Chapter 2, Problem 26E

a.

Explanation of Solution

Number of bits required:

“7” bits are required to represent “-64” in a 2’s complement representation.

Explanation:

• Using “2” bits, “$2^2$”, which means “4” numbers can be produced.
• Using “3” bits, “$2^3$”, which means “8” numbers can be produced and the maximum value that can be represented using “3” bits is “7”.
• Hence, using “n” bits, “$2^n$” numbers can be produced and the maximum positive value of “${\text{2}}^{\text{n}}-\text{1}$” can be represented

b.

Explanation of Solution

Largest positive number in 7-bit 2’s complement code:

• The Left most Significant Bit (LSB) denotes that the number is positive or negative.
• The number is positive, if LSB is “0”.
• The number is negative, if LSB is “1”.
• Here is to find, the largest positive number, hence the LSB in binary representation must be “0”.
• The remaining “6” digits must be “1”.
• Hence the largest positive number in 7-bit binary representation is “0111111”, which means “+63”.

Representation in binary and decimal:

• The value “63” is represented in binary as follows,

$2^6$

$$

c.

Explanation of Solution

Largest unsigned number using “7” bits:

• For unsigned numbers, the sign bit is not used.
• The largest unsigned number can be obtained by placing “1” in all the 7-bit positions.

$2^6$

$2^5$

$2^4$

$2^3$

$2^2$

$2^1$

$2^0$

Integer