Chapter 3 part i data representation

Data Representation
Chapter 3 Part I

Contents
 Introduction to number system
 Different bases numbers
 Converting different bases numbers
into decimal
 Converting decimal numbers into
different bases
 Converting floating points

Objectives
 At the end of this chapter, students
should be able to:
 Explain the different bases number
system
 Convert numbers into different bases
 Convert floating numbers into different
bases

Introduction
 Human
 use decimal (base 10) number system to
count and perform arithmetic.
 Computer
 uses binary (base 2) number system.
 each digit is known as a bit (0 and 1).

Introduction
 Computer stores and manipulates the
bits in group of 8(byte), 16 (halfword),
32 (word) and 64 (double word).
 Number of bits used in calculations
affects the accuracy and size
limitations of numbers manipulated by
the computer.

Introduction
 Counting in base 10
 E.g.: 0, 1, 2, 3, …, 10, …, 100,…
10 (1 x 101
+ 0 = 10 + 0)
63 (6 x 101
+ 3 x 100
= 60 + 3)
747 (7 x 102
+ 4 x 101
+ 7 x 100 =
700 + 40 + 7)

Different bases numbers
 Number can be represented in different bases.
 Different name of bases:
 Base 2  binary
 Base 3  ternary
 Base 4  quaternary
 Base 5  quinary
 Base 6  senary
 Base 7  septenary
 Base 8  octal
 Base 9 nonary
 Base 10  decimal
 Base 11  undenary
 Base 12  duodecimal
 Base 16 hexadecimal
 Base 20  vigesimal
 Base 60 sexagesimal

 Base 2 includes (0, 1)
 Base 3 includes (0,1,2)
 Base 4 includes (0,1,2,3)
 Base 9 includes ?
 Base 12 includes ?
 Base 16 includes?

 Base 2  0,1, 10, 11,100, 101, …
 Base 3  0,1,2, 10, 11, 12, 20, 21, …
 Base 4  0,1,2,3,10,11,12, 13, 20, …
 Base 9  ?
 Base 12  ?
 Base 16  ?

Converting different bases
numbers into decimal

number into decimal
 Counting in base 8
 Example: 1278
(1 x 82
) + (2 x 81
)+ (7 x 80
) = 64 + 16 + 7
= 8710

numbers into decimal
 Counting in base 2:
 E.g.:10112
(1 x 23
) + 0 + (1 x 21
) + (1 x 20
) = 8+0+2+1
= 1110

Converting decimal numbers
into different bases

 Converting to base 8
 E.g.: 30
30 / 8 = 3 Balance 6
3 / 8 = 0  Balance 3
3010 = 368

 E.g.: 30
30 / 4 = 7 Balance 2
7 / 4 = 1  Balance 3
1 / 4 = 0  Balance 1
3010 = 1324

 E.g.: 30
30 / 2 = 15 Balance 0
15 / 2 = 7  Balance 1
7 / 2 = 3  Balance 1
3 / 2 = 1  Balance 1
1 / 2 = 0  Balance 1
3010 = 111102

Converting floating points
10
10/2 = 5  Balance 0
5/2 = 2  Balance 1
2/2 =1  Balance 0
1/2 =0  Balance 1
1010
.25
0.25 X 2 = 0.5  0
0.5 X 2 = 1  1
01
•Convert decimal to base 2 floating number
•E.g.: 10.25
1010.01

101
(1x22
) + 0 + (1x20
)
4 + 1
5
1101
(1 x ½)+(1x ¼)+(1x 1/16)
0.5 + 0.25 + 0.0625
0.8125
 E.g.: 101.1101
 Convert base 2 floating number to decimal
5.8125

Chapter 3 part i data representation

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Chapter 3 part i data representation