All information about numbers

1. Number System
• Decimal Number System
• The decimal system contains ten unique
symbols 0,1,2,3,4,5,6,7,8 and 9
• Since counting in decimal involves ten
symbols, we say that it’s base or radix is 10

2. • The left most digit in any number
representation which has the greatest
positional weight out of all the digits present
in that number, is called the most significant
digit (MSD)
• The right most digit in any number
representation which has the least positional
weight out of all the digits present in that
number, is called the least significant digit
(LSD)

3. • The digits on the left side of the decimal point
form the integer part of a decimal number
• The digit on the right side of the decimal point
form the fractional part of a decimal number
• The digit to the right of the decimal point have
weights which are negative powers of 10 and
the digits to the left of the decimal point have
weights which are positive powers of 10
eg 9256.26
Integer part Fractional part

4. • The value of a decimal number is the sum of
the products of the digits of the number with
their respective column weights
• In general
dn
dn-1
dn-2
… d1
d0
. d-1
d-2
d-3
…
• Value of decimal number is
(dn
* 10n
) + (dn-1
* 10n-1
) + … + (d1
* 101
) +
(d0
* 100
) + (d1
* 10-1
) + (d-2
* 10-2
) + …

5. • EXAMPLE : 9256.26
• (9 * 103
) + (2 * 102
) + (5 * 101
) + (6 * 100
) + (2
* 10-1
) + (6 * 10-2
)
• = (9 * 1000) + (2 * 100) + (50) + (6 * 1) +
(2 / 10) + (6 * 1/ 100)
• = 9000 + 200 + 50 + 6 + (1/5 + 6/100)
• = 9256 + 0.26
• = 9256.26

6. Binary Number System
• The base or radix is 2. hence it has 2 symbols, 0 an
1
• A binary digit is called a bit
• A binary number consist of a sequence of bits,
each of which is either 0 or 1
• The binary point separates the integer and
fraction parts

7. • The first bit to the left of the binary point has a weight
of 20
and that column is called units column.
• The second bit to the left has a weight of 21
and it is in
the 2’s column and so on…
• The first bit to the right of the binary point has a
weight of 2-1
and it is said to be in the 1/ 215
column
and so on
• In general; dn
dn-1
dn-2
… d1
d0
. d-1
d-2
d-3
… dk
its
decimal equivalent is;
(dn
* 2n
) + (dn-1
* 2n-1
) + … + (d1
* 21
) + (d0
* 20
) +
(d-1
* 2-1
) + (d-2
* 2-2
) + ...

8. • Binary number system is used in digital computers
because the switching circuits used in these
computers use 2-state devices such as transistors,
diodes, etc.
• A transistors can be OFF or ON
• A switch can be OPEN or CLOSED
• A diode can be OFF or ON
• These devices have to exist in one of the two
possible states. These 2 states can be represented
by 0 and 1.

9. Decimal to Binary Conversion
• The decimal integer number is converted to binary
integer number by successive division by 2, and the
decimal fraction is converted to binary fraction by
successive multiplication by 2. This is known as double
– dabble method.
• In this method, the given decimal integer number is
successively divided by 2 till the quotient is zero. The
remainder read from bottom to top give the
equivalent binary integer number.
• In the successive multiplication by 2 method, the given
decimal fraction and the subsequent decimal fractions
are successively multiplied by 2, till the fraction part of
the product is 0. The integers are read from top to
bottom give the equivalent binary fraction

10. • Convert 163.87510
to binary
• Conversion of 163 (integer)
2 163 (163)10
= (10100011)2
2 81 1
2 40 1
2 20 0
2 10 0
2 5 0
2 2 1
1 0

11. • Conversion of fraction (0.875)10
• 0.875 0.750 0.500
x 2 x 2 x 2
-------- --------- ---------
1.750 1.500 1.000
1 1 1
• (0.875)10
= (111)2
• (163.875) = (10100011.111)

