1. Data Representation in Computers
Student :Ahmed Alsaqqa
MIS: Rasha Ragheb Atallah

2. Session Objectives Describe what a Number system is
Explain the decimal, octal and hexadecimal number systems
Convert a number from one number system to another
Practice binary arithmetic
List the various standard codes used to represent the unit of information
ASCII
EBCDIC

3. Session Objectives (Contd.)
Explain Data Representation
Explain Data Storage
Explain Packed Decimal
Binary Arithmetic Calculation
Explain CPU and its speed

4. Number systems The additive approach - Number
earlier consisted of symbols e.g.
Roman number system - I for 1,
II for 2, III for 3 etc.
Positional numbering - Symbols
represent different values depending
on the position they occupy e.g. the
Decimal system

5. Decimal Number System
In the decimal number system the successive position to the left of the decimal point represent units, tens, hundreds, thousands etc.
(3 * 100) + (6*10) + (5*1) = 365
The position of the number affects its value.
These kind of number systems therefore are called positional number system.
Base
Position number
(6*10)

6. Decimal Number System (Contd.)
The value of each digit in the
number system is determined by:
a) The digit itself
b) The position of the digit
in the number
c) The base/radix of the system

7. Binary Number System
The binary number system has a base of two and symbols used are 0 and 1.
In this number system, as we move to the left, the value of the digit will be two times greater than its predecessor because the base is two.
Thus the value of the places are :
  64  32  16  8  4  2  1
Binary Number
Least Significant bit
Most Significant bit

8. Octal number systems Binary Octal 000 0 001 1 010 2 011 3 100 4 101 5
Uses a base of 8
Values increase
from right to left 1, 8, 64,
Binary Octal

9. Octal Number System
The octal system has the base of 8. The value increase from right to left as 1, 8, 64, 512, 4096.
The decimal value of an octal number 1204 can be computed as :
1204 = (1 * 512) + (2 * 64) + (0 * 8) + (4 * 1) 
=  
= 644

10. Octal Number System
To convert a number from binary to octal and vice versa, the following table must be kept in mind: 
Binary Octal  
000 0
001 1
010 2
011 3
100             4
101 5
110             6
111             7

11. Hexadecimal Number Systems
Hexadecimal Decimal A B C D E F

12. Hex. Number Systems(Contd.)
Uses a base of 16
The 16 symbols required for the
hexadecimal number system obtained
by using the alphabets A, B, C,
D, E and F
Converting hexadecimal to decimal
decimal equivalent of a hexadecimal
number A0119
(10 * 65,536)+(0 * 4,096)+(1 * 256)+
( 1 * 16) + ( 9 * 1)
= 6,55,
= 6, 55, 641

13. Binary to Decimal Conversion
Converting binary numbers to decimal
value
<64 <32 <16 <8 <4 <2 <1
The decimal equivalent of is
= (1 * 32 ) + (1 * 16) + (0 * 8) +
( 1 * 4) + ( 0 * 2) + (0 * 1) =
= 52

14. Decimal to Binary Conversion
Divide the decimal number by the
base of the required number system
Note the remainder in one column
and divide the quotient again with the base
Keep repeating this process until quotient is
reduced to a zero
Reading remainders in the reverse
order gives the binary equivalent

15. Decimal to Binary Conversion
E.g. Converting the decimal number 52
to its binary equivalent.
Remainder
2 |__52
2 |__26 | 0
2 |__13 | 0
2 |__06 | 1
2 |__03 | 0
2 |__01 | 1
2 |__00 | 1
Thus the binary equivalent of the decimal
number 52 is

16. Binary to Hexadecimal Each hexadecimal digit is represented
by 4 binary digits.
Binary Hexadecimal A B C D E F

17. Binary to Hexadecimal (Contd.)
Split the quantity into groups of four
outwards from right to left
Each group of four is directly converted into
its hexadecimal equivalent
Add zeros to the left of the number if
necessary
E.g. Binary
Hexadecimal Equivalent
A C

18. Hexadecimal to Binary Write binary equivalent of each
hexadecimal digit in groups of four
E.g. hexadecimal 191A412C
Thus the required binary number can be
written as :
The leading zeroes are omitted

19. Converting from Binary to Octal
The binary number must be divided into groups of three from the octal point- to the right in case of the fractional portion and to the left in case of the integer portion.
Each group can then be replaced with their octal equivalent.
We may add zero to the left of the number if required.
For example :
Binary  
52524 is the octal equivalent of the given binary number.

20. Converting from Octal to Binary
Each octal digit is replaced with the appropriate ‘triple’ of binary digits.
For example :
  6         5
Similarly the binary equivalent of the octal number 65 is

21. DATA (in binary Digits)
Binary Concepts 1
-- OFF
-- ON
DATA (in binary Digits)

22. Data Representation Main() { printf(“ Hello”);
printf(“We are enjoying a world of alphabetical coding”); }

23. Data Representation Digital computers use binary code to
represent characters.
Binary code is made up of binary
digits or bits.
A string of "0s" and "1s" is used to
Byte is a sequence of 8 bits.
Most computers have words that
consist of 8 or 16 bits.
In large computers the number of bits
per word could be 16 or 32 bits.

