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Data Representation

 

Data in computer is represent in binary form i.e 0 and 1 .


To know data representation in computer we must know following number system:

 

a)  Decimal number system:- 0,1,2,3,4,5,6,7,8,9 Base=10
 

b)  Binary number system:- 0,1 Base=2
 

c)   Octal number System:- 0,1,2,3,4,5,6,7 Base=8
 

d)  Hexa Decimal System:- 0,1,2,3,4,5,6,7,8,9,A(10),B(11),C(12),D(13),E(14),F(15); Base =16

 

A repeated division and remainder algorithm can convert decimal to binary, octal, or hexadecimal.
 

1.  Divide the decimal number by the desired target radix (2, 8, or 16).
 

2.  Append the remainder as the next most significant digit.
 

3.  Repeat until the decimal number has reached zero.

 

Decimal to Binary

Here is an example of using repeated division to convert 1792 decimal to binary:

 

Decimal Number

  Operation

  Quotient

  Remainder

  Binary Result

1792

÷ 2 =

896

0

0

896

÷ 2 =

448

0

00

448

÷ 2 =

224

0

000

224

÷ 2 =

112

0

0000

112

÷ 2 =

56

0

00000

56

÷ 2 =

28

0

000000

28

÷ 2 =

14

0

0000000

14

÷ 2 =

7

0

00000000

7

÷ 2 =

3

1

100000000

3

÷ 2 =

1

1

1100000000

1

÷ 2 =

0

1

11100000000

0

done.

 

Decimal to Octal

Here is an example of using repeated division to convert 1792 decimal to octal:
 

Decimal Number

  Operation

  Quotient

  Remainder

  Octal Result

1792

÷ 8 =

224

0

0

224

÷ 8 =

28

0

00

28

÷ 8 =

3

4

400

3

÷ 8 =

0

3

3400

0

done.

 

 

Decimal to Hexadecimal

Here is an example of using repeated division to convert 1792 decimal to hexadecimal:
 

Decimal Number

  Operation

  Quotient

  Remainder

  Hexadecimal Result

1792

÷ 16 =

112

0

0

112

÷ 16 =

7

0

00

7

÷ 16 =

0

7

700

0

done.


The only addition to the algorithm when converting from decimal to hexadecimal is that a table must be used to obtain the hexadecimal digit if the remainder is greater than decimal 9.

 

Decimal:

0

1

2

3

4

5

6

7

Hexadecimal:

0

1

2

3

4

5

6

7

Decimal:

8

9

10

11

12

13

14

15

Hexadecimal:

8

9

A

B

C

D

E

F

 


 

The addition of letters can make for funny hexadecimal values. For example, 48879 decimal converted to hex is:
 

Decimal Number

  Operation

  Quotient

  Remainder

  Hexadecimal Result

48879

÷ 16 =

3054

15

F

3054

÷ 16 =

190

14

EF

190

÷ 16 =

11

14

EEF

11

÷ 16 =

0

11

BEEF

0

done.

 

Other fun hexadecimal numbers include: AD, BE, FAD, FADE, ADD, BED, BEE, BEAD, DEAF, FEE, ODD, BOD, DEAD, DEED, BABE, CAFE, C0FFEE, FED, FEED, FACE, BAD, F00D, and my initials DAC.
 

Octal To Binary

Converting from octal to binary is as easy as converting from binary to octal. Simply look up each octal digit to obtain the equivalent group of three binary digits.

 

Octal:
 
0
 
1
 
2
 
3
 
4
 
5
 
6
 
7
 
Binary:
 
000 001 010 011 100 101 110 11
 

Octal: 3 4 5  
Binary: 011 100 101 =011100101 binary

 

Octal to Hexadecimal

When converting from octal to hexadecimal, it is often easier to first convert the octal number into binary and then from binary into hexadecimal. For example, to convert 345 octal into hex:

(from the previous example)
 

Octal  =

3

4

5

Binary =

011

100

101

= 011100101 binary


Drop any leading zeros or pad with leading zeros to get groups of four binary digits (bits):
Binary 011100101 = 1110 0101

Then, look up the groups in a table to convert to hexadecimal digits.

 

Binary:

0000

0001

0010

0011

0100

0101

0110

0111

Hexadecimal:

0

1

2

3

4

5

6

7

Binary:

1000

1001

1010

1011

1100

1101

1110

1111

Hexadecimal:

8

9

A

B

C

D

E

F

 

 

Binary =

1110

0101

Hexadecimal =

E

5

= E5 hex


Therefore, through a two-step conversion process, octal 345 equals binary 011100101 equals hexadecimal E5.

Octal to Decimal


Converting octal to decimal can be done with repeated division.

 

1.  Start the decimal result at 0.
 

2.  Remove the most significant octal digit (leftmost) and add it to the result.
 

3.  If all octal digits have been removed, you’re done. Stop.
 

4.  Otherwise, multiply the result by 8.
 

5.  Go to step 2.

 

Octal Digits

  Operation

  Decimal Result

  Operation

  Decimal Result

345

+3

3

× 8

24

45

+4

28

× 8

224

5

+5

229

done.

 

The conversion can also be performed in the conventional mathematical way, by showing each digit place as an increasing power of 8.

345 octal = (3 * 82) + (4 * 81) + (5 * 80) = (3 * 64) + (4 * 8) + (5 * 1) = 229 decimal

Converting from hexadecimal is next...

Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.

 

Hexadecimal:

0

1

2

3

4

5

6

7

Binary:

0000

0001

0010

0011

0100

0101

0110

0111

Hexadecimal:

8

9

A

B

C

D

E

F

Binary:

1000

1001

1010

1011

1100

1101

1110

1111

 

 

Hexadecimal =

A

2

D

E

Binary =

1010

0010

1101

1110

= 1010001011011110 binary

Hexadecimal to Octal

When converting from hexadecimal to octal, it is often easier to first convert the hexadecimal number into binary and then from binary into octal. For example, to convert A2DE hex into octal:
 

(from the previous example)
 

Hexadecimal =

A

2

D

E

Binary =

1010

0010

1101

1110

= 1010001011011110 binary


Add leading zeros or remove leading zeros to group into sets of three binary digits.

Binary: 1010001011011110 = 001 010 001 011 011 110

Then, look up each group in a table:

 

Binary:

000

001

010

011

100

101

110

111

Octal:

0

1

2

3

4

5

6

7

 

Binary =

001

010

001

011

011

110

Octal =

1

2

1

3

3

6

= 121336 octal


Therefore, through a two-step conversion process, hexadecimal A2DE equals binary 1010001011011110 equals octal 121336.

Hexadecimal to Decimal


Converting hexadecimal to decimal can be performed in the conventional mathematical way, by showing each digit place as an increasing power of 16. Of course, hexadecimal letter values need to be converted to decimal values before performing the math.

 

 

Hexadecimal:

0

1

2

3

4

5

6

7

Decimal:

0

1

2

3

4

5

6

7

Hexadecimal:

8

9

A

B

C

D

E

F

Decimal:

8

9

10

11

12

13

14

15


A2DEhexadecimal:
= ((A) * 163) + (2 * 162) + ((D) * 161) + ((E) * 160)
= (10 * 163) + (2 * 162) + (13 * 161) + (14 * 160)
= (10 * 4096) + (2 * 256) + (13 * 16) + (14 * 1)
= 40960 + 512 + 208 + 14
= 41694 decimal





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