Data Representation

 

All data on digital computers is represented as a sequence of 0s and 1s. This includes numeric data, text, executable files, images, audio, and video. The ASCII standard associates a seven bit binary number with each of 128 distinct characters. The MP3 file format rigidly specifies how to encode each raw audio file as a sequence of 0s and 1s. All data are numbers, and all numbers are data.

In this section we describe how to represent integers in binary, decimal, and hexadecimal and how to convert between different representations. We also describe how to represent negative integers.

Number systems.

There are many ways to represent integers: the number of days in the month of October can be represented as 31 in decimal, 11111 in binary, 1F in hexadecimal, or XXXI in Roman Numerals. It is important to remember than an integer is an integer, no matter whether it is represented in decimal or with Roman Numerals.

Data Representation

Binary Number

In mathematics and computer science, the binary numeral system, or base-2 numeral system, represents numeric values using two symbols: 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Numbers represented in this system are commonly called binary numbers

Computers are based on the binary (base 2) number system because each wire can be in one of two states (on or off)

Examples:

101012 = 10101B = 1×24+0×23+1×22+0×21+1×20 = 16+4+1= 21

101112 = 10111B = 1×24+0×23+1×22+1×21+1×20 = 16+4+2+1= 23

1000112 = 100011B = 1×25+0×24+0×23+0×22+1×21+1×20 =32+2+1= 35

 

decimal

We are most familiar with performing arithmetic with the decimal (base 10) number system. This number system has been widely adopted, in large part because we have 10 fingers. However, other number systems still persist in modern society.

decimal number system, also called Hindu-Arabic, or Arabic, number system,  in mathematics, positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot (decimal point) to represent decimal fractions. In this scheme, the numerals used in denoting a number take different place values depending upon position. In a base-10 system the number 543.21 represents the sum (5 × 102) +  (4 × 101) + (3 × 100) + (2 × 10−1) + (1 × 10−2).

 

Decimal numbers uses digits from 0..9.

These are the regular numbers that we use.

Example:

253810 = 2×103+5×102+3×101+8×100

 

Octal

Octal (pronounced AHK-tuhl , from Latin octo or “eight”) is a term that describes a base-8 number system. An octal number system consists of eight single-digit numbers: 0, 1, 2, 3, 4, 5, 6, and 7. The number after 7 is 10. The number after 17 is 20 and so forth.

The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010, which can be grouped into (00)1 001 010 – so the octal representation is 112.

In the decimal system each decimal place is a power of ten. For example:

Decimal numberIn the octal system each place is a power of eight. For example:

Octal NumberBy performing the calculation above in the familiar decimal system we see why 112 in octal is equal to 64+8+2 = 74 in decimal.

Examples:

278 = 2×81+7×80 = 16+7 = 23

308 = 3×81+0×80 = 24

43078 = 4×83+3×82+0×81+7×80 = 2247

 

Hexadecimal

In mathematics and computer science, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a–f) to represent values ten to fifteen. For example, the hexadecimal number 2AF3 is equal, in decimal, to (2 × 163) + (10 × 162) + (15 × 161) + (3 × 160), or 10995.

Each hexadecimal digit represents four binary digits (bits), and the primary use of hexadecimal notation is a human-friendly representation of binary-coded values in computing and digital electronics. One hexadecimal digit represents a nibble, which is half of an octet or byte (8 bits)

Hex numbers uses digits from 0..9 and A..F.

H denotes hex prefix.

Examples:

2816 = 28H = 2×161+8×16= 40

2F16 = 2FH = 2×161+15×16= 47

BC1216 = BC12H = 11×163+12×162+1×161+2×16= 48146

Decimal

Base-10

Binary

Base-2

Octal

Base-8

Hexadecimal

Base-16

0

0

0

0

1

1

1

1

2

10

2

2

3

11

3

3

4

100

4

4

5

101

5

5

6

110

6

6

7

111

7

7

8

1000

10

8

9

1001

11

9

10

1010

12

A

11

1011

13

B

12

1100

14

C

13

1101

15

D

14

1110

16

E

15

1111

17

F

16

10000

20

10

17

10001

21

11

18

10010

22

12

19

10011

23

13

20

10100

24

14

21

10101

25

15

22

10110

26

16

23

10111

27

17

24

11000

30

18

25

11001

31

19

26

11010

32

1A

27

11011

33

1B

28

11100

34

1C

29

11101

35

1D

30

11110

36

1E

31

11111

37

1F

32

100000

40

20

 

 

 

Reference

http://www.willamette.edu/~gorr/classes/cs130/lectures/data_rep.htm

http://introcs.cs.princeton.edu/java/51data/

http://en.wikipedia.org/wiki/External_Data_Representation

http://guide.dhcuration.org/representation/

http://en.wikibooks.org/wiki/A-level_Computing/AQA/Problem_Solving,_Programming,_Data_Representation_and_Practical_Exercise/Fundamentals_of_Data_Representation/Binary_number_system

 

 

 

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