Question

Convert the binary expansion of each of these integers to

an octal expansion.

a) (1111 0111)2

b) (1010 1010 1010)2

c) (111 0111 0111 0111)2

d) (101 0101 0101 0101)2

Short answer

$$\begin{gathered} a\left)\right(367{)}_8 \\ \text{b)}\left(5252{)}_8 \\ \text{c)}\right(73567{)}_8 \\ \text{d)}(52525{)}_8 \end{gathered}$$

Step-by-step solution

Step 1:

$n=a_k{b}^k+a_{k1}{b}^{k1}+\dots .+a_{1}b+a_0$

The octal expansion has base

$\begin{array}{r}(11110111{)}_2={1.2}^7+{1.2}^6+{1.2}^5+{1.2}^4+{0.2}^3+{1.2}^2+{1.2}^1+{1.2}^0\\ \\ =128+64+16+4+2+1\\ \\ =247\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 8 until you obtain

$\begin{array}{r}247=8.30+7\\ \\ 30=8.3+6\\ \\ 3=8.0+3\end{array}$

The successive reminders of each division represents the binary expansion from right to left

${\left(367\right)}_8$

Step 2:

The octal expansion has base

$\begin{array}{r}(101010101010{)}_2={1.2}^{11}+{0.2}^{10}+{1.2}^9+{0.2}^8+{1.2}^7+{0.2}^6+{1.2}^5+{0.2}^4+{1.2}^3+{0.2}^2+{1.2}^1+{0.2}^0\\ \\ =2048+512+128+32+2\\ \\ =2730\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 8 until you obtain 0

$\begin{array}{r}2730=8.314+2\\ \\ 341=8.42+5\\ \\ 42=8.5+2\\ \\ 5=8.0+5\end{array}$

The successive reminders of each division represents the binary expansion from right to left

${\left(5252\right)}_8$

Step 3:

The octal expansion has base

$\begin{array}{r}(111011101110111{)}_2={1.2}^{14}+{1.2}^{13}+{1.2}^{12}+{0.2}^{11}+{1.2}^{10}+{1.2}^9+{1.2}^8+{0.2}^7+{1.2}^6+{1.2}^5+{1.2}^4+{0.2}^3+{1.2}^2+{1.2}^1+{1.2}^0\\ \\ =16384+8192+4096+1024+512+256+64+32+16+4+2+1\\ \\ =30583\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 8 until you obtain 0

$\begin{array}{r}30583=8.3822+7\\ \\ 3822=8.477+6\\ \\ 477=8.59+5\\ \\ 59=8.7+3\\ \\ 7=8.0+7\end{array}$

The successive reminders of each division represents the binary expansion from right to left

${\left(73567\right)}_8$

Step 4:

The octal expansion has base $b=2$

$\begin{array}{r}(101010101010101{)}_2={1.2}^{14}+{0.2}^{13}+{1.2}^{12}+{0.2}^{11}+{1.2}^{10}+{0.2}^9+{1.2}^8+{0.2}^7+{1.2}^6+{0.2}^5+{1.2}^4+{0.2}^3+{1.2}^1+{0.2}^1+{1.2}^0\\ \\ =16384+4096+1024+256+64+16+4+1\\ \\ =21845\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 8 until you obtain 0

$\begin{array}{r}21845=8.2730+5\\ \\ 3822=8.341+2\\ \\ 477=8.42+5\\ \\ 59=8.5+2\\ \\ 7=8.0+5\end{array}$

The successive reminders of each division represents the binary expansion from right to left

${\left(52525\right)}_8$