Question

Convert (BADFACED)16 from its hexadecimal expansion to its binary expansion.

Short answer

$(10111010110111111010110011101101{)}_2$

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

$\mathbf{(}\mathbf{\text{BADFACED}}{\mathbf{)}}_{\mathbf{16}}={11.16}^7+{10.16}^6+{13.16}^5+{15.16}^4+{10.16}^3+{12.16}^2+{14.16}^1+{13.16}^0$

localid="1668489074549" $\begin{array}{r}=2952790016+167772160+13631488+983040+40960+3072+224+13\\ \\ =3135220973\end{array}$

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

$\begin{array}{r}3135220973=2.1567610486+1\\ \\ 1567610486=2.783805243+0\\ \\ 783805243=2.391902621+1\\ \\ 391902621=2.195951310+1\\ \\ 195951310=2.97975655+0\\ \\ 97975655=2.48987827+1\\ \\ 48987827=2.24493913+1\end{array}$

localid="1668489404657" $\begin{array}{r}24493913=2.12246956+1\\ \\ 12246956=2.6123478+0\\ \\ 6123478=2.3061739+0\\ \\ 3061739=2.1530869+1\\ \\ 1530869=2.765434+1\\ \\ 765434=2.382717+0\\ \\ 382717=2.191358+1\\ \\ 191358=2.95679+0\\ \\ 95679=2.47839+1\\ \\ 47839=2.23919+1\\ \\ 23919=2.11959+1\\ \\ 11959=2.5979+1\\ \\ 5979=2.2989+1\\ \\ 2989=2.1494+1\\ \\ 1494=2.747+0\\ \\ 373=2.186+1\\ \\ 186=2.93+0\\ \\ 93=2.46+1\\ \\ 23=2.11+1\\ \\ 11=2.5+1\\ \\ 5=2.2+1\\ \end{array}$

$$\begin{gathered} 2=+2.1+0 \\ 1=2.0+1 \end{gathered}$$

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

$(10111010110111111010110011101101{)}_2$