Question

Convert (1 1000 0110 0011)2from its binary expansion

to its hexadecimal expansion.

Short answer

$(1863{)}_{16}$

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

$\begin{array}{r}(1100001100011{)}_2={1.2}^{12}+{1.2}^{11}+{0.2}^{10}+{0.2}^9+{0.2}^8+{0.2}^7+{1.2}^6+{1.2}^5+{0.2}^4+{0.2}^3+{0.2}^2+{1.2}^1+{1.2}^0\\ \\ =4096+2048+64+32+2+1\\ \\ =6243\end{array}$

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

$\begin{array}{r}6243=16.390+3\\ \\ 390=16.24+6\\ \\ 24=16.1+8\\ \\ 1=16.0+1\end{array}$

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

$(1863{)}_{16}$