12. • Convert 5210
to Binary
2 52
2 26 0
2 13 0
2 6 1
2 3 0
1 1
(52)10
= (110100)2

13. • Convert (0.75)10
to binary
• 0.75 0.50
x 2 x 2
------ --------
1.50 1. 00
1 1
• (0.75)10
= (0.11)2

14. • Convert (105.15)10
to Binary
• Conversion of integer (105)10
2 105
2 52 1
2 26 0
2 13 0
2 6 1
2 3 0
1 1
(105)10
= (1101001)

15. • Conversion of fraction (0.15)10
• 0.15 0.30 0.60 0.20 0.40 0.80
x 2 x 2 x 2 x 2 x 2 x 2
------- ------- ------- ------- ------- -------
0.30 0.60 1.20 0.40 0.80 1.60
0 0 1 0 0 1
• This particular fraction can never be expressed
exactly in binary. This process may be terminated
after a few steps
• (0.15)10
= (001001)2
• (105.15)10
= (1101001.001001)2

16. • (13)10
2 13
2 6 1
2 3 0
1 1
• (13)10
= (1101)2

17. • (0.65625)10
• 0.65625 0.31250 0.62500 0.25000 0.50000
x 2 x 2 x 2 x 2 x 2
----------- ----------- ----------- ----------- -----------
1.31250 0.62500 1.25000 0.50000 1.00000
1 0 1 0 1
• (0.65625)10
= (010101)2

18. • (10.625)10
2 10
2 5 0
2 2 1
1 0
• 0.625 0.250 0.500
x 2 x 2 x 2
-------- -------- --------
1.250 0.500 1.000
1 0 1
• (0.625)10
= (0.101)2
• (10.625)10
= (1010.101)2

19. • (25.5)10
2 25
2 12 1
2 6 0
2 3 0
1 1
(25)10
= (11001)2
0.5
x 2
-----
1.0
1
(0.5)10
= (0.1)2
(25.5)10
= (11001.1)2

20. Binary to Decimal Conversion
• Binary numbers may be converted to their
decimal equivalents by the positional weights
method. In this method the, each binary digit
of the number is multiplied by its positional
weight and the product terms are added to
obtain the decimal number.

21. • Convert (10101)2
To decimal
• 24
23
22
21
20
• (
1*24
)+(0*23
)+(1*22
)+(0*21
)+(1*20
)
=24 +
0 + 22
+ 0 + 2 0
=16+4+1
(21)
10

22. • Convert (11011.101)2 to decimal
• 1 1 0 1 1 . 1 0 1
• 24
23
22
21
20
2-1
2-2
2-3
• =(2
4
*1) +(2
3
*1)+(2
2
*0)+(2
1
*1)+(2
0
*1)+(2
-1
*1)+(2
-2
*0)+(2
-3
*1)
• =2
4
+2
3
+0+2
1
+2
0
+1/2+0+1/2
3
=16+18+2+1+1/2 +1/8
=27+(8+2)/16
=27+5/8
=27+0.625

23. • Convert (1001011)2
to decimal
• 1 0 0 1 0 1 1
• 26
25
24
23
22
21
20
• =(2
6*1)+(2
5*0)+(2
4*0)+(2
3*1)+(2
2*0)+(2
1*1)+(2
0*1)
• =64+0+0+8+0+2+1
• =(75)10

24. • Convert (11111)2
to decimal
• 1 1 1 1 1
• 24
23
22
21
20
• =(2
4*1)+(2
3*1)+(2
2*1)+(2
1*1)+(2
0*1)
• =16+8+4+2+1
• =(31)10

25. Octal Number System
• Base or radix is 8
• Has 8 independent symbols 0,1,2,3,4,5,6,7,8
• Since its base is 8 = 23 ,
every 3 bit group of binary
can be represented by an octal digit.
• When dealing with large binary numbers of many
bits,it is convenient and more efficient for us to
write the numbers in octal rather than binary.

