Data Representation

transcript

Data Representation

CS2052 Computer Architecture

Computer Science & Engineering

University of Moratuwa

Dilum BandaraDilum.Bandara@uom.lk

Outline

Representing numbers

Unsigned

Signed

Floating point

Representing characters & symbols

Unicode

Data Representation in Computers

Data are stored in Registers

Registers are limited in number & size

With a n-bit register

Min value 0

Max value 2n-1

n-1 n-2 ...… ... 2 01

n-bits

Data Representation

Representation

• Represents quality or

characteristics

• Not proportional to a

• Name, NIC no, index no,

Address

Qualitative

Quantitative

• Quantifiable

• Proportional to value α

• No of students, marks for

CS2052, GPA

Data Representation (Cont.)

Quantitative

Integers

Signed

Unsigned

Non-integers

Signed

UnsignedQualitative

Number Systems

Decimal number system

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Binary number system

Octal number system

0, 1, 2, 3, 4, 5, 6, 7

Hexadecimal number system

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

Quantitative Numbers

Integers

Unsigned 20

Signed +20, -20

Non-integers

Floating point numbers - 10.25, 3.33333…, 1/8 = 0.125

Signed Integers

We need a way to represent negative values

3 representations

Sign & Magnitude representation (S&M)

Complement method

Bias notation or Excess notation

1. Sign & Magnitude Representation

n-bit unsigned magnitude & sign bit (S)

0 – Integer is positive or zero

1 – Integer is negative or zero

Range –(2n-1) to +(2n-1)

sign n-1 ...n-2 ... 2 01

Magnitude (n-bits)

Example – Sign & Magnitude

If 8-bit register is used what are min & max

numbers?

What are 0000 0000 and 1000 0000 in decimal?

Representation of zero is not unique

Sign & Magnitude (Cont.)

Advantages

Sign reversal

Finding absolute value |a|

Flip sign bit

Disadvantage

Adding a negative of a number is not the same as

subtraction

e.g., add 2 and -3

Need different operations

Zero is not unique

2. Complement Method

Base = Radix

Radix r system means r number of symbols

e.g., binary numbers have symbols 0, 1

2 types

r’s complement

(r – 1)’s complement

Where r is radix (base) of number system

Examples

Decimal 9’s & 10’s complement

Binary 1’s & 2’s complement

Complement Method – Definition

Given a number m in base/radix r & having n

digits

(r – 1)’s complement of m is

(rn – 1) – m

r ’s complement of n is

(rn – 1) – m + 1 = rn – m

Example – Complement Method

If m = 5982 & n = 4 digits 9’s complement is

9 9 9 9 - maximum representable no

5 9 8 2 –

4 0 1 7

10’s complement

9 9 9 9 or 1 0 0 0 0

5 9 8 2 – 5 9 8 2 –

4 0 1 7 4 0 1 8

4 0 1 814

Example – Complement Method

If m = 382 & n = 3

-n = -382 =

9 9 9 9 9 9

3 8 2 – 3 8 2 -

6 1 7 6 1 7

-382 = 617 or 618

Depending on which complement we use

These are called complementary pair

1’s complement

Calculated by

(2n – 1) – m

If m = 0101

1’s complement of m on a 4-bit system

1 1 1 1

0 1 0 1 –

1 0 1 0

This represents -5 in 1’s complement

Finding 1’s Complement – Short Cut

Invert each bit of m

Example

m = 0 0 1 0 1 0 1 1

1’s complement of m = 1 1 0 1 0 1 0 0

m = 0 0 0 0 0 0 0 0 = 0

1’s complement of m = 1 1 1 1 1 1 1 1 = -0

Values range from -127 to +127

Addition with 1’s Complement

If results has a carry add it to LSB (Least

Significant Bit)

Example

Add 6 and -3 on a 3-bit system

6 = 1 1 0

-3 = 1 0 0 +

= 1 0 1 0

2’s Complement

Doesn’t require end-around carry operation as in

1’s complement

2’s complement is formed by

Finding 1’s complement

Add 1 to LSB

New range is from -128 to +127

-128 because of +1 to negative value

Example – 2’s Complement

Find 2’s complement of 0101011

m = 0 1 0 1 0 1 1

= 1 0 1 0 1 0 0

________ 1 +

2’s = 1 0 1 0 1 0 1

Short-cut

1. Search for the 1st bit with value 1 starting from LSB

2. Inver all bits after 1st one

Example – 2’s Complement (Cont.)

