1. DIGITAL DESIGN THIRD EDITION M. MORRIS MANO
CHAPTER 1 : BINARY SYSTEMS PROBLEMS

2. 1. 1-) List the octal and the hexadecimal numbers from 16 to 32
1.1-) List the octal and the hexadecimal numbers from 16 to 32. Using A and B for the last two digits, list the numbers from 10 to 26 in base 12 .
Octal :
16 = 8¹ x 2 + 8º x 0 => (16)10 = (20)8
32 = 8¹ x 4 + 8º x 0 => (32)10 = (40)8
20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 40

3. Hexadecimal : Base-12 : 16 = 16¹ x 1 + 16º x 0 => (16)10 = (10)16
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B , 1C, 1D, 1E, 1F, 20
Base-12 :
10 = 12º x A => (10)10 = (A)12
26 = 12¹ x º x 2 => (26)10 = (22)12
A, B, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 20, 21, 22

4. 1.2-) What is the exact number of bytes in a system that contains (a) 32K byte, (b)64M bytes, and (c)6.4G byte ?
(a) 32K byte:
1K = 2¹º = 1,024
32K = 32 x 2¹º = 32 x 1,024 = 32,768
32K byte = 32,768 byte

5. (b) 64M byte: (c) 6.4G byte: 1M = 2²º = 1,048,576
64M = 64 x 2²º = 64 x 1,048,576 = 67,108,864
64M byte = 67,108,864 byte
(c) 6.4G byte:
1G = 2³º = 1,073,741,824
6.4G = 6.4 x 2³º = 6.4 x 1,073,741,824 = 6,871,747,674
6.4G byte = 6,871,747,674 byte

6. 1.3-) What is the largest binary number that can be expressed with 12 bits? What is the equivalent decimal and hexadecimal ?
Binary:
( )2
Decimal:
( )2 = 1x 2º+ 1 x 2¹ + 1 x 2² +…..+ 1 x 2¹¹ + 1 x 2¹²
( )2 = 4,095
Hexadecimal:
( )2
F F F =
(FFF)16

7. 1.4-) Convert the following numbers with the indicated bases to decimal : (4310)5 , and (198)12 .
(4310)5 = 0 x 5º + 1 x 5¹ + 3 x 5² + 4 x 5³ =
(4310)5 = 580
(198)12 = 8 x 12º + 9 x 12¹ + 1 x 12² =
(198)12 = 260

8. 1.5-) Determine the base of the numbers in each case for the following operations to be correct : (a) 14/2 = 5 ; (b) 54/4 = 13 ; (c) = 40 .
(a) (14)a / (2)a = (5)a
(4 x aº + 1 x a¹) / (2 x aº) = 5 x aº
(4 + a) / 2 = 5
4 + a = 10
a = 6

9. (b) (54)b / (4)b = (13)b
(4 x bº + 5 x b¹) / (4 x bº) = 3 x bº + 1 x b¹
(4 + 5b) / 4 = 3 + b
4 + 5b = b
b = 8
(c) (24)c + (17)c = (40)c
(4 x cº + 2 x c¹) + (7 x cº + 1 x c¹) = 4 x c¹
4 + 2c c = 4c
c = 11

10. 1.6-) The solution to the quadratic equation x² - 11x + 22 = 0 is x=3 and x=6. What is the base of the numbers?
x² - 11x + 22 = (x – 3) . (x – 6)
x² - 11x + 22 = x² - (6 + 3)x + (6.3)
(11)a = (6)a + (3)a
1 + a = 6 + 3
a = 8

11. 1. 7-) Express the following numbers in decimal : (10110. 0101)2 , (16
( )2
( )2 = (1/4) + (1/16)
( )2 =
= 2¹ + 2² + (2^4) +( 2^-2) + (2^-4)

12. ( )16
(16.5)16 = (5/16)
(16.5)16 =
( )8
(26.24)8 = (2/8) + (4/64)
(26.24)8 =
= 6 x 16º + 1 x 16¹ + 5 x (16^-1)
= 6 x 8º + 2 x 8¹ + 2 x (8^-1) + 4 x (8^-2)

13. 1.8-) Convert the following binary numbers to hexadecimal and to decimal : (a) , (b) Explain why the decimal answer in (b) is 8 times that of (a) .
( )2 = ( 1 . D )16 = 1 x 16º + D x (16^-1)
1 D

14. 1.9-) Convert the hexadecimal number 68BE to binary and then from binary convert it to octal .
Binary form:
( )2=( ) B E
Octal form:
( ) =(064276)8

15. 1.10-) Convert the decimal number 345 to binary in two ways : Convert directly to binary; Convert first to hexadecimal, then from hexadecimal to binary. Which method is faster ?

