Question

Convert the binary expansion of each of these integers to a decimal expansion.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}{\mathbf{\left(}\mathbf{1101}\mathbf{\right)}}_{\mathbf{2}} \\ \mathbf{b}\mathbf{)}\mathbf{}{\mathbf{\left(}\mathbf{1010110101}\mathbf{\right)}}_{\mathbf{2}}\mathbf{} \\ \mathbf{c}\mathbf{)}\mathbf{}{\mathbf{\left(}\mathbf{1110111110}\mathbf{\right)}}_{\mathbf{2}}\mathbf{} \\ \mathbf{d}\mathbf{)}\mathbf{}{\mathbf{\left(}\mathbf{111110000011111}\mathbf{\right)}}_{\mathbf{2}} \end{gathered}$$

Short answer

a) 27

b) 693

c) 958

d) 31775

Step-by-step solution

Concept of binary and decimal expansion

If the $\left\{\text{base b representation of}\mathrm{n}\right\}\text{is}{\mathrm{a}}_0{a}_1{a}_2\dots a_k\text{, then}n=a_k{b}^k+a_{k-1}{b}^{k-1}+\dots +a_{1}b+a_0$

Step 2(a): Converting the binary expansion of the integer (11011)2 to a decimal expansion 

The binary expansion has base$b=2$

$\begin{array}{r}(11011{)}_2=1\cdot 2^4+1\cdot 2^3+0\cdot 2^2+1\cdot 2^1+1\cdot 2^0\\ =16+8+2+1\\ =27\end{array}$

Hence the solution is 27.

Step 3(b): Converting the binary expansion of the integer (1010110101)2 to a decimal expansion 

The binary expansion has base $b=2$

$\begin{array}{r}(1010110101{)}_2=1\cdot 2^9+0\cdot 2^8+1\cdot 2^7+0\cdot 2^6+1\cdot 2^5+1\cdot 2^4+0\cdot 2^3+1\cdot 2^2+0\cdot 2^1+1\cdot 2^0\\ =512+128+32+16+4+1\\ =693\end{array}$

Hence the solution is 693.

Step 4(c): Converting the binary expansion of the integer (1110111110)2 to a decimal expansion 

$\begin{array}{r}(1110111110{)}_2=1\cdot 2^9+1\cdot 2^8+1\cdot 2^7+0\cdot 2^6+1\cdot 2^5+1\cdot 2^4+1\cdot 2^3+1\cdot 2^2+1\cdot 2^1+0\cdot 2^0\\ =512+256+128+32+16+8+4+2\\ =958\end{array}$

Hence the solution is 958.

Step 5(d): Converting the binary expansion of the integer (110100100010000)2 to a decimal expansion

$\begin{array}{r}(111110000011111{)}_2=1\cdot 2^{14}+1\cdot 2^{13}+1\cdot 2^{12}+1\cdot 2^{11}+1\cdot 2^{10}+0\cdot 2^9+0\cdot 2^8+0\cdot 2^7+0\cdot 2^6+0\cdot 2^5+1\cdot 2^4+1\cdot 2^3+1\cdot 2^2+1\cdot 2^1+1\cdot 2^0\\ =16384+8192+4096+2048+1024+16+8+4+\\ =31775\end{array}$Hence the solution is 31775.