1 data representation characters, integers and real numbers binary number system octal number system...

1

Data Representation

Characters, Integers and Real Numbers Binary Number System Octal Number System Hexadecimal Number System

Powered by DeSiaMore

2

Data Representation

Binary Representation Only two things, need only 1 bit Want to represent Sweet and Sour

1 for Sweet 0 for Sour

Want to represent more than 2 things Need more bits

For 4 things, like 4 colors need 2 bits 00 - blue yellow - 10 01 - red black - 11


3

Data Representation

Binary Representation If you have

More than 4 things need more than 2 bits 8 things, need 3 bits 16 things, need 4 bits 32 things, need 5 bits

In general, if you want to represent 2n things need n bits

Binary means each time we add a bit, we double the things we can represent


4

Data Representation

Representing Numbers Integer - is a whole number, 0, 1, 2, 3, 4 … n To store integers in the computer, we only need the

number of bits for the magnitude of the number So, for integers up to 256 which = 28

we need 8 bits Higher integers will require more bits

9 bits can store integers up to 512 10 bits can store integers up to 1024


5

Representing Numbers

For integers `always include 0 Zero takes up one of our choices

Means we can actually store numbers up to 255 in bits

Binary Decimal

0000 0000 0

0000 0001 1

0000 0010 2

0000 0011 3

0000 0100 4 …

1111 1111 255 Maximum in bits

+ 1 Add 1

10000 0000 256


6

Representing Numbers Real Numbers or “Floating Point”

More complicated to store real numbers Will have a value and a fractional part Example:

104.3 -35.21

3.5346 0.999

10000.98 8.6547 Numbers that have decimal points are real numbers


7

Representing Numbers Real Numbers

Example: 3 2 7 5 . 6 1 2 103 102 101 100 10-1 10-2 10-3

To the left of decimal point: 1’s,10’s,100’s,1000’s To the right of point: 10th’s, 100th’s, 1000th’s The decimal point has a special name it’s called the

Radix point Inside the computer we store the value as an integer

and then include information about Radix Point

Radix Point


8


Composed of three parts

1. Sign + or -

2. Mantissa - the number value

3. Exponent - tells how the radix point gets moved Example: 148.69

Sign: + Exponent: -2 Mantissa: 14869


9


More Examples: -2345.1 Sign: - Exponent: -1 Mantissa: 23451

Example: 677.00 Sign: + Exponent: 0 Mantissa: 677


10


Equation for defining a real number:

Sign * mantissa * 10exponent

Decimal point “floats” according to the exponent

Examples using the above equation

(assume 5 digits per number):12001.00 +12001 X 100

-520.05 -52005 X 10-2

0.62200 +62200 X 10-5

21.998 +21998 X 10-3


11


When storing a real number, PC uses 64 bits 1 bit for the sign 11 bits for the exponent 52 bits for the mantissa


12

Representing Characters Storing Characters

A text document that you type in a word processor is decomposed into paragraphs, sentences and words

Individual characters are actually stored in the computer


13

Representing Characters To represent a document

Must represent every character How do you do this?

Documents are in a given language In any language there are a finite number of characters Want to list each character and assign it a unique binary

string or pattern


14

Representing Characters English - How many characters?

26 - Upper case letters 26 - Lower case letter 0 - digits Punctuation characters Non-printing control codes The number of unique things determines the number of bits

needed


15

Representing Characters Computer manufacturers found it easier to standardize

the way characters are represented Recall: ASCII, EBCDIC and Unicode

ASCII - Allows for 256 characters That means can represent 256 different characters How many bits then per character?


16

Representing Characters

ASCII Needs bits to represent all the characters Charts are printed in different ways

Hex, binary and decimal Easiest to read in decimal

However, numbers are stored in binary in the computer

Note: ASCII codes are in order Means can sort characters and numbers A, B, C, D, E, F, … a, b, c, d, e, f, … 1, 2, 3, 4, 5, ….