16. Method 1: (345)10 Number Divided by 2 Remainder 345 345/2=172 1 172
172/2=86 86
86/2=43 43
43/2=21 21
21/2=10 10
10/2=5 5
5/2=2 2
2/2=1
(345)10

17. Method 2: (345)10=(159)16 (1 101 1001)2 Number Divided by 16 Remainder
345/16=21 9 21
21/16=1 5
(345)10=(159)16
( )2

18. 1. 11-) Do the following conversion problems : (a) Convert decimal 34
1.11-) Do the following conversion problems : (a) Convert decimal to binary . (b) Calculate the binary equivalent of 1/3 out to 8 places. Then convert from binary to decimal. How close is the result to 1/3 ? (c) Convert the binary result in (b) into hexadecimal. Then convert the result to decimal . Is the answer the same ?

19. 34
0.4375
34:2=17 r=0
17:2=8 r=1
8:2=4 r=0
4:2=2 r=0
2:2=1 r=0
34=(100010)2
0.4375*2=0.875 r=0
0.875*2= r=1
0.75*2= r=1
0.5*2= r=1
0*2= r=0
0.4375=( )2
=( )2

20. (b) 1/3=0.3333… *2= r=0
*2= r=1
*2= r=0
*2= r= .
0.3333…=( ….)= 0+ ¼ / /32 +… =~ …

21. (c) …=
( )16=5/16 +5/256 +5/4096 +…=~

22. 1.12-) Add and multiply the following numbers without converting them to decimal. (a) Binary numbers 1011 and (b) Hexadecimal numbers 2E and 34 .
(a) (11) (11) (5) (5) +__________ x_____ (16) _________ (55)

23. (b)
2E (46) E
34 (52)
+____ x____
62 (98) B8 8A
+____
958(2392)

24. 1.13-) Perform the following division in binary : 1011111 ÷ 101 .
( )2=95
(101)2=5
95/5=19 (10011)2
0101
101
0000

25. 1.14-) Find the 9’s- and the 10’s-complement of the following decimal numbers : (a) (b) (c) (d)
9’s comlements : = = = =

26. 10’s complements
(a) =
(b) =
(c) =
(d) =

27. 1. 15-) (a) Find the 16’s-complement of AF3B
1.15-) (a) Find the 16’s-complement of AF3B (b) Convert AF3B to binary (c) Find the 2’s-complement of the result in (b) (d) Convert the answer in (c) to hexadecimal and compare with the answer in (a)
16^5-AF3B=50C5
(AF3B)16=
(c)
(d) = 50C5

28. 1.16-) Obtain the 1’s and 2’S complements of the following binary numbers : (a) (b) (c) (d) (e)
1’s complements: (a) (b) (c) (d) (e)
2’s complement : (a) (b) (c) (d) (e)

29. 1.17-) Perform subtraction on the following unsigned numbers using the 2’s-complement of the subtrahend. Where the result shoud be negative, 10’s complement it and affix a minus sign. Verify your answers . (a) (b) (c) (d)

30. (a)7188+6951=4139 One carry out so answer is correct.
(b) = correct answer=-1950
(c) =5005 correct answer=-4995
(d) = One carry out so answer is correct.

31. 1.18-) Perform subtraction on the following unsigned binary numbers using the 2’s-complement of the subtrahend. Where the result should be negative, 2’s complement it and affix a minus sign . (a) (b) (c) (d)

32. =00010(27-25=2)
=011111(52-21=31)
(c) =  (11-48=-37)
(d) =111111 (42-43=-1)

33. 1.19-) The following decimal numbers are shown in sign- magnitude form : and Convert them to signed 10’s-complement form and perform the following operations : (Note that the sum is and requires six digits). (a) (+9826)+(+801) (b)(+9826)+(-801) (c)(-9826)+(+801) (d)(-9826)+(-801) 00

34. =010627
(b) =09025
(c) =990975-09025
(d) =989373-10627

35. 1.20-) Convert decimal +61 and +27 to binary using the signed-2’s complement representation and enough digits to accomodate the numbers. Then perform the binary equivalent of (+27) + (- 61) , (- 27) + (+61) and (-27) + (- 61) . Convert the answers back to ecimal and verify that they are correct .

