Consider $Z = X - Y$, where $X$, $Y$ and $Z$ are all in sign-magnitude form. $X$ and $Y$ are each represented in $n$ bits. To avoid overflow, the representation of $Z$ would require a minimum of:

In an $n$-bit sign-magnitude representation:

• $1$ bit is used for the sign.

• $(n-1)$ bits are used for the magnitude.

• The range of values represented by $n$ bits is $[-(2^{n-1} - 1), +(2^{n-1} - 1)]$.

For $Z = X - Y$:

• The maximum possible value is:

  $$Z_{\max} = X_{\max} - Y_{\min} = (2^{n-1} - 1) - (-(2^{n-1} - 1)) = 2(2^{n-1} - 1) = 2^n - 2$$

• The minimum possible value is:

  $$Z_{\min} = X_{\min} - Y_{\max} = -(2^{n-1} - 1) - (2^{n-1} - 1) = -(2^n - 2)$$

To represent values up to magnitude $2^n - 2$ in sign-magnitude form:

• We need $k$ magnitude bits such that $2^k - 1 \ge 2^n - 2$, which gives $k = n$ magnitude bits.

• Adding $1$ bit for the sign gives a total of $n + 1$ bits.

Thus, the representation of $Z$ requires a minimum of $n + 1$ bits.