Question

Convert (ABCDEF)16from its hexadecimal expansion to

its binary expansion.

Short answer

$(101010111100110111101111{)}_2$

Step-by-step solution

Step 1:

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

in hexadecimal notation

$\begin{array}{r}A=10\\ B=11\\ C=12\\ D=13\\ E=14\\ F=15\end{array}$

The hexadecimal expansion has base

localid="1668489972665" $\begin{array}{r}(\mathrm{ABCDEF}{)}_{16}={10.16}^5+{11.16}^4+{1216}^3+{13.16}^2+{14.16}^1+{15.16}^0\\ \\ =10485760+720896+49152+3328+224+15\\ \\ =11259375\end{array}$

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

localid="1668490069131" $\begin{array}{r}11259375=2.5629587+1\\ \\ 5629687=2.2814843+1\\ \\ 2814843=1407421+1\\ \\ 1407421=2.703710+1\\ \\ 703710=2.351855+0\\ \\ 351855=2.175927+1\\ \\ 175927=2.87963+1\\ \\ 87963=43981+1\\ \\ 43981=2.21990+1\\ \\ 21990=2.10995+0\\ \\ 10995=2.5497+1\end{array}$

localid="1668490255084" $\begin{array}{r}5497=2.2748+1\\ \\ 2748=2.1374+0\\ \\ 1374=2.687+0\\ \\ 687=2.343+1\\ \\ 343=2.171+1\\ \\ 171=2.85+1\\ \\ 85=2.42+1\\ \\ 42=2.21+0\\ \\ 21=2.10+1\\ \\ 10=2.5+0\\ \\ 5=2.2+1\\ \\ 2=2.1+0\\ \\ 1=2.0+1\end{array}$

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

$(101010111100110111101111{)}_2$