17

ASCII

Codes


18

Representing Characters ASCII

First 32 characters are non-printing ESC key, Backspace key, Tab, Enter key plus others

ASCII is ok for English, but what about other languages? Like Chinese?

Chinese has many more characters far greater than 256


19

Representing Characters Unicode

Goal of the Unicode representation is to create a character set that can represent every language used in the world

Unicode is a large extension of the ASCII character set Comparison between ASCII and UnicodeUnicode ASCII 16 bits/character 8 bits/character65,000 characters 256 charactersSuperset of ASCII


20

Representing Characters Unicode

Superset of ASCII ASCII takes up the first 256 characters of Unicode

Still evolving, currently 40,000 characters have been encoded!


21

Number Systems


22

Number Systems All number systems have a base or radix Decimal system – the base is 10 Different number systems have different bases

Binary the base = 2 Time the base = 60

Base affects counting Counting in Decimal – ranges from 0 - Counting in base 60 – ranges from 0 - 59


23

Number Systems Number systems are positional Know how big the number is from its position in the

number Example: 943

You know that 9=900, 4 = 40, 3 = 3


24

Number Systems

Positional Notation A more structured way to say this:

Rightmost digit = value x base0 power Next digit to the left = value x base1 power Next digit to the left = value x base2 power …..

Example: 943 base 109 X 102 = 9 X 100 = 9004 X 101 = 4 X 10 = 40 +3 X 100 = 3 X 1 = 3 943

We will extend this concept to the other number systems:Octal and Hexadecimal


25

Computer Number Systems Actually 3 number systems used for computers

Binary - base 2 Octal – base 8 Hexadecimal - base 16

Digits in each system Binary: 0,1 Octal: 0 - 7 Hexadecimal - 0 -, and A,B,C,D,E,F

A = 10, B=11, C=12, D=13, E=14, F=15


26

Computer Number Systems Why other number systems?

Binary numbers are hard to read – ok for computers Even small numbers need lots of digits Example:

Decimal number = 14 Binary number = 1110

Other number systems are for people Makes it easier to read and work with digital numbers


27

Binary Number Chart


28

Computer Number Systems Why other number systems?

Turns out that Octal and Hexadecimal make it easy to convert between

the different number systems Binary to Octal, Hexadecimal Octal, Hexadecimal to Decimal


29

Number System Conversions

Binary Number System and Decimal Decimal and Binary Binary and Octal and hexadecimal Binary Arithmetic


30

Converting Between Number Systems

Useful to convert between number systems People more familiar with decimal numbers Should know how to do this even if you always use a

calculator to do the conversion Binary to octal, hex or decimal Decimal to binary Octal or hex to decimal


31

Converting Between Number Systems Example: Binary to Decimal

1 0 1 1

23 22 21 20

1 x 23 = 1 x 8 = 8

0 x 22 = 0 x 4 = 0

1 x 21 = 1 x 2 = 2

1 x 20 = 1 x 1 = + 1

11


32

Converting Between Number Systems Octal Conversion

3 2 18

210

3 x 82 = 3 x 64 = 1922 x 81 = 2 x 8 = 16 +

1 x 80 = 1 x 1 = 1

20910


33

Number Systems Converting Between Number Systems

Octal Conversion

2 3 3 18

3210

2 x 83 = 2 x 512 = 1024

3 x 82 = 3 x 64 = 192 +

3 x 81 = 3 x 8 = 24

1 x 80 = 1 x 1 = 1

124110


34


Hexadecimal Conversion

A B C16

162 161 160

A x 162 = 10 x 256 = 2560B x 161 = 11 x 16 = 176 +C x 160 = 12 x 1 = 1

274810


35


Binary to decimal Number is large, takes a long time Shortcut:

