lec7 data representation

8/6/2019 Lec7 Data Representation

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Chapter 3Data Representation

Dr. Bernard Chen Ph.D.

University of Central ArkansasSpring 2009


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Data Types The data types stored in digital computers may be

classified as being one of the following categories:

1. numbers used in arithmetic computations,2. letters of the alphabet used in data processing, and3. other discrete symbols used for specific purposes.

All types of data are represented in computers in binary-

coded form.


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Radix representation of

numbers Radix or base : is the total number

of symbols used to represent avalue. A number system of radix ruses a string consisting of r distinctsymbols to represent a value.


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Binary NumbersBinary numbers are made of b inary dig it s ( bit s):0 and 1

Convert the following to decimal(1011) 2 = 1x2 3 + 0x2 2 + 1x2 1 + 1x2 0

= (11) 10


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Example

Use radix representation to convert the binary number (101.01)into decimal.

The position value is power of 2

1 0 1 . 0 1

22 21 20 2-1 2-2

4 + 0 + 1 + 0 + 1/2 2 = 5.25(101.01) 2 (5.25) 10

= 1 x 2 2 + 0 x 2 + 1 + 0 x 2 -1 + 1 x 2 -2


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Binary Addition

1 1 1 1 0 1

+ 1 0 1 1 1

---------------------

0

1

0

1

1

1111

1 1 00

carries

Example Add (11110) 2 to

(10111) 2

(111101) 2 + (10111) 2 = (1010100) 2carry


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Binary SubtractionWe can also perform subtraction (with borrows).Example: subtract (10111) from (1001101)

1 10

0 10 10 0 0 10

1 0 0 1 1 0 1

- 1 0 1 1 1

------------------------

0 1 1 0 1 1 0

borrows

1+1=2

(1001101) 2 - (10111) 2 = (0110110) 2


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The Growth of BinaryNumbers

n 2n

0 2 0=11 2 1=22 2 2=43 2 3=84 2 4=165 2 5=32

6 26

=647 2 7=128

n 2n

8 2 8=2569 2 9=512

10 2 10 =1024

11 2 11 =2048

12 2 12 =4096

20 2 20 =1M30 2 30 =1G

40 240

=1T

Mega

Giga

Tera


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Octal Numbers

Octal numbers (Radix or base=8) are made of octaldigits: (0,1,2,3,4,5,6,7)

How many items does an octal number represent?

Convert the following octal number to decimal(465.27) 8 = 4x8 2+ 6x8 1+ 5x8 0 + 2x8 -1+ 7x8 -2


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Counting in Octal

0 1 2 3 4 5 6 7

10 11 12 13 14 15 16 17

20 21 22 23 24 25 26 27


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Conversion Between Number

Bases

Decimal(base 10)

Octal(base 8)

Binary(base 2)

Hexadecimal(base16)

We normally convert to base 10because we are naturally used to the

decimal number system.

We can also convert to other number systems


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Converting an Integer from Decimal to AnotherBase

1. Divide the decimal number by the base (e.g. 2)2. The remainder is the lowest-order digit

3. Repeat the first two steps until no divisor remains.

4. For binary the even number has no remainder 0, while the odd has 1

For each digit position:


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Converting an Integer from Decimalto

Another Base

Example for (13) 10 :Integer Quotient

13/2 = (12+1) a 0 = 1

6/2 = ( 6+0 ) a 1 = 03/2 = (2+1 ) a 2 = 11/2 = (0+1) a 3 = 1

Remainder Coefficient

Answer (13) 10 = (a 3 a 2 a 1 a 0)2 = (1101) 2


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Converting a Fraction from

Decimalto

Another Base

1. Multiply decimal number by the base (e.g. 2)2. The integer is the highest-order digit

3. Repeat the first two steps until fraction becomes zero.



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DECIMAL TO BINARY CONVERSION(INTEGER+FRACTION)

(1) Separate the decimal number into integer and fraction parts.

