Question

1a. Convert 67 (base 10) to 8-bit binary using signed magnitude. Show your work.

1b. Convert 69 (base 10) to 8-bit binary using one’s complement. Show your work

1c. Convert 70 (base 10) to 8-bit binary using two’s complement. Show your work.

1d. Convert - 67 (base 10) to 8-bit binary using signed magnitude.

1e. Convert - 67 (base 10) to 8-bit binary using ones compliment. Show your work.

1f. Convert - 67 (base 10) to 8-bit binary using 2s compliment. Show your work.

Answer 1

Solution:

1(a)

Given,

=>Number = (67)₁₀

Explanation:

Converting number into signed magnitude binary:

=>Positive numbers are represented in signed magnitude same as unsigned numbers.

=>67 % 2 => quotient = 33, remainder = 1

=>33 % 2 => quotient = 16, remainder = 1

=>16 % 2 => quotient = 8, remainder = 0

=>8 % 2 => quotient = 4, remainder = 0

=>4 % 2 => quotient = 2, remainder = 0

=>2 % 2 => quotient = 1, remainder = 0

=>1 % 2=> quotient = 0, remainder = 1

=>Arranging remainders from bottom to top = 1000011

=>Hence signed magnitude number representation of (67)₁₀ = (1000011)₂

=>Hence signed magnitude number representation of (67)₁₀ in 8 bits = (01000011)₂

1(b)

Given,

=>Number = (69)₁₀

Explanation:

Converting decimal number into 1's complement form:

=>Positive numbers are represented in 1's complement form same as unsigned numbers.

=>69 % 2 => quotient = 34, remainder = 1

=>34 % 2 => quotient = 17, remainder = 0

=>17 % 2 => quotient = 8, remainder = 1

=>8 % 2 => quotient = 4, remainder = 0

=>4 % 2 => quotient = 2, remainder = 0

=>2 % 2 => quotient = 1, remainder = 0

=>1 % 2 => quotient = 0, remainder = 1

=>Arranging remainders from bottom to top = (1000101)₂

=>Hence 1's complement number representation of (69)₁₀ = (1000101)₂

=>Hence 1's complement number representation of (69)₁₀ in 8 bits = (01000101)₂

1(c)

Given,

=>Number = (70)₁₀

Explanation:

Converting decimal number into 2's complement form:

=>Positive numbers are represented in 2's complement form same as unsigned numbers.

=>70 % 2 => quotient = 35, remainder = 0

=>35 % 2 => quotient = 17, remainder = 1

=>17 % 2 => quotient = 8, remainder = 1

=>8 % 2 => quotient = 4, remainder = 0

=>4 % 2 => quotient = 2, remainder = 0

=>2 % 2 => quotient = 1, remainder = 0

=>1 % 2 => quotient = 0, remainder = 1

=>Arranging remainders from bottom to top = (1000110)₂

=>Hence 2's complement number representation of (70)₁₀ =(1000110)₂

=>Hence 2's complement number representation of (70)₁₀ in 8 bits = (01000110)₂

1(d)

Given,

=>Number = (-67)₁₀

Explanation:

Converting number into signed magnitude binary:

=>In case of negative decimal numbers representation is different

=>MSB of the signed magnitude number represents the sign of the number. If MSB = 1 then negative number otherwise positive number.

=>Remaining bits after MSB bits represents the modulus decimal value.

=>We know that (67)₁₀ from part (a) = (1000011)₂

=>Hence (-67)₁₀ in signed magnitude representation = (11000011)₂

1(e)

Given,

=>Number = (-67)₁₀

Explanation:

Converting decimal number into 1's complement form:

=>In 1's complement form we flip each bit of the binary number of modulus value of decimal number.

=>We know that (67)₁₀ from part (a) = (01000011)₂

=>Hence (-67)₁₀ in 1's complement form = (10111100)₂

1(f)

Given,

=>Number = (-67)₁₀

Explanation:

Converting decimal number into 2's complement form:

=>In case of negative numbers representation is different.

=>MSB of 2's complement number represents the sign of number, if MSB = 1 then number is negative otherwise positive.

=>All the bits represents the value of the 2's complement number.

=>We know that (67)₁₀ from part (a) = (01000011)₂

=>Hence (-67)₁₀ in 1's complement form = (10111100)₂

=>Hence (-67)₁₀ in 2's complement form = (10111100)₂ + (00000001)₂

=>Hence (-67)₁₀ in 2's complement form = (10111101)₂

I have explained each and every part with the help of statements attached to it.