36. +61= =
+27= =
(a)27+(-61)= =
(b)-27+(+61)= =
(c)-27+(-61)= = (overflow) =

37. 1. 21-) Convert decimal 9126 to both BCD and ASCII codes
1.21-) Convert decimal 9126 to both BCD and ASCII codes. For ASCII, an odd parity bit is to be appended at the left .
BCD:
ASCII:

38. 1.22-) Represent the unsigned decimal numbers 965 and 672 in BCD and then show the steps necessary to form their sum .
965= = ___ ___ ____ _________________  (1637)10

39. 1.23-) Formulate a weighted binary code for the decimal digits using weights 6, 3, 1, 1 .

40. 6 3 1
Decimal 2
4(0101) 5
7(1010) 8 9

41. 1.24-) Represent decimal number 6027 in (a) BCD, (b) excess-3 code, and (c) 2421 code .

42. 1. 25-) Find the 9’s complement of 6027 and express it in 2421 code
1.25-) Find the 9’s complement of 6027 and express it in 2421 code. Show that the result is the 1’s complement of the answer to (c) in Problem This demonstrates that the 2421 code is self-complementing .
9’s complement of 6027 is 3972
6027 as 2421 code is 
3972 as 2421 code is 

43. 2^4 =16 2^5 =32 2^6=64  6 bits are necessary.
1.26-) Assign a binary code in some orderly manner to the 51 playing cards. Use the minimum number of bits.
2^4 =16
2^5 =32
2^6=64  6 bits are necessary.

44. 1.27-) Write the expresion “G. Boole” in ASCII using an eight-bit code. Include the period and the space. Treat the leftmost bit of each character as a parity bit. Each 8-bit code shouls have even parity.
G B O O L E ( )( ) ( ) ( ) ( ) ( ) ( ) 11

45. 1.28-) Decode the following ASCII code :
Jane Doe

46. 1.29-) The following is a string of ASCII characters whose bit patterns have benn converted into hexadecimal for compactness : 4A EF E 20 C4 EF E5 . Of the 8 bits in each pair of digits, the leftmost is a parity bit. The remaining bits are the ASCII code.
J O H N (space) D O E

47. 62 of them are numbers and letters. 32 of them are special characters.
1.30-) How many printing characters are there in ASCII ? How many of them are special characters (not letters or numerals) ?
94 characters
62 of them are numbers and letters.
32 of them are special characters.

48. 1.31-) What bit must be complemented to change an ASCII letter from capital to lowercase, and vice versa ?
Cevap: Bir ASCII karakteri büyük harften küçük harfe çevirmek için sağdan 6. bit 0 iken 1 yapılır. Küçükten büyüğe çevrilecekse 1 iken 0 yapılır.

49. 1. 32-) The state of a 12-bit register is 100010010111
1.32-) The state of a 12-bit register is What is its content if it represents (a) three decimal digits in BCD? (b) three decimal digits in the excess-3 code? (c) three decimal digits in code? (d) binary number?

50. Three Decimal Digits in BCD: Three Decimal Digits in Exces-3 Code:
Three Decimal Digits in Exces-3 Code:
(8-3) (9-3) (7-3)
Three Decimal Digits in the Code:
Binary Code:
897
564
897
2^11+2^7+2^4+2^2+2+1=2199

51. 1.33-) List the ASCII code for the 10 decimal digits with an even parity bit in the leftmos position.

52. 1.34-) Assume a 3-input AND gate with output F and a 3-input OR gate with output G. Inputs are A, B, and C . Show the signals (by means of a timing diagram) of the outputs F and G as functions of three inputs ABC. Use all possible combinations of ABC.

53. F :  A , B , C F :  A , BX , CX  AX , B , CX  AX , BX , C
NOT: X’ler HIGH ya da LOW olabilir