Chapter 2.4, Problem 2.45PP

Program Plan Intro

Fractional binary numbers:

The decimal notation is represented by the below form:

$\text{d }={\displaystyle \sum _{\text{i }=\text{ -n}}^{\text{m}}{\text{10}}^{\text{i}}\times {\text{d}}_{\text{i}}}$

From the above equation, each decimal digit “d_i” ranges between “0” and “9”.

Example:

The example for above equation is given below.

Consider the number 12.34₁₀

$\begin{array}{l}\text{12}{\text{.34}}_{\text{10}}=\text{1}\times {\text{10}}^{\text{1}}+\text{2}\times {\text{10}}^{\text{0}}+\text{3}\times {\text{10}}^{-\text{1}}+\text{4}\times {\text{10}}^{-\text{2}}\\ =\text{1}\times \text{10}+\text{2}\times \text{1}+\text{3}\times \frac{\text{1}}{\text{10}}+\text{4}\times \frac{\text{1}}{\text{100}}\\ =\text{10}+\text{2}+\frac{\text{3}}{\text{10}}+\frac{\text{4}}{\text{100}}\end{array}$

$\begin{array}{l}=\text{1}2+\frac{\text{300}+\text{40}}{\text{1000}}\\ =\text{1}2+\frac{\text{340}}{\text{1000}}\\ =\text{1}2\frac{\text{34}}{\text{10}}\end{array}$

If each binary digit or bit b_i ranges between “0” and “1”, then the notation represented by the below equation

$\text{b }={\displaystyle \sum _{\text{i }=\text{ -n}}^{\text{m}}{\text{2}}^{\text{i}}\times {\text{b}}_{\text{i}}}$

Example:

The example for given equation is given below:

Consider, the bit “b” is 101.11₂

$\begin{array}{l}\text{101}{\text{.11}}_{\text{2}}=\text{1}\times {\text{2}}^{\text{2}}+\text{0}\times {\text{2}}^{\text{1}}+\text{1}\times {\text{2}}^{\text{0}}+\text{1}\times {\text{2}}^{-\text{1}}+\text{1}\times {\text{2}}^{-\text{2}}\\ =\text{1}\times \text{4}+\text{0}+\text{1}\times \text{1}+\text{1}\times \frac{\text{1}}{{\text{2}}^{\text{1}}}+\text{1}\times \frac{\text{1}}{{\text{2}}^{\text{2}}}\\ =\text{4}+\text{1}+\frac{\text{1}}{\text{2}}+\frac{\text{1}}{\text{4}}\end{array}$

$\begin{array}{l}=5+\frac{2+1}{4}\\ =5\frac{3}{4}\end{array}$

Simple method to determine the conversion of fractional value to binary number is given below:

Considering a fraction of the form is $\frac{\text{x}}{{\text{2}}^{\text{k}}}$

• Now, write the binary value by using the binary representation of “x”, with the point inserted by “k” positions from the right of given binary value.
• Consider an example, if fractional value is $\frac{\text{25}}{\text{16}}$

$\frac{\text{25}}{\text{16}}=\frac{\text{25}}{{\text{2}}^{\text{4}}}$

• The binary value of “25” is “11001”. Then place the binary point four positions from the right. Finally, the result is 1.1001.