Chapter Questions
For $n=8$, let $A$ and $B$ be two fixed-point binary numbers in $2 \mathrm{~s}$ complement representation, where $A=01101101$ and $B=11001111$. Determine $A+B^{\prime}+$ $1+B$, where $B^{\prime}$ is the 1 s complement.
Perform the indicated operations on the fixed-point numbers shown below in $2 \mathrm{~s}$ complement representation. In each case, indicate if there is an overflow.(a)(b)$$\text { +) } \begin{array}{rrrrrrrr}0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\\hline\end{array}$$(c)$$\begin{array}{rrrrrrrr}1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\\hline\end{array}$$(d)$$\begin{array}{rrrrrrrr}0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\-) & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\\hline\end{array}$$
Let $A$ and $B$ be two fixed-point binary numbers in $2 \mathrm{~s}$ complement representation as shown below, where $A^{\prime}$ and $B^{\prime}$ are the Is complement of $A$ and $B$, respectively. Perform the indicated operations.$$A=10110001 \quad B=11100100$$(a) $A-(B+1)$(b) $A^{\prime}+1+B^{\prime}+1$(c) $A^{\prime}+1-\left(B^{\prime}+1\right)$
Obtain the sum of the following unsigned hexadecimal numbers:$$\begin{array}{rrrrr}1 & 2 & 3 & 4 & 5 \\6 & 7 & 8 & 9 & \mathrm{~A} \\\text { +) } & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} \\\hline\end{array}$$
Obtain the sum of the following $2 \mathrm{~s}$ complement radix 2 numbers:$$\begin{array}{rrrrrr}1 & 1 & 1 & 1 & 1 & 0 \\1 & 1 & 1 & 1 & 0 & 0 \\1 & 1 & 1 & 0 & 0 & 0 \\+ & 1 & 0 & 0 & 0 & 0 \\\hline\end{array}$$
Perform the following radix 2 subtraction using the 1 s complement method:$$\begin{array}{rrrrrrrr}1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\\hline\end{array}$$
Obtain the sum of the following radix 5 numbers:$$\begin{array}{llll}1 & 2 & 3 & 4 \\4 & 3 & 2 & 1 \\3 & 3 & 3 & 3 \\\hline\end{array}$$
Perform a subtraction on the operands shown below, which are in radix complementation for radix 3 .$$\begin{array}{rrrrr}0 & 2 & 0 & 2 & 1 \\-) & 2 & 1 & 0 & 0 \\\hline\end{array}$$
Write the equations for two ways to detect overflow for two $n$-bit operands in fixed-point addition assuming radix complementation for radix 2 .
Indicate whether overflow occurs for the following decimal numbers when they are converted to 8-bit fixed-point binary numbers for radix 2 for the indicated operations.(a) $(+35)+(+42)$(b) $(-62)-(+67)$(c) $(-31)+(-34)$(d) $(-31)-(+33)$
Add the following numbers, which are shown in radix complementation. Obtain the sum in radix $4: 0201_{10}+321_4$.
Show the radix 3 result of the following subtraction:$$\begin{aligned}& \begin{array}{llllllll}2 & 0 & 2 & 2 & 0 & 2 & 0 & 1_3\end{array} \\& \begin{array}{lllllllll}-) & 2 & 1 & 0 & 2 & 0 & 2 & 0 & 23\end{array} \\&\end{aligned}$$
Use the paper-and-pencil method to multiply the following numbers, which are in $2 \mathrm{~s}$ complement representation:$$\begin{aligned}& 001110 \\& 000111\end{aligned}$$
Use the paper-and-pencil method to multiply the following numbers, which are in $2 \mathrm{~s}$ complement representation:111111001011
Use the sequential add-shift method to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:$$\begin{aligned}\text { Multiplicand } A & =1111 \\\text { Multiplier } B & =0111\end{aligned}$$
Use the sequential add-shift method to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:Multiplicand $A=0101$Multiplier $B=0101$
Use the Booth algorithm to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:Multiplicand $A=010111$Multiplier $B=001101$
Use the Booth algorithm to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:Multiplicand $A=010110$Multiplier $B=110110$
Use the Booth algorithm to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:Multiplicand $A=110001$Multiplier $B=100110$
Use bit-pair recoding to determine the multiplicand multiples to be added for the following multipliers, which are in 2 s complement representation.(a) 100111(b) 110011(c) 001101
Use bit-pair recoding to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:$$\begin{aligned}\text { Multiplicand } A & =110010 \\\text { Multiplier } B & =011001\end{aligned}$$
Use bit-pair recoding to multiply the following operands, which are in $2 \mathrm{~s}$ complement representation:Multiplicand $A=0100110$Multiplier $B=1001101$
Multiply the following two unsigned fixed-point operands using an array multiplier:Multiplicand $A=0111$Multiplier $B=1100$
Use the sequential restoring method to perform a divide operation on the following operands:Dividend $A=00110000$Divisor $B=0111$
Use the sequential nonrestoring method to perform a divide operation on the following operands:Dividend $A=00000111$Divisor $B=0100$
Use the sequential nonrestoring method to perform a divide operation on the following operands:Dividend $A=01100011$Divisor $B=0111$
The decimal operands shown below are to be added using decimal (BCD) addition. Obtain the answer that correctly represents the intermediate sum; that is, the sum before correction (adjustment) is applied.$$(+725)+(+536)$$
Perform the indicated decimal operation on the operands shown below.$$(+20)+(-32)$$
Perform decimal multiplication using a read-only memory (ROM) on the following operands:$$\begin{aligned}\text { Multiplicand } A= & 736 \\\text { Multiplier } B= & 48\end{aligned}$$
Perform the following floating-point addition operation for positive operands:$$\begin{aligned}& A=0,1 \quad 0 \quad 0 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad \times 2^6 \\& \text { +) } B=0,11110000 \times \times 2^2 \\&\end{aligned}$$
Under what conditions can there be an exponent overflow during floatingpoint arithmetic?
Perform the operation listed below for normalized floating-point numbers using 8-bit fractions.$$\begin{array}{r}-10 \\+\quad 33 \\\hline\end{array}$$
Multiply the floating-point fractions shown below using the sequential addshift method.$$\begin{array}{rllllllllll}\text { Multiplicand } A=0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & \times 2^5 \\\text { Multiplier } B= & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \times 2^4\end{array}$$
Multiply the floating-point fractions shown below using the sequential addshift method.
Multiply the floating-point fractions shown below using the sequential addshift method.$$\begin{array}{rllllllllll}\text { Multiplicand } A= & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & \times 2^5 \\\text { Multiplier } B= & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & \times 2^3\end{array}$$
Multiply the floating-point fractions shown below using the sequential addshift method.$$\begin{aligned}& \text { Multiplicand } A=1,110000000 \times 2^4 \\& \text { Multiplier } B=0, \quad 1 \quad 0 \quad 0 \quad 0 \quad 1 \quad 0 \quad 0 \quad 0 \quad \times 2^5 \\&\end{aligned}$$
How is quotient overflow determined in floating-point division and how is the problem solved?
Explain the biasing problem that occurs during floating-point multiplication and division.