The document discusses different number systems including decimal, binary, octal, and hexadecimal. It provides details on how each system works including the base and symbols used. Additionally, it describes how to convert between the different number systems using methods like repeated division, multiplying by place values, and looking up equivalent values in tables. Conversions covered include decimal to and from the other systems as well as converting between binary, octal, and hexadecimal.

1. DATA REPRESENTATION
CONVERSION

2. NUMBER SYSTEM
Decimal Number System
Binary Number System
Octal Number System
Hexadecimal Number System

3. Decimal Number System
 The decimal system is composed of 1- numerals or
symbols (Deca means 10, that is why this is called decimal
system). These 10 symbols are 0,1,2,3,4,5,6,7,8,9 ; using
these symbols as digit as number, we can express any
quantity. The decimal system, also called the base-10
system

4. Binary Number System
 Binary System, there are only two symbols or possible
digit values, 0 and 1. This base-2 system can be used to
represent any quantity that can be represented in
decimal or other number systems.
 The Binary system is also a positional-value system,
wherein each binary digit has its own value expressed as
a power of 2.

5. Octal Number System
 The Octal number system is very important in digital
computer work. The octal number system has a base of
eight, meaning that it has eight unique symbols :
0,1,2,3,4,5,6,7 . Thus each digit of an octal number can
have any value from 0 to 7.
 The octal system is a positional value system, wherein
each octal digit has its own value expressed as a power of
8.

6. Hexadecimal Number System
 The Hexadecimal System uses base 16. Thus, it has 16
possible digit symbols. It uses the digits 0 – 9 and the
letter A, B, C, D, E & F as the 16 digit symbols.
 Hexadecimal is a positional value System has its own
value expressed as a power of 16.

7. NUMBER CONVERSIONS
CONVERSIONS WITH BINARY
Decimal To Binary
Decimal Fraction To Binary
Binary To Decimal
Binary Fraction To Decimal

8. Decimal To Binary
 To converting decimal to Binary we use Repeated division
method. In this the no. is successively divide by 2 and its
remainder recorded.
 For Example convert decimal to Binary 4310
2 43
2 21 1
2 10 1
2 5 0
2 2 1
2 1 0
1 1
CONVERSIONS WITH BINARY
WRITE IN
THIS ORDER
From Down to Up
Your Answer 4310
= 1010112

9. Decimal Fraction To Binary
 To Convert a decimal fraction into binary, multiply the
decimal fraction by the base that’s 2. Do untill you will
get zero at fractional part.
 For Example Convert 0.37510 to Binary Integer Part
Multiply(fractional part)0.375 * 2 = 0.750 0
0.75 * 2 = 1.50 1
0.50 * 2 = 1.00 1
Your Answer is 0.37510 = 0.0112
Write
From
Up to
Down
CONVERSIONS WITH BINARY

10. Binary To Decimal
 To convert Binary to Decimal, Add positional weights or
values with power of 2 start from right side.
 For Example Convert 11011 to Decimal.
24 23 22 21 20
1 * 24 + 1 * 23 + 0 * 22 + 1 * 21 + 1 * 20
= 16 + 8 + 0 + 2 + 1
= 2710 (decimal)
1 1 0 1 1
CONVERSIONS WITH BINARY

11. Binary Fraction To Decimal
 To find binary fraction, take the sum of products of each
digit value (0 – 1) and its positional value. Starts from left
side.
 For Example convert 0.0101 to Decimal.
2-1 2-2 2-3 2-4
0 * 2-1 + 1 * 2-2 + 0 * 2-3 + 1 * 2-4
= 0 + 0.25 + 0 + 0.0625
0.01012 = 0.312510 (decimal)
CONVERSIONS WITH BINARY
. 0 1 0 1

12. NUMBER CONVERSIONS
CONVERSIONS WITH OCTAL
Decimal To Octal
Decimal Fraction To Octal
Octal To Decimal
Octal To Binary
Binary To Octal

