Question

Convert (1011 0111 1011)2from its binary expansion to

its hexadecimal expansion.

Short answer

$(B7B{)}_{16}$

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 b = 2

localid="1668492004701" $\begin{array}{r}(101101111011{)}_2={1.2}^{11}+{0.2}^{10}+{1.2}^9+{1.2}^8+{1.2}^7+{1.2}^6+{1.2}^5+{1.2}^4+{1.2}^3+{0.2}^2+{1.2}^1+{1.2}^0\\ \\ =2.48+512+256+64+32+16+8+2+1\\ \\ =2939\end{array}$

You obtain binary expansion of the decimal expansion of an integer by consecutively dividing the integer by 16 until you obtain 0

$\begin{array}{r}2939=16.183+11\\ \\ 183=16.11+7\\ \\ 11=16.0+11\end{array}$

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

$(B7B{)}_{16}$