Question

Now, suppose you have the following two 8-bit hexadecimal numbers, both of which use two's complement: ef 4a

4. What is the decimal equivalent of each of these numbers?

6. What is their sum in hexadecimal? (Note that the sum must also be confined to 8 bits in order for two's complement to work.)

Answer 1

4)
Hexadecimal     Binary
    0           0000
    1           0001
    2           0010
    3           0011
    4           0100
    5           0101
    6           0110
    7           0111
    8           1000
    9           1001
    A           1010
    B           1011
    C           1100
    D           1101
    E           1110
    F           1111
Use this table to convert from hexadecimal to binary
Converting EF to binary
E => 1110
F => 1111
So, in binary EF is 11101111
EF in binary is 11101111

Converting 4A to binary
4 => 0100
A => 1010
So, in binary 4A is 01001010
4A in binary is 01001010

first = 11101111
since left most bit is 1, this number is negative number.
so, follow these steps below to convert this into a decimal value.
I. first flip all the bits. Flip all 0's to 1 and all 1's to 0.
   11101111 is flipped to 00010000
II. Add 1 to above result
00010000 + 1 = 00010001
III. Now convert this result to decimal value
=> 10001
=> 1x2^4+0x2^3+0x2^2+0x2^1+1x2^0
=> 1x16+0x8+0x4+0x2+1x1
=> 16+0+0+0+1
=> 17
so, 11101111 in 2's complement decimal is -17

second = 01001010
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1001010
=> 1x2^6+0x2^5+0x2^4+1x2^3+0x2^2+1x2^1+0x2^0
=> 1x64+0x32+0x16+1x8+0x4+1x2+0x1
=> 64+0+0+8+0+2+0
=> 74
so, 01001010 in 2's complement decimal is 74

Answer: numbers in decimal are -17 and 74

6)
Adding 11101111 and 01001010 in binary
    11101111
    01001010
-------------
 (1)00111001
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00111001

Verification
---------------
sum = 00111001
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 111001
=> 1x2^5+1x2^4+1x2^3+0x2^2+0x2^1+1x2^0
=> 1x32+1x16+1x8+0x4+0x2+1x1
=> 32+16+8+0+0+1
=> 57
This is correct since we can verify that -17+74 = 57
So, there was no overflow.

Now, let's convert the sum 00111001 back to hexadecimal

Converting 00111001 to hexadecimal
0011 => 3
1001 => 9
So, in hexadecimal 00111001 is 0x39

Answer: 0x39