26. • 28
27
26
25
24
23
22
21
20
• 256 128 64 32 16 8 4 2 1
• Octal Number Binary Number
• 0 000
• 1 001
• 2 010
• 3 011
• 4 100
• 5 101
• 6 110
• 7 111

27. • Binary To octal conversion
• To convert a binary number to an octal
number, starting from the binary point make
groups of 3 bits each ,on either side of the
binary point and replace each 3 bit binary
group by the equivalent octal digit.

28. • Convert (110101.101010)2
to Octal
• 110 101 . 101 010
• 6 5 5 2
• (65.52)8

29. • Convert (10101111001.0111)2
to Octal
• 10 101 111 001 .011 1
• Since we need 3 bits ,we add 0 to make it 3
bits wherever required
• 010 101 111 001. 011 100
• 2 5 7 1 3 4
• =(2571.34)8

30. • Convert (1001110)2
to octal
• 1 001 110
• 001 001 110
• 1 1 6
• (116)8

31. • Convert Binary number (111110011001)2
to its
octal equivalent
•

32. Decimal to Octal
• To convert a mixed decimal number to a mixed
octal number, convert the integer and the
fraction parts separately.
• To convert the given decimal integer number
to octal,succefully divide the given number by
8 till the quotient is 0.The remainder read
upwards gives the equivalent octal number.

33. • To convert the given decimal fraction to octal
successively multiply the decimal fraction by 8
till the product is 0 .
• The integers to the left of the octal point read
downwards give the octal fraction.

34. • Convert (378)10
to octal
8 378
8 472
8 5 7
0 5
• (378)10
=(572)8

35. • Convert
(5497)10 to octal
8 5497
8 687 1
8 857
8 10 5
8 1 2
0 1
• (5497)10=(12571)8

36. • Convert (0.6875) to octal
• 0.6875 0.5000
x 8 x 8
-------- --------
5.5000 4.0000
•
• 5 4

37. • Convert (112)10
to ( )8
Soln:
(160)8
• Convert (2048)10
to ( )8
• Soln:
(4000)8
• Convert (891)10
to ( )8
• Soln:
(1573)8

38. Octal Number to Decimal Conversion
• To convert an octal number to a decimal
number,multiply each digit in the octal
number by the weight of its position and add
all the product terms.

39. • Convert (4057)8
to decimal
• =(4*83
)+(0*82
)+(5*81
)+(7*80
)
• =(4*512)+0+40+7
• =2048+47
• =(2095)10

40. • Convert 23228
to its decimal equivalent
• =(2*83
)+(2*82
)+(2*81
)+(2*80
)
• =1024+192+16+2
• =123410

41. Hexadecimal Number System
• Binary Numbers are too lengthy to be handled
by humans so there is a need to represent the
binary numbers concisely.Hence haxadecimal
number system was developed.
• Base or radix is 16.
• The 16 independent sysmbols are
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

42. • Since its base is 16 that is 24
every 4 binary
digit combination can be represented by one
hexadecimal digit.
• A 4 bit group is called a nibble.
Since computer
words come in 8 bits ,16 bits ,32 bits and so on
.i.e multiples of 4 bits,they can be represented
in hexadecimal.

43. • Bit:A bit short for bit is the smallest unit of
data that can be stored in the computer
• 1 (True) 0 (False)
• Byte:A byte contains 8 bits.For example
11101001.
• (A single character that you type ,such as the
letter A or letter T takes up one byte of
storage.