Add 6 and -5 on a 4-bit system

5 = 0101

-5 = 1011

6 = 0 1 1 0

-5 = 1 0 1 1+

1 0 0 0 1 0 0 0 1

Discard

1’s vs. 2’s Complement

1’s complement has 2 zeros (+0, -0)

Value range is less than 2’s complement

2’s complement only has a single zero

Value range is unequal

No need of a separate subtract circuit

Doing a NOT operation is much more cost effective

interms of circuit design

However, multiplication & division is slow

S&M, 1’s, & 2’s

Source: www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html

4-bit, 2’s Complement Adder

Subtractor

Source: www.instructables.com/id/How-to-Build-an-8-Bit-Computer/step7/The-ALU/

Detecting Negative Numbers &

Overflow

Check for MSB

To find magnitude

1’s complement

Flip all bits

2’s complement

Flip all bits + 1

Rules to detect overflows in 2’s complement

If sum of 2 positive numbers yields a negative result,

sum has overflowed

If sum of 2 negative numbers yields a positive result,

sum has overflowed

Else, no overflow 25

3. Bias Notation

Number range to be represented is given a

positive bias so that smallest unsigned value is

equal to -B

If bias is B

Value B in number range becomes zero

If bias is 127 for 8-bit number range is

-127 (0 - 127) to 128 (255 - 127)26

Bias Notation (Cont.)

Bit pattern Unsigned Biased-127

0000 0000 +0 -127

0000 0001 +1 -126

0000 0010 +2 -125

… … …

0111 1110 +126 -1

0111 1111 +127 +0

1000 0000 +128 +1

… … …

1111 1111 +255 +128

Floating Point Numbers

We needed to represent fractional values &

values beyond 2n – 1

+3207.23 = 3.20723x103

-0.000321 = -3.21x10-4

Mantissa

(base)

Exponent

Floating Point Numbers (Cont.)

es mN 2.)1(

Mantissa

Exponent

sign en-1 ...… e0 … m0m1mn-1 …mn-2

MantissaExponent

IEEE Floating Point Standard (FPS)

2 standards

1. Single precision

32-bits

23-bit mantissa

8-bit exponent

1-bit sign

2. Double precision

64-bits

52-bit mantissa

11-bit exponent

1-bit sign

31 30 23 22 0

S Exponent Mantissa (bits 0-22)

63 62 52 51 0

S Exponent Mantissa (bits 0-51)

IEEE Floating Point Standard (Cont.)

E – 127 = e

E = e + 127

What is stored is E

s31 e30 ...… e23 … m0m1m22 …mn-2

MantissaExponent

1272].1[)1( Es mN

Single Precision

23-bit mantissa

m ranges from 1 to (2 – 2-23 )

8-bit exponent

E ranges from 1 to 254

0 & 255 reserved to represent -∞, +∞, NaN

e ranges from -126 to +127

Maximum representable number

2 x 2127 = 2128

Review – Binary Fractions

What is 101.11012 in decimal? 22 x 1 + 21 x 0 + 20 x 1 + 2-1 x 1 + 2-2 x 1 + 2-3 x 0 + 2-4 x 1

= 4 x 1 + 2 x 0 + 1 x 1 + 0.5 x 1 + 0.25 x 1 + 0.125 x 0 + 0.0625 x 1

= 4 + 1 + 0.5 + 0.25 + 0.0625

= 5.8125

What is 5.812510 in binary?

Integer – 5 = 101

Fraction – Keep multiplying fraction until answer is 1

0.8125 x 2 = 1.625

0.625 x 2 = 1.25

0.25 x 2 = 0.5

0.5 x 2 = 1.0

0.812510 = .1101233

Example – Single Precision

IEEE Single Precision Representation of 17.1510

Find 17.1510 in binary

17.1510 = 10001.00100112

= 1.00010010011 x 24

E = e + 127

E = 4 + 127 = 131

0 1000 0011 0001 0010 0110 0000 ……..

MantissaExponent

Character Representation

Belong to category of qualitative data

Represent quality or characteristics

Includes

Letters a-z, A-Z

Digits 0-9

Symbols !, @ , *, /, &, #, $

Control characters <CR>, <BEL>, <ESC>, <LF>

Character Representation (Cont.)

With a single byte (8-bits) 256 characters can be

represented

Standards

ASCII – American Standard Code for Information

Interchange

EBCDIC – Extended Binary-Coded Decimal

Interchange Code

Unicode

De facto world-wide standard

Used to represent

Upper & lower-case Latin letters

Numbers

Punctuations

Control characters

There are 128 standard ASCII codes

Can be represented by a 7 digit binary number

000 0000 through 111 1111

Plus parity bit

ASCII Hex Symbol

(space)

ASCII Table

ASCIIHex Symbol

ASCII Hex Symbol

ASCII Table (Cont.)

ASCII Hex Symbol

ASCII – Things to Note

ASCII codes for digits aren’t equal to numeric

Uppercase & lowercase alphabetic codes differ

by 0x20

Shift key clears bit 5

0x20 = 32 = 0010 0000

Example:

A = 65 = 0100 0001

a = 97 = 0110 0001

Most languages need more than 128 characters

Unicode (www.unicode.org)

Designed to overcome limitation of number of characters

Assigns unique character codes to characters in a wide range of languages

A 16-bit character set

UCS-4 is 32-bit

65,536 (216) distinct Unicode characters

Unicode provides a unique number for every character,no matter what the platform,no matter what the program,no matter what the language

Unicode (Cont.)

Source: www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html

Unicode – Sinhala & Tamil