Question

Convert the octal expansion of each of these integers to a

binary expansion.

a) (572)8 b) (1604)8

c) (423)8 d) (2417)8

Short answer

$$\begin{gathered} a)(101111010{)}_2 \\ b)(1110000100{)}_2 \\ c)(100010011{)}_2 \\ d)(10100001111{)}_2 \end{gathered}$$

Step-by-step solution

Step 1:

$\begin{array}{r}n=a_k{b}^k+a_{k1}{b}^{k1}+\dots .+a_{1}b+a_0\\ \\ (572{)}_8={5.8}^2+{7.8}^1+{2.2}^0\\ \\ =320+56+2\\ \\ =378\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

localid="1668484434683" $$\begin{gathered} \begin{array}{r}378=2.189+0\\ \\ 189=2.94+1\\ \\ 94=2.47+0\\ \\ 47=2.23+1\\ \\ 23=2.11+1\\ \\ 11=2.5+1\\ \\ 5=2.2+1\\ \end{array} \\ \begin{array}{r}2=2.1+0\\ \\ 1=2.0+1\end{array} \end{gathered}$$

Step 2:

The octal expansion has base

$\begin{array}{r}(1604{)}_8={1.8}^2+{6.8}^1+{0.8}^0+{4.2}^0\\ \\ =512+384+4\\ \\ =900\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{r}900=2.450+0\\ \\ 450=2.225+0\\ \\ 225=2.112+11\\ \\ 112=2.56+0\\ \\ 56=2.28+0\\ \\ 28=2.14+0\\ \\ 14=2.7+0\\ \\ 7=2.3+1\\ \\ 3=2.1+1\\ \\ 1=2.0+1\end{array}$

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

$(1110000100{)}_2$

Step 3:

The octal expansion has base

$\begin{array}{r}(423{)}_8={4.8}^2+{2.8}^1+{3.2}^0\\ \\ =256+16+3\\ \\ =275\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{r}275=2.137+1\\ \\ 137=2.68+1\\ \\ 68=2.34+0\\ \\ 34=2.17+0\\ \\ 17+2.8+1\\ \\ 8=2.4+0\\ \\ 4=2.2+0\\ \\ 2=2.1+0\\ \\ 1=2.0+1\end{array}$

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

$(100010011{)}_2$

Step 4:

The octal expansion has base

$\begin{array}{r}(2417{)}_8={2.8}^3+{4.8}^2+{1.8}^1+{7.2}^0\\ \\ =1024+256+8+7\\ \\ =1295\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{r}1295=2.647+1\\ \\ 647=2.323+1\\ \\ 323=2.161+0\\ \\ 161=2.80+0\\ \\ 80=2.40+0\\ \\ 40=2.20+0\\ \\ 20=2.10+0\\ \\ 10=2.5+01\\ \\ 5=2.2+1\\ \\ 2=2.1+0\\ \\ 1=2.0+1\end{array}$

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

${\left(10100001111\right)}_2$