44. • 28
27
26
25
24
23
22
21
20
• 256 128 64 32 16 8 4 2 1
• Hex Binary Hex Binary
• 0 0000 B 1011
• 1 0001 C 1100
• 2 0010 D 1101
• 3 0011 E 1110
• 4 0100 F 1111
• 5 0101
• 6 0110
• 7 0111
• 8 1000
• 9 1001
• A 1010

45. • Binary to hexadecimal conversion
• (0100101000000001)2
= ( )16
• 0100 1010 0000 0001
4 A 0 1
=(4A01)16

46. (100011011011110101000100001)2
100 0110 1101 1110 1010 0010 0001
0100 0110 1101 1110 1010 0010 0001
binary: 0100 0110 1101 1110 1010 0010 0001
hex: 4 6 D E A 2 1
Convert Binary to Hexadecimal

47. • Hexadecimal to Binary
• To convert a hexadecimal number to
binary,replace each hex digit by its 4 bit binary
group.

48. • Convert 4BAC 16
to binary
• 4 B A C
• 0100 1011 1010 1100
• (0100101110101100)2

49. • 3 A 9 E . B 0 D
• 0011 1010 1001 1110 1011 0000 1101
• (0011101010011110101100001101)
2

50. • Decimal to Hexadecimal
• Successively divide the given decimal number
by 16 till the quotient is 0.
• The remainder read from bottom to top gives
the equivalent hexadecimal integer.
• To convert a decimal fraction to hexadecimal
successively multiply the given decimal
fraction by 16,till the product is 0.

51. • Convert (2598)10
to hexadecimal
• Remainder
• 16 2598 Decimal Hex
16 162 6 6
16 10 2 2
0 10 A
(A26)16

52. • Convert (1728)10
to hexadecimal
•
• 16 1728
• 16 108 0
• 16 06 12
• 0 06
• (6C0)16

53. • Hexadecimal to Decimal
(5C7)16
to decimal
=(5*162
)+(12*161
)+(7*160
)
=(5*16*16)+(192)+7
=1280+192+7
=(1479)10

54. ASCII CODE
• ASCII stands for American Standard Code for
Information Interchange.
• Computers can only understand numbers, so
an ASCII code is the numerical representation
of a character such as 'a' or '@' or an action of
some sort.

55. Binary Arithmetic
• Binary Addition
• Binary Subtraction
• Binary Multiplication
• Binary Division

56. • Binary Addition
• The rules for binary Addition are
• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (0 With a carry 1)
• ( 1+1+1=1 with carry 1)

57. 1 1
1 0 1 0
1 + 1 1 1
1 0 0 0 1

58. • Add 1101.101 and 111.011
1 1 1 1 1 1
1 1 0 1 . 1 0 1
1 + 1 1 1 . 0 1 1
1 0 1 0 1 . 0 0 0

59. • 1 1 1
+ 1 1 0 0
1 0 1 1

60. • Add 10011 and 1111101
1 1 1 1
1 1 1 0 0 1 1
+ 11 1 1 1 1 0 1
10 0 1 0 0 0 0

61. • 10111 + 110101
• 1 1 1
1 1 0 1 1 1
• 1 1 1 0 1 0 1
• 1 0 0 1 1 0 0
• 11011 + 1001010 = 1100101

62. Binary Subtraction
• The rules for Binary Subtraction are
• 0 - 0 = 0
• 1 - 1 = 0
• 1 - 0 = 1
• 0 - 1 = 1 (with a borrow of 1)

63. • Subtract 1000-10
•
• 1 0 0 0
- 1 0
0 1 1 0

64. 1 1 1 0 1 0 1 0
- 0 0 1 1 1 0 0 1
1 0 1 1 0 0 0 1

66. •
• 0 0 1 1 0 0 0 1
- 0 0 0 1 1 0 1 1
0 0 0 1 0 1 1 0

67. 1 0 1 0 . 0 1 0
- 1 1 1 . 1 1 1
0 0 1 0 . 0 1 1

71. Binary Multiplication
• The rules for Binary Multiplication are
• 0 * 0 = 0
• 1 * 1 = 1
• 1 * 0 = 0
• 0 * 1 = 0

72. • 1 1 0 1
• 1 1 0
0 0 0 0
1 1 1 0 1
• 1 1 1 0 1
• 1 0 0 1 1 1 0

75. Binary Division