Homework I Solutions:
1.1
List the octal and hexadecimal numbers from 16 to 32. Using A, B, and C for the last three
digits, list the numbers from 8 to 28 in base 13.
Decimal
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Octal
20 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37 40
Hexadecimal 10 11 12 13 14 15 16 17 18 19 1A 1A 1C 1D 1E 1F 20
Decimal
Base 13
1.4
8
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
9 A B C 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 20 21 22
What is the largest binary number that can be expressed with 14 bits? What are the
equivalent decimal and hexadecimal numbers?
14-bit binary: 11 1111 1111 1111
Decimal: 214 -1 = (16,383)10
Hexadecimal: (3FFF)16
1.8 Convert the decimal number 431 to binary in two ways: (a) Convert directly to binary; (b)
convert first to hexadecimal and then from hexadecimal to binary. Which method is faster?
(431)10 = (110101111)2 = (1AF)16
Conversion from decimal to hexadecimal
is faster than conversion from decimal to
binary.
Quotients Remainders
431/2
215
1
LSB
215/2
107
1
107/2
53
1
53/2
26
1
26/2
13
0
13/2
6
1
6/2
3
0
3/2
1
1
1/2
0
1
MSB
431/16
26
15 (F)
LSD
26/16
1
10 (A)
1/16
0
1
MSD
1.10 Convert the following binary numbers to hexadecimal and to decimal: (a) 1.10010, (b) 110.010. Explain why the decimal answer in
(b) is 4 times that in (a).
(a) 1.100102 = 0001.10012 = 1.916 = 1 + 9/16 = 1.56310
(b) 110.0102 = 0110.01002 = 6.416 = 6 + 4/16 = 6.2510
Reason: 110.0102 is the same as 1.100102 shifted to the left by two places.
1.11 Perform the following division in binary: 111011 ÷ 10l.
111011.0000 101
101
1011.110011
01001
101
1001
101
1000
101
0110
101
001000
101
0110
101
1
1.15 Find the 9's and the 10's complement of the following decimal numbers:
(a) 52,784,630
(c) 25,000,000
(b) 63,325,000
(d) 00,00,000
(a)
52,784,630
9’s comp: 47,215,369
10’s comp: 47,215,370
(b)
63,325,600
9s comp: 36,674,399
10s comp: 36,674,400
(c)
25,000,000
9’s comp: 74,999,999
10s comp: 75,000,000
(d)
00,000,000
9s comp: 99,999,999
10s comp: 00,000,000
1.18 Perform subtraction on the given unsigned binary numbers using the 2's complement of the subtrahend. Where the result should be negative,
find its 2's complement and affix a minus sign.
(a) 10011 - 10001
(b) 100010 - 100011
(c) 1001 - 101000
(a)
10011
-10001
-------
(d) 110000 – 10101
(b)
10011
2’s comp. 01111
----Result
100010
100010
100010
-100011 2’s comp. 011101
------------111111 Æ-(000001)
(c)
001001
001001
-101000 2’s comp: 011000
-----------100001 Æ -(011111)
(d)
110000
110000
-010101 2’s comp: 101011
-----------1011011
1.25 Represent the decimal number 5,137 in (a) BCD, (b) excess-3 code, (c) 2421 code, and (d) a 6311 code.
(a) 5,13710
(b)
(c)
(d)
BCD: 0101_0001_0011_0111
Excess-3: 1000_0100_0110_1010
2421: 1011_0001_0011_0111
6311: 0111_0001_0100_1001
1.30 The following is a string of ASCII characters whose bit patterns have been converted into hexadecimal for compactness; 73 F4 E5 76 E5 4A
EF 62 73. Of the eight bits in each pair of digits, the leftmost is a parity bit. The remaining bits are the ASCII code.
(a) Convert the string to bit form and decode the ASCII.
(b) Determine the parity used: odd or even.
73 F4 E5 76 E5 4A EF 62 73
73: 0_111_0011 s
F4: 1_111_0100 t
E5: 1_110_0101 e
76: 0_111_0110 v
E5: 1_110_0101 e
4A: 0_100_1010 j
EF: 1_110_1111 o
62: 0_110_0010 b
73: 0_111_0011 s
Odd parity is used.
1.35 By means of a timing diagram similar to Fig. 1.5, show the signals of the outputs f and g in Fig.P1.35 as functions of the three inputs
a, b, and c. Use all eight possible combinations of a, b, and c.
a
b
c
f
g
a
0
1
0
1
0
1
0
1
0
1
0
1
b
0
0
1
1
0
0
1
1
0
0
1
1
c
0
0
0
0
1
1
1
1
0
0
0
0
f
1
1
1
1
1
1
1
0
1
1
1
1
g
1
0
0
0
0
0
0
0
1
0
0
0