Chapter 1, Problem 43CRP

Explanation of Solution

$8\text{bit}$ Floating point and the two’s complement:

The $8\text{bit}$ floating point is the representation in which $8\text{bit}$ of the memory is used. The value stored is positive if the sign bit is zero and the value is negative if the sign bit is one. The two’s complement of the integer is a way of representation in which the sign of the bits on the left hand tells the sign of the integer that is positive or negative.

$\begin{array}{cc}\text{Bit}\text{patttern}& \text{Value}\text{Represented}\\ 111& 3\\ 110& 2\\ 101& 1\\ 100& 0\\ 011& -1\\ 010& -2\\ 001& -3\\ 000& -4\end{array}$

The bit patterns $10000010,01101000\text{and}00000010$ are converted into $8\text{bit}$ Floating point and the two’s complement:

• The first bit pattern is $10000010$, the leftmost value is one so the integer is negative.
• The conversion of the negative integer to the positive is shown below:

$\text{Bit}\text{conversion}=\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 1& 0\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 0& 1& 1& 1& 1& 1& 1& 0\end{array}$

$0\times 2^7+1\times 2^6+1\times 2^5+1\times 2^4+1\times 2^3+1\times 2^2+1\times 2^1+0\times 2^0=126$

The two’s compliment of $10000010$ is ${\left(-126\right)}_{10}$.

• The second bit pattern is $01101000$, the leftmost value is zero so the integer is positive.

  $0\times 2^7+1\times 2^6+1\times 2^5+0\times 2^4+1\times 2^3+0\times 2^2+0\times 2^1+0\times 2^0=104$

  The two’s compliment of $01101000$ is ${\left(104\right)}_{10}$.

• The third bit pattern is $00000010$, the leftmost value is zero so the integer is positive.

  $0\times 2^7+0\times 2^6+0\times 2^5+0\times 2^4+0\times 2^3+0\times 2^2+1\times 2^1+0\times 2^0=2$

  The two’s compliment of $00000010$ is ${\left(2\right)}_{10}$