For each of the following scenarios, describe the range of numbers that can be r

Question

For each of the following scenarios, describe the range of numbers that can be represented. That is to say, what is the lowest number that can be represented, and what is the largest number that can be represented. For the last two, the solution will be two algebraic formulas (use a-f to see the pattern) (a) Unsigned 4-bit binary (b) Two's Complement 4-bit binary (c) Unsigned 5-bit binary (d) Two's Complement 5-bit binary (e) Unsigned 6-bit binary (f Two's Complement 6-bit binary (g) Unsigned n-bit binary (h) Two's Complement n-bit binary

Explanation / Answer

a. Unsigned 4-bit binary:

All the four bits are used to represent the number. Hence there are 2^4 possible numbers can be represented using 4 bits. The range is 0 to 2^4 -1, i.e it starts from 0-15(0 to 15)

b. Two's complement 4-bit binary:

In two's complement only 3 least significant bits are used to represent the number and the last most significant bit is used to represent the sign of the number. Therefore with 3 bits to represent the number the range will from

-(2^(4-1)) to 2^(4-1) - 1

-2^3 to 2^3-1 i.e., -8-7(-8 to 7)

c. Unsigned 5-bit binary:

All the five bits are used to represent the number. Hence there are 2^5 possible numbers can be represented using 5 bits. The range is 0 to 2^5 -1, i.e it starts from 0-31(0 to 31).

d. Two's complement 5-bit binary:

In two's complement only 4 least significant bits are used to represent the number and the last most significant bit is used to represent the sign of the number. Therefore with 4 bits to represent the number the range will from

-(2^(5-1)) to 2^(5-1) - 1

-2^4 to 2^4-1 i.e., -16-15(-16 to 15)

e. Unsigned 6-bit binary:

All the six bits are used to represent the number. Hence there are 2^6 possible numbers can be represented using 6 bits. The range is 0 to 2^6 -1, i.e it starts from 0-63(0 to 63).

f. Two's complement 6-bit binary:

In two's complement only 5 least significant bits are used to represent the number and the last most significant bit is used to represent the sign of the number. Therefore with 5 bits to represent the number the range will from

-(2^(6-1)) to 2^(6-1) - 1

-2^5 to 2^5-1 i.e., -32-31(-32 to 31)

g. Unsigned n-bit binary:

All the 'n' bits are used to represent the number. Hence there are 2^n possible numbers can be represented using 'n' bits. The range is 0-2^n-1 (0 to 2^n -1).

h. Two's complement n-bit binary:

In two's complement only n-1 least significant bits are used to represent the number and the last most significant bit is used to represent the sign of the number. Therefore with n-1 bits to represent the number the range will be from

-(2^(n-1)) to 2^(n-1) - 1

i.e. -(2^(n-1)) - 2^(n-1) - 1

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