Question

Convert the following decimal numbers to 16-bit 2’s complement binary. Display your result in hexadecimal.

a.3030

b.404

c.5050

d.-5050

e.-20000

Show work with steps

Answer 1

a)
Since this is a positive number. we can directly convert this into binary
Divide 3030 successively by 2 until the quotient is 0
3030/2 = 1515, remainder is 0
1515/2 = 757, remainder is 1
757/2 = 378, remainder is 1
378/2 = 189, remainder is 0
189/2 = 94, remainder is 1
94/2 = 47, remainder is 0
47/2 = 23, remainder is 1
23/2 = 11, remainder is 1
11/2 = 5, remainder is 1
5/2 = 2, remainder is 1
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 101111010110
Adding 4 zeros on left hand side of this number to make this of length 16
so, 3030 in 2's complement binary is 0000101111010110

Converting 0000101111010110 to hexadecimal
0000 => 0
1011 => B
1101 => D
0110 => 6
So, in hexadecimal 0000101111010110 is 0x0BD6

b)
Since this is a positive number. we can directly convert this into binary
Divide 404 successively by 2 until the quotient is 0
404/2 = 202, remainder is 0
202/2 = 101, remainder is 0
101/2 = 50, remainder is 1
50/2 = 25, remainder is 0
25/2 = 12, remainder is 1
12/2 = 6, remainder is 0
6/2 = 3, remainder is 0
3/2 = 1, remainder is 1
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 110010100
Adding 7 zeros on left hand side of this number to make this of length 16
so, 404 in 2's complement binary is 0000000110010100

Converting 0000000110010100 to hexadecimal
0000 => 0
0001 => 1
1001 => 9
0100 => 4
So, in hexadecimal 0000000110010100 is 0x0194

c)
Since this is a positive number. we can directly convert this into binary
Divide 5050 successively by 2 until the quotient is 0
5050/2 = 2525, remainder is 0
2525/2 = 1262, remainder is 1
1262/2 = 631, remainder is 0
631/2 = 315, remainder is 1
315/2 = 157, remainder is 1
157/2 = 78, remainder is 1
78/2 = 39, remainder is 0
39/2 = 19, remainder is 1
19/2 = 9, remainder is 1
9/2 = 4, remainder is 1
4/2 = 2, remainder is 0
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 1001110111010
Adding 3 zeros on left hand side of this number to make this of length 16
so, 5050 in 2's complement binary is 0001001110111010

Converting 0001001110111010 to hexadecimal
0001 => 1
0011 => 3
1011 => B
1010 => A
So, in hexadecimal 0001001110111010 is 0x13BA

d)
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 5050 successively by 2 until the quotient is 0
5050/2 = 2525, remainder is 0
2525/2 = 1262, remainder is 1
1262/2 = 631, remainder is 0
631/2 = 315, remainder is 1
315/2 = 157, remainder is 1
157/2 = 78, remainder is 1
78/2 = 39, remainder is 0
39/2 = 19, remainder is 1
19/2 = 9, remainder is 1
9/2 = 4, remainder is 1
4/2 = 2, remainder is 0
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 1001110111010
Adding 3 zeros on left hand side of this number to make this of length 16
So, 5050 in normal binary is 0001001110111010
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   0001001110111010 is flipped to 1110110001000101
Step 3:. Add 1 to above result
1110110001000101 + 1 = 1110110001000110
so, -5050 in 2's complement binary is 1110110001000110

Converting 1110110001000110 to hexadecimal
1110 => E
1100 => C
0100 => 4
0110 => 6
So, in hexadecimal 1110110001000110 is 0xEC46

e)
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 20000 successively by 2 until the quotient is 0
20000/2 = 10000, remainder is 0
10000/2 = 5000, remainder is 0
5000/2 = 2500, remainder is 0
2500/2 = 1250, remainder is 0
1250/2 = 625, remainder is 0
625/2 = 312, remainder is 1
312/2 = 156, remainder is 0
156/2 = 78, remainder is 0
78/2 = 39, remainder is 0
39/2 = 19, remainder is 1
19/2 = 9, remainder is 1
9/2 = 4, remainder is 1
4/2 = 2, remainder is 0
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 100111000100000
Adding 1 zeros on left hand side of this number to make this of length 16
So, 20000 in normal binary is 0100111000100000
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   0100111000100000 is flipped to 1011000111011111
Step 3:. Add 1 to above result
1011000111011111 + 1 = 1011000111100000
so, -20000 in 2's complement binary is 1011000111100000

Converting 1011000111100000 to hexadecimal
1011 => B
0001 => 1
1110 => E
0000 => 0
So, in hexadecimal 1011000111100000 is 0xB1E0