *decimal numbers still use this system of expansion* ## Base Conversion > [!summary] **Base Conversion** > To construct the base $b$ expansion of an integer $n$: > 1. Divide $n$ by $b$ to obtain a quotient ($q$) and remainder ($r$). > - $n=bq_0 + a_0$, where $0 \le a_0 < b$ > - The remainder, $a_0$, is the rightmost digit in the base $b$ expansion of $n$ > 2. Next, divide $q_0$ by $b$: > - $q_0 = bq_1 + a_1$, where $0 \le a_1 < b$ > - The remainder, $a_1$, is the second digit from the right in the base $b$ expansion of $n$. > 3. Continue by successively dividing the quotients by $b$, obtaining the additional base $b$ digits as the remainder. > - The process terminates when the quotient is 0. >