24. Data representation (Contd.)
When data is keyed in, each keystroke
is converted to a binary character
code and transmitted to the computer
Each character to the printer, screen,
disk is communicated in binary code.
While displaying or printing, the character
is converted back to human readable form

25. Data Storage Data is stored and processed in computers in
the binary form. These symbols 0 and 1, are
called bits
2 bits give four unique combinations
i.e. 00, 01, 10 , 11.
A string of 8 bits is called a byte
Words are stored one character/byte.

26. Data Storage (Contd.)
During calculation the decimal number is converted to its binary equivalent.
After calculation the result is converted back to its decimal equivalent.

27. The Packed Decimal Packed decimal - data is stored in
a half- way house between pure binary
and one byte per digit
Four bits are required to store all 10
symbols that constitute the decimal
number system
One byte would store the representation
of two decimal digits

28. The Packed Decimal (Contd.)
e.g. The number 34 would be stored
in character form as:
Byte Byte 2
Using packed decimal the same number
would be stored as :
Byte

29. Binary Arithmetic Addition
The following rules of binary addition
are to be remembered:
0 + 0 = 0
0 + 1 = 1 = 1 + 0
1 + 1 = 0 carry 1 to the next column to
the left
= 1 carry 1 to the next column
e.g. Carry
11011
100010

30. Complementary Subtraction
Three steps to perform subtraction :
Find the complement of the number you are
subtracting
To the complement of the number add the
number we are subtracting from
If there is a carry of 1 add the carry to the
result of the addition
else
re-complement the sum and attach a
negative sign

31. Complementary Subtraction (Contd.)
To find the complement of a binary
number invert all the bits
e.g.
Number Complement

32. Complementary Subtraction (Contd.)
Example of subtraction :
e.g.1
Step 1. Find the complement of
It is

33. Complementary Subtraction (Contd.)
Step 2. Add the number you are
subtracting from
carry
Since there is a carry of 1, Add the carry
+ _______ 1

34. Complementary Subtraction (Contd.)
e.g
Step 1.
Complement of is
Step 2.
Carry
Step 3. Since there is no carry we
re-complement the result and add a
negative sign
Thus the answer is

35. Multiplication 10101 x11001 ------- 00000 ----------- 1000001101
Rules for Multiplication:
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
E.g * 11001

36. Division 1. Start from the left of the dividend
2. Perform subtraction i.e. divisor should be
subtracted from the dividend
a) if subtraction is possible put 1 in the
quotient and subtract the divisor
from digits of the dividend
else
put 0 in the quotient
b) bring down the next digit to the
right of the remainder
3. Do step 2 till no more digits remain in
the dividend

37. Division The complete table for binary division is: 0/1 = 0 1/1 = 1
For example:
/ 110
Then,
0101 (Quotient)
________
(Dividend)
(Divisor) Step 1
Step 2b
Step 2a
Step 2b
Step 2b
11 (Remainder)

38. Unit of Information Most computers use a coded version of true
binary numbers to represent letters, special
symbols, decimal numbers etc.
7 digits are required to uniquely represent
all 128 characters
Standardised coding to enable transfer
of data between computers

39. ASCII Common standard is the American Standard
Code for Information Interchange (ASCII)
ASCII uses 7 bits per character
possible to provide 128 different
arrangements
Separate Codes are used to convey -
end of file, end of page etc. to the computer
These codes are called non-printable
control characters
ASCII code is used to represent data
internally in personal computers

40. EBCDIC Extended Binary Coded Decimal Interchange Code ( EBCDIC)
EBCDIC uses 8 bits per character
Thus 256 characters can be represented
using EBCDIC
The EBCDIC code is used in IBM
mainframe models and other similar
machines
Electronic Circuits available to transform
characters from ASCII to EBCDIC and
vice- versa
One can achieve the same results
using a computer program

41. An insight into CPU
CPU also referred, as Microprocessor is actually the brain of a computer.
There are lot of chips on the motherboard and they all kind of look alike.
There are several companies who manufacture microprocessor chips : Advanced Micro Devices (AMD), Cyrix, Intel …

42. CPU Properties
The following are some of CPUs properties:

43. What is meant by CPU speed?
Every computer contains an internal clock that regulates the rate at which each instruction is executed.
The clock speed is measured in terms of MegaHertz (MHz).
1 Mhz is equal to 1 million cycles per second. Hence a computer with 120 MHz speed means 120 million cycles per second.
It must be noted that a faster CPU with less hard disk space would result into a mediocre machine performance.

44. What if I increase the speed of clock?
Running a microprocessor much faster than the speed for which it has been tested and approved is also called as “Overclocking”.
In many cases you can force your CPU to run faster by setting the jumper on the motherboard. But it may cause overheating.
Normally, running a CPU too fast will not damage the chip but the computer would not function properly.

45. What is meant by Word Size and how is it related with speed?
Every computer has internal work areas, kind of little workbenches. These workbenches are called as registers.
Registers are special high-speed storage area within the CPU. All data must be represented in the register before it can be processed.
The number of the registers a CPU has and the size of each (in terms of bits) determine the speed and performance of the CPU.
The largest number a computer can process in one operation is determined by size of a word.