Convert from binary to octal (or hex) Convert from octal (or hex) to decimal


36

Number Systems

Converting Between Number Systems Convert a binary number to Octal - Easy

111101100 = 482 in decimal

1. Break up the number into groups of three starting at the right

2. Each group of three forms a binary number

3. For each digit - you only need to know binary numbers up through 7

4. Memorize or look up on a chart


37

Converting from Binary

to Octal

Example: 111|101|100100 = 4101 = 5111 = 7

754 octal

Example: 1|010|110

110 = 6

010 = 2

001 = 1

126 octal


38

Convert from Octal to Decimal 754 octal

7 x 82 = 7 x 64 = 448

5 x 81 = 5 x 8 = 40

4 x 80 = 4 x 1 = 4

48210

126 octal

1 x 82 = 1 x 64 = 64

2 x 81 = 2 x 8 = 16

6 x 80 = 6 x 1 = 6

8610


39

Converting from Binary to Hex1. Starting from the rightmost digit, group into 4 instead of

3, convert each group of 4 into a hex digit

2. Example: 1010110 0110 = 6

0101 = 5

3. Now convert to decimal as before5 x 161 = 5 x 16 = 80

6 x 160 = 6 x 1 = 6

8610


40

Converting Decimal to Other Base Systems Rules for converting decimals involve division Divide the decimal by the other base

Example: Convert to binary, divide by 2 Get a quotient and a remainder Remainder becomes the new digit in the number Quotient replaces the original number Do this until quotient = 0


41

Converting Decimal to Other Base Systems Algorithm for Converting Decimal to other Base

Systems While quotient is not zero

Divide decimal number by new base Make remainder the next digit to the left in the answer Replace decimal number with the quotient


42

Converting Decimal to Other Base Systems Decimal to Binary

Example: 45 decimal

45 / 2 = 22 quotient with 1 remainder

22/ 2 = 11 quotient with 0 remainder

11/2 = 5 quotient with 1 remainder




Final answer: 101101 binary


43

Converting Decimal to Other Base

Systems

Decimal to octal Example: 321 decimal

321/8 = 40 is quotient and 1 is remainder



Final Answer: 501 octal


44

Converting Decimal to Other Base

Systems

Decimal to hex Example: 43 decimal

43/16 = 2 is quotient and 11 is remainder, 11=B in hex


Final answer: 2B in hex


45

Binary Arithmetic Similar to decimal - same rules, different digits Basic idea is: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3 until we

have 9 + 1 = what? No symbol for 10 in decimal, reuse the same digits Rely on position to tell what the meaning is Rightmost digit becomes 0 and carry 1 to next position:

9 + 1

10


46

Binary Arithmetic In Binary addition there is the same idea Run out of digits a lot sooner 0 + 1 = 1 and 1 + 1 = 0 with a carry of 1 Repeat this for a large number with multiple columns Rules of Binary Addition:

0 + 1 = 1

1 + 1 = 0 with a carry of 1

1 + 1 + 1 = 1 with a carry of 1


47

Binary Arithmetic Add two numbers Examples 10110 46 1010 10+ 11011 27 + 1111 15

101 5 1110 14

+ 101 5 + 111 7


48

Binary Arithmetic Binary Subtraction

Similar to decimal Rules for borrowing

0 - 0 = 0

1 - 1 = 0

0 - 1 = Can’t do this unless you borrow from the next column with a 1


49

Binary Arithmetic Binary Subtraction

Cross out the 1 and put a zero in its place In decimal, you reduce the number by 1 In binary, you cross out the 1 and put a zero

The 1 you borrowed becomes: 10 - 1 = 1 Why?


50

Binary Arithmetic

Binary Subtraction 10-1 = 1 because 10 is 2 in binary and 2 - 1 = 1 Examples: 1011 11 110010 50- 100 4 - 10101 21

1000 8 1001001 73

- 11 3 - 10111 23


51

Binary Arithmetic

Multiplication Rules

1 x 0 = 0 0 x 0 = 0 1 x 1 = 1

11010 26 101110 46 10101 21x 101 5 x 10 2 x 111 7------------------ ------------------- -----------------


52

Converting Numbers with Fractions


53

Signed Numbers


54

BCD


1 data representation characters, integers and real numbers binary number system octal number system...

Documents

number of bits

number value3

nto store integers

bits32 things

bits16 things

bits8 things

exponent52 bits

bits9 bits