13. Decimal To Octal
 A decimal integer can be converted to octal by
repeated-division method with division factor of 8.
 Example Convert 26610 to Octal
remainder
8 266 2
8 33 1
8 4 4
0
26610 = 4128
WRITE IN
THIS ORDER
From Down to Up
CONVERSIONS WITH OCTAL

14. Decimal Fraction To Octal
 To convert Decimal fraction into Octal, multiply
fractional part with 8 till you get fractional part 0.
 Example : convert 0.37510 to Octal
Integer Part
0.375 * 8 = 3.0 3
 0.37510 = 0.38
CONVERSIONS WITH OCTAL
Write
From
Up to
Down

15. Octal To Decimal
 It can easily converted into decimal by multiplying
each octal digit by its positional weight.
 For Example 3728 to Decimal
82 81 80
3 * 82 + 7 * 81 + 2 * 80
= 3 * 64 + 7 * 8 + 2 * 1
= 25010
CONVERSIONS WITH OCTAL
3 7 2

16. Octal To Binary
 To convert Octal To Binary is easy. This converting is
performed by converting each octal digit to its 3 bit
binary. Possible digits converted as indicated in Table
 Example : 4728 to binary
From table , 4 = 100 , 7 = 111 & 2 = 010
We get 4728 = 1001110102
CONVERSIONS WITH OCTAL
Octal
Digit
0 1 2 3 4 5 6 7
Binary 000 001 010 011 100 101 110 111

17. Binary To Octal
 Its simply the reverse of octal to binary. Make the three
bits group starting from LSB. Then convert it with
using Table
 For Example: 110101102 to Octal
Make group of three 011 , 010 & 110
011 = 3 , 010 = 2 & 110 = 6
110101102 = 3268
CONVERSIONS WITH OCTAL
Octal
Digit
0 1 2 3 4 5 6 7
Binary 000 001 010 011 100 101 110 111
Add Zero
To Make it
group of
3 bit.

18. NUMBER CONVERSIONS
CONVERSIONS WITH HEX
Decimal To HEX
Decimal Fraction To HEX
HEX To Decimal
HEX To Binary
Binary To HEX

19. Decimal To HEX
 A decimal integer can be converted to hex by repeated-
division method with division factor of 16.
 Example Convert 26610 to Hex
remainder
16 423 7
16 26 A
16 1 1
0
42310 = 1A716
1010 = A16
WRITE IN
THIS ORDER
From Down to Up
CONVERSIONS WITH HEX

20. Decimal Fraction To Hex
 To convert Decimal fraction into Hex, multiply
fractional part with 16 till you get fractional part 0.
 Example : convert 0.0312510 to Hex
Integer Part
0. 03125 * 16 =0.5 0
0. 5 * 16 = 8.0 8
 0.0312510 = 0.0816
CONVERSIONS WITH HEX
Write
From
Up to
Down

21. HEX To Decimal
 It can easily converted into decimal by multiplying
each Hex digit by its positional weight has power of 16.
 For Example 2AF16 to Decimal
162 161 160
2 * 162 + A * 161 + F * 160
= 2 * 256 + 10 * 8 + 15 * 1
= 60710
CONVERSIONS WITH HEX
2 A F
Decimal Hex
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
Decimal Hex
8 8
9 9
10 A
11 B
12 C
13 D
14 E
15 F

22. HEX To Binary
 To convert Hex To Binary is easy. This converting is
performed by converting each hex digit to its 4 bit
binary. Possible digits converted as indicated in Table
Example : 3A616 to binary
From table, 3 = 0011 ,
A = 1010 & 6 = 0110
We get 3A616 = 0011101001102
CONVERSIONS WITH HEX
Binary Hex
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
Binary Hex
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

23. Binary To HEX
 Its simply the reverse of Hex to binary. Make the four
bits group starting from LSB. Then convert it with
using Table
 For Example: 10101110102 to Hex
Make group of four 0010
, 1011 & 1010
0010 = 2 , 1011 = B & 1010 = A
10101110102 = 2BA16
CONVERSIONS WITH HEX
Add Zero to Make it
group of 4 bit.
Binary Hex
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
Binary Hex
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F