Answer
a. 100.1
b. 10.11
c. 1.001
d. 0.0101
e. 101.101
Work Step by Step
a.
There are two parts in the given base ten representations. First
is integer part and the other is the fractional part.
$4\frac{1}{2}$ is equivalent to $4.5$
Step $1 :$ To convert the integer part of the base ten representations which is $(4)$ to its equivalent binary representations, we have to divide the number by $(2)$ until the quotient is zero, as shown in the table below.
\begin{equation}
\begin{array}{|c|c|c|}\hline \text { remainder } & {\text { divide by } 2} & {\text { number }} \\ \hline 0 & {2} & {4} \\ \hline 0 & {2} & {2} \\ \hline 1 & {2} & {1} \\ \hline & {2} & {0\ (stop)} \\ \hline\end{array}
\end{equation}
Therefore, the binary representation is the sequence of the remainder from bottom to top: $(4)_{10} = (100)_{2}$
Step $2 :$ To convert the fractional part of the base ten representations which is $( \frac{1}{2} = 0.5)$ to its equivalent binary representation, we have to multiply the fractional number by $(2)$ and keep track of the resulting integer and fractional part. Continue multiplying by $(2)$ until you get a resulting fractional part equal to zero. Then just write out the integer parts from the results of each multiplication from the top to bottom as shown below:
$0.5 * 2 = 1.0$
fractional part equal to zero. Then just write out the integer parts from the results of each multiplication from the top to bottom.
Therefore, $(0.5)_{10} = (0.1)_{2}$
Step $3 :$ concatenate integer part and fractional part
Therefore, $(4.5)_{10} = 100 + 0.1 = (100.1)_{2}$
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b.
$2\frac{3}{4}$ is equivalent to $2.75$
Step $1 :$ convert integer part (2) to binary
\begin{equation}
\begin{array}{|c|c|c|}\hline \text { remainder } & {\text { divide by } 2} & {\text { number }} \\ \hline 0 & {2} & {2} \\ \hline 1 & {2} & {1} \\ \hline & {2} & {0 (stop)} \\ \hline\end{array}
\end{equation}
Therefore, $(2)_{10} = (10)_{2}$
Step $2 :$ convert fractional part (0.75) to binary
$0.75 * 2 = 1.5 = 1 + 0.5$
$\ \ 0.5 * 2 = 1 \ \ \ = 1 + 0.0$
fractional part equal to zero. Then just write out the integer parts from the results of each multiplication from top to bottom.
Therefore, $(0.75)_{10} = (0.11)_{2}$
Step $3 :$ concatenate integer part and fractional part
Therefore, $(2.75)_{10} = 10 +0 .11 = (10.11)_{2}$
