Question

1. Convert the following decimal numbers to binary (You may use as many binary digits as necessary)
   1. 256¹⁰
   2. 2467¹⁰
   3. 1023¹⁰
   4. 23¹⁰
   5. 7¹⁰
2. Convert the following binary numbers to hexadecimal
   1. 1101 0010 1101 1111
   2. 1100 0101 0010 1001
   3. 1000 0000 0000 0001
   4. 0001 1000 0000 0000
3. Given  bits, how many unsigned binary numbers can be represented?
4. Given  bits, how many signed binary numbers can be represented using the two’s compliment method?
5. In the two’s complement method, why is there one more negative number than there are positive numbers?

Answer 1

1)
a)
Divide 256 successively by 2 until the quotient is 0
256/2 = 128, remainder is 0
128/2 = 64, remainder is 0
64/2 = 32, remainder is 0
32/2 = 16, remainder is 0
16/2 = 8, remainder is 0
8/2 = 4, remainder is 0
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 100000000
Answer: 100000000

b)
Divide 2467 successively by 2 until the quotient is 0
2467/2 = 1233, remainder is 1
1233/2 = 616, remainder is 1
616/2 = 308, remainder is 0
308/2 = 154, remainder is 0
154/2 = 77, remainder is 0
77/2 = 38, remainder is 1
38/2 = 19, remainder is 0
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 100110100011
Answer: 100110100011

c)
Divide 1023 successively by 2 until the quotient is 0
1023/2 = 511, remainder is 1
511/2 = 255, remainder is 1
255/2 = 127, remainder is 1
127/2 = 63, remainder is 1
63/2 = 31, remainder is 1
31/2 = 15, remainder is 1
15/2 = 7, remainder is 1
7/2 = 3, remainder is 1
3/2 = 1, remainder is 1
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 1111111111
Answer: 1111111111

d)
Divide 23 successively by 2 until the quotient is 0
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 10111
Answer: 10111

e)
Divide 7 successively by 2 until the quotient is 0
7/2 = 3, remainder is 1
3/2 = 1, remainder is 1
1/2 = 0, remainder is 1
Read remainders from the bottom to top as 111
Answer: 111