1. Numbers and Number Systems
Instructor : Eng. Haya Sammaneh

2. Introduction Real World Data Computer Data Input device Dear Mom:
Keyboard …
Digital camera …

3. Number Systems Decimal -- 10 symbols (0,1,2,3,4,5,6,7,8,9)
Binary -- 2 symbols (0,1)
Octal -- 8 symbols (0,1,2,3,4,5,6,7)
Hexadecimal symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) , A = 10 , B = 11 ….F = 15

4. Converting with fraction, examples
Binary to Decimal
Ex: (1101.1)2  1*23 + 1*22 + 0*21 + 1*20 + 1*2-1 = (13.5)10
Octal to Decimal
Ex: (673)8  6* *81 + 3*80 = (443)10
Hexadecimal to Decimal
Ex. (A9C)16  A* *161 + C * 160=
10* * * 160 =(2788)10

5. Converting, example Binary to Octal Ex: (011011.101100)2(33.54)8
Binary to Hexadecimal
Ex: ( )2(FB.D8)16
Hexadecimal to Binary
Ex. (A3.B)16  ( )2

6. Converting, example Decimal to Binary EX: (12.3)10  (1100.01001)2
12 / 2 = 6 ( Remainder 0 (right) )
6 / 2 = 3 ( Remainder 0 )
3 / 2 = 1 ( Remainder 1 )
1 / 2= 0 ( Remainder 1 (left))
0.3 * 2 = 0.6 (left)
0.6 * 2 = 1.2
0.2 * 2 = 0.4
0.4 * 2 = 0.8
0.8 * 2 = 1.6 (right)
Using the same method to convert the Decimal number to any base.

7. Signed Numbers
In general a N-bit integer can store numbers in the range of -2N-1 -1 to 2N -1

8. Ways to represent negatives
Negative integers are stored via two’s complement representation
1’s complement
reverse the bits to get the negative
Ex: 1101  1’s complement  0010
2’s complement
It happens by reversing the bits (1’s complement) then adding 1.
Ex: 1101  2’s complement  0011
Get 1’s complement  0010
then add 1  0011

9. Binary arithmetic – addition and multiplication
0 + 0 = 0
0 + 1 = 1
Overflow: If there is not enough room to hold the result correctly.
If the two numbers are of opposite signs, no overflow can occur. (Why not?)
multiplication
0 * 0 = 0
0 * 1 =0
1 * 0 = 0
1 * 1 = 1
1 + 0 = 1
1 + 1 = 0 and carry 1
(Result is smaller than one of them)

10. Concepts of bit, byte and word
Bit is the smallest data item in computers (short for "binary digit" . Each data item, or bit, can assume either the value 0 or the value 1.
Computer circuitry performs various simple bit manipulations, such as examining the value of a bit, setting the value of a bit and reversing a bit (from 1 to 0 or from 0 to 1).
Bytes are composed of eight bits which is the smallest grouping of numbers . Large amounts of memory are indicated in terms of kilobytes (1,024 bytes), megabytes (1,048,576 bytes), and gigabytes (1,073,741,824 bytes).
Words: The size of a word varies from one computer to another, depending on the CPU. For computers with a 16-bit CPU, a word is 16 bits (2 bytes). On large mainframes, a word can be as long as 64 bits (8 bytes) or 128 bit (16 byte).
Some computers and programming languages distinguish between short-words and long-words. A short-word is usually 2 bytes long, while a long-word is 4 bytes.

11. Standard Alphanumeric Formats
Problem : Representing text strings, such as “Hello, world”, in a computer
The standards for representing letters (alpha) and numbers
ASCII – American standard code for information interchange
Unicode

12. Character Code :ASCII and Unicode
Why do computers use numbers for the names of letters?
Because they store all information in number form.
Technical detail: they store information as ‘bytes’; each ‘byte’ consists of 8 ‘bits’; each ‘bit’ is either the number ‘0’ or ‘1’’

13. Character Code :ASCII and Unicode
ASCII and Unicode are two computer ‘languages’ for naming letters
The ASCII name for ‘a’ is ‘61’
The Unicode name for ‘a’ is ‘U+0061’

14. ASCII Reference Table Control , Numeric, Alphabetic, Punctuations Codes
MSD
LSD 1 2 3 4 5 6 7
NUL
DLE SP @ P p
SOH
DC1 ! A Q a W
STX
DC2 “ B R b r
ETX
DC3 # C S c s
EOT
DC4 $ D T d t
ENQ
NAK % E U e u
ACJ
SYN
& F V f v
BEL
ETB ‘ G g w 8 BS
CAN ( H X h x 9 HT EM ) I Y i y LF
SUB * : J Z j z VT
ESC + ; K [ k { FF FS ,
< L \ l | CR GS - = M ] m } SO RS .
> N ^ n ~ SI US / ? O _ o
DEL
7416

15. ASCII Most widely used coding scheme
Computer systems can represent up to 256 letters
Technical detail: with one 8-bit byte (28 = 256)
ASCII only uses 7 bits (27 = 128)
The first are called ASCII letters (characters)
1-32: That’s for control characters like the ‘option’ key.

16. ASCII Problem
No one agrees on what letters the numbers stand for.

17. Unicode: fixing the ASCII problem
Most common 16-bit form represents 65,536 characters.
Multilingual: These characters cover the principal written languages of the Americas, Europe, the Middle East, Africa, India, Asia, and Pacifica.