(2) Repeatedly divide the integer part by 2 to give a quotientand a remainder and

Remove the remainder. Arrange the sequence of remaindersright to left from the period. (Least significant bit first)

(3) Repeatedly multiply the fraction part by 2 to give an integerand a fraction partand remove the integer. Arrange the sequence of integers

left to right from the period. (Most significant fraction bitfirst)


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(Example) (41.6875) 10 (?)2

Integer = 41, Fraction = 0.6875

The first procedure produces

41 = 32+8+1

= 1 x 2 5 + 0 x 2 4 + 1 x 2 3 + 0 x 2 2 + 0 x 2 + 1

Integer remainder 41 /2 120 0

10 05 12 01 1

Overflow Fraction

X by 2 .6875

1 .3750

0 .750

1 .51 0

.Closer tothe point


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Converting an Integer fromDecimal to Octal

1. Divide decimal number by the base (8)

2. The remainder is the lowest-order digit

3. Repeat first two steps until no divisor remains.



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Example for (175) 10 :Integer Quotient

175/8 = 21 + 7/8 a 0 = 721/8 = 2 + 5/8 a 1 = 5

2/8 = 0 + 2/8 a 2 = 2

Remainder Coefficient

Answer (175) 10 = (a 2 a 1 a 0)2 = (257) 8


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1. Multiply decimal number by the base (e.g. 8)2. The integer is the highest-order digit

3. Repeat first two steps until fraction becomeszero.



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Hexadecimal NumbersHexadecimal numbers are made of 16 symbols:

(0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F)

Convert a hexadecimal number to decimal

(3A9F) 16 = 3x163

+ 10x162+

9x161+

15x160

= 14999 10 Hexadecimal with fractions:

(2D3.5) 16 = 2x16 2+ 13x16 1+ 3x16 0 + 5x16 -1 = 723.3125 10Note that each hexadecimal digit can be representedwith four bits.

(1110) 2 = (E) 16Groups of four bits are called a nibble.

(1110) 2


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ExampleConvert the decimal number(107.00390625)10 into hexadecimalnumber.

(107.00390625)10 (6B.01)16

Integer remainder

107 Divide/16

6 11=B

0 6

Overflow Fraction

X by 16 .00390625

0 . 06251 . 0000

.Closer to

the period


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One to one comparisonBinary, octal, andhexadecimal similarEasy to build circuitsto operate on theserepresentationsPossible to convertbetween the threeformats


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Converting between Base 16and Base 2

Conversion is easy!Determine 4-bit value for each hex digit

Note that there are 2 4 = 16 different values of four bits which means each 16 value is

converted to four binary bits. Easier to read and write in hexadecimal.

Representations are equivalent!

3A9F 16 = 0011 1010 1001 1111 23 A 9 F


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Converting between Base 16and Base 8

1. Convert from Base 16 to Base 22. Regroup bits into groups of three starting from

right

3. Ignore leading zeros

4. Each group of three bits forms an octal digit (8 is

35237 8 = 011 101 010 011 111 25 2 3 73

3A9F 16 = 0011 1010 1001 1111 23 A 9 F


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ExampleConvert 101011110110011 to

a. octal number

b. hexadecimal number a. Each 3 bits are converted to octal :

(101) (011) (110) (110) (011)

5 3 6 6 3

101011110110011 = (53663) 8 b. Each 4 bits are converted to hexadecimal:

(0101) (0111) (1011) (0011)

5 7 B 3101011110110011 = (57B3) 16

Conversion from binary to hexadecimal is similar except that the bitsdivided into groups of four.

Bi C d d


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Binary CodedDecimal Binary coded decimal (BCD) represents each decimal digit with four bits

Ex. 0011 0010 1001 = 329 10

This is NOT the same as 001100101001 2 Why use binary coded decimal? Because people think in decimal.

Digit BCD Code Digit BCD Code

0 0000 5 0101

1 0001 6 01102 0010 7 01113 0011 8 10004 0100 9 1001

3 2 9


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BCD versus other codes BCD not very efficient

Used in earlycomputers (40s, 50s)

Used to encodenumbers for seven-segment displays.

Easier to read?

(Example)The decimal 99 is represented

by 1001 1001.


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