Given two decimal numbers A = 78 and B = 29, find A - B and B - A using the following methods: a) Signed 2's complement format using 8 bits. b) 9's complement format.

Steps to Solve

1. Convert A and B to binary (8 bits)

$A = 78_{10} = 01001110_2$ and $B = 29_{10} = 00011101_2$

2. Calculate A - B using 2's complement To calculate $A - B$, we can calculate $A + (-B)$, where $-B$ is the 2's complement of $B$. First, find the 1's complement of B by inverting all bits: $1's\ complement\ of\ B = 11100010_2$ Then, add 1 to find 2's complement: $2's\ complement\ of\ B = 11100010_2 + 1_2 = 11100011_2$ Now, add A and the 2's complement of B: $A + (-B) = 01001110_2 + 11100011_2 = 00110001_2$ (ignoring the carry-out) $00110001_2 = 49_{10}$, so $A - B = 49$

3. Calculate B - A using 2's complement To calculate $B - A$, we can calculate $B + (-A)$, where $-A$ is the 2's complement of $A$. First, find the 1's complement of A by inverting all bits: $1's\ complement\ of\ A = 10110001_2$ Then, add 1 to find 2's complement: $2's\ complement\ of\ A = 10110001_2 + 1_2 = 10110010_2$ Now, add B and the 2's complement of A: $B + (-A) = 00011101_2 + 10110010_2 = 11001111_2$ Since the result is negative (the leftmost bit is 1), we take the 2's complement of the result to find its magnitude: $1's\ complement\ of\ 11001111_2 = 00110000_2$ $2's\ complement\ of\ 11001111_2 = 00110000_2 + 1_2 = 00110001_2 = 49_{10}$ Since the sign bit was 1, $B - A = -49$

4. Calculate A - B using 9's complement $A - B = 78 - 29$. Since both numbers are positive, we don't need to find their 9's complement. We can directly subtract them: Since we are using 9's complement, we need consider how 9's complement works with negative numbers. However, since $A > B$, we can perform direct subtraction: $78 - 29 = 49$

5. Calculate B - A using 9's complement $B - A = 29 - 78$. Since $B < A$, we will get a negative result. Subtract A from B: $29 - 78 = -49$ To represent -49 in 9's complement, first find the 9's complement representation of 49. Since we're not given the number of digits to use for the 9's complement, we are assuming the numbers are padded to the same length when calculating the 9's complement. In this case both numbers have two digits, so there's no need to pad. 9's complement of $49 = (9-4)(9-9) = 50$ Therefore, $B - A = -49$ is represented as $50$ in 9's complement.