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GO Classes DPP | DIGITAL LOGIC |BINARY SYSTEM

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11 11 votes

A $6$-bit CPU uses $\mathbf{2}$'s complement representation for signed integers. Two numbers, $X= (101101)_2$ and $Y=(011010)_2$, are fed into an Arithmetic Logic Unit (ALU) to perform the operation $Z=X-Y$.

Which of the following statements is TRUE regarding the result $Z$ ?

  1. The result $Z$ in decimal is $-45$ and no overflow occurred.
     
  2. The result $Z$ in decimal is $+19$ and an overflow occurred.
     
  3. The result $Z$ in decimal is $-19$ and no overflow occurred.
     
  4. The result $Z$ in decimal is $-27$ and an overflow occurred.

2 Answers

4 4 votes

$$
\begin{aligned}
X & =-19 \\\\
Y & =+26 \\\\
Z & =(-19)-(+26)=-45
\end{aligned}
$$
In a $6$-bit $2$'s complement system, the range of representable integers is:

  • Range $:\left[-2^{n-1}, 2^{n-1}-1\right]$
     
  • For $n=6:\left[-2^5, 2^5-1\right]=[-32,+31]$.
     
  • Since the theoretical result $-45$ falls outside the range $[-32,31]$, an overflow occurs.

$$
\begin{aligned}
& X=101101 \\\\
& -Y=2^{\prime} \text { s complement of } 011010=100110 \\\\
& Z=X+(-Y)=101101+100110=(1) 010011
\end{aligned}
$$
The $6$-bit result is $010011$, which is $+19$ (the "incorrect" result due to overflow).

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