Chapter 1, Problem 30CRP

a.

Explanation of Solution

Convert two’s complement representation $01111$ to its equivalent base ten forms:

• To find the conversion of two’s complement representation $01111$ to its equivalent base ten form, identify the leftmost bit as $0$ or $1$. When the leftmost bit is $0$, the result is positive and in this case, multiply value of each digit by the quantity associated with its position and add the result. When the leftmost bit is $1$, the result is negative and in this case, copy the bits from right to left until $1$ is copied, then complement the remaining bits. To complement the pattern, change $0$ to $1$ and $1$ to $0$. And then multiply value of each digit by the quantity associated with its position and add the result to find the equivalent base ten form.
• The leftmost bit is $0$ which gives a positive result

Explanation of Solution

Convert two’s complement representation $10100$ to its equivalent base ten forms:

• The leftmost bit is $1$ which gives a negative result. To find the conversion of two’s complement representation $10100$ to its equivalent base ten form, copy the bits from right to left until $1$ is copied, then change $0$ to $1$ and $1$ to $0$ complement the remaining bits:

$\begin{array}{ccccc}1& 0& 1& 0& 0\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 0& 1& 1& 0& 0\end{array}$

• To find the conversion of two’s complement representation to its equivalent base ten form, multiply value of each digit by the quantity associated with its position and add the result

c.

Explanation of Solution

Convert two’s complement representation $01100$ to its equivalent base ten forms:

• The leftmost bit is $0$ which gives a positive result. To find the conversion of two’s complement representation $01100$ to its equivalent base ten form, multiply value of each digit by the quantity associated with its position and add the result:

$\begin{array}{c}0\times 2^4+1\times 2^3+1\times 2^2+0�\end{array}$

d.

Explanation of Solution

Convert two’s complement representation $10000$ to its equivalent base ten forms:

• The leftmost bit is $1$ which gives a negative result. To find the conversion of two’s complement representation $10000$ to its equivalent base ten form, copy the bits from right to left until $1$ is copied, then change $0$ to $1$ and $1$ to $0$ to complement the remaining bits:

$\begin{array}{ccccc}1& 0& 0& 0& 0\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 1& 0& 0& 0& 0\end{array}$

• To find the conversion of two’s complement representation to its equivalent base ten form, multiply value of each digit by the quantity associated with its position and add the result

e.

Explanation of Solution

Convert two’s complement representation $10110$ to its equivalent base ten forms:

• The leftmost bit is $1$ which gives a negative result. To find the conversion of two’s complement representation $10110$ to its equivalent base ten form, copy the bits from right to left until $1$ is copied, then change $0$ to $1$ and $1$ to $0$ to complement the remaining bits:

$\begin{array}{ccccc}1& 0& 1& 1& 0\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 0& 1& 0& 1& 0\end{array}$

• To find the conversion of two’s complement representation to its equivalent base ten form, multiply value of each digit by the quantity associated with its position and add the result