Definition For \(b\) an integer greater than \(1\) and \(n\) a positive integer, the base \(b\) expansion of \(n\) is \[(a_{k-1} \cdots a_1 a_0)_b\] where \(k\) is a positive integer, \(a_0, a_1, \ldots, a_{k-1}\) are (symbols for) nonnegative integers less than \(b\), \(a_{k-1} \neq 0\), and \[n = \sum_{i=0}^{k-1} a_{i} b^{i}\]
Notice: The base \(b\) expansion of a positive integer \(n\) is a string over the alphabet \(\{x \in \mathbb{N} \mid x < b\}\) whose leftmost character is nonzero.
| Base \(b\) | Collection of possible coefficients in base \(b\) expansion of a positive integer |
|---|---|
| Binary (\(b=2\)) | \(\{0,1\}\) |
| Ternary (\(b=3\)) | \(\{0,1, 2\}\) |
| Octal (\(b=8\)) | \(\{0,1, 2, 3, 4, 5, 6, 7\}\) |
| Decimal (\(b=10\)) | \(\{0,1, 2, 3, 4, 5, 6, 7, 8, 9\}\) |
| Hexadecimal (\(b=16\)) | \(\{0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F\}\) |
| letter coefficient symbols represent numerical values \((A)_{16} = (10)_{10}\) | |
| \((B)_{16} = (11)_{10} ~~(C)_{16} = (12)_{10} ~~ (D)_{16} = (13)_{10} ~~ (E)_{16} = (14)_{10} ~~ (F)_{16} = (15)_{10}\) |
Examples:
\((1401)_{2}\)
\((1401)_{10}\)
\((1401)_{16}\)
Find and fix any and all mistakes with the following:
\((1)_2 = (1)_8\)
\((142)_{10} = (142)_{16}\)
\((20)_{10} = (10100)_2\)
\((35)_8 = (1D)_{16}\)
Convert \((2A)_{16}\) to
binary (base )
decimal (base )
octal (base )
ternary (base )