Number System

Number System

Personal Computer (PC)

 Designed for one user at a time.

 general-purpose.

 Cost-effective.

 For naïve user than a computer expert.

Components of PC

 Computer case

 Processor

 Motherboard

 Main memory

 Storage drive

 Visual display unit

 Video card

 Keyboard

Number System

 A number system is a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

 The number system can be seen as the context that allows the symbols "11"

to be interpreted as the binary symbol for three, the decimal symbol for eleven,....

 The number system consists of

 Base

 Set of digits (From 0 to Base-1)

 Representation for a set of numbers (e.g. all integers)

 Unique representation to every number in the set

Types of Number System

 Non-Positional number system

 Positional number system

Non-Positional Number System

• Symbol represents the value regardless of its position.

• Difficult to perform arithmetic operation.

• For example:-

I, II, III, IV, V, VI, VII, VIII, IX, X

XI,XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX

Positional Number System

• Symbols (Digits) represent different values depending upon the position.

• The values of each digit is determined by:-

• Digit itself

• Position of the digit

• Base of the number system

Continuing with our example…

642 in base 10 positional notation is:

6 x 10

= 6 x 100 = 600 + 4 x 10

= 4 x 10 = 40

+ 2 x 10º = 2 x 1 = 2 = 642 in base 10

This number is in base 10

The power indicates the position of

the number

Positional Number System

Decimal Number System

 The base is equal to 10

 Uses 10 different symbols.

For example:

(2*1000) + (5*100) + (8*10) + (6*1)

=2000 + 500 + 80 + 6

=2586

Binary Number System

• Binary digit 0 or 1

• The base is 2.

• Each position represents a power of the base 2.

Example:-Conversion from 00111101 to decimal is-

Conversion of decimal representation to binary

Divide decimal number to base 2 and Take remainders in reverse order. Example : Convert 36₁₀ to binary number

2 36 2 18 2 9 2 4 2 2 2 1 0

Remainder 0

0 1 0 0

Least Significant Bit (LSB)

Conversion of binary representation to decimal

 Covert (11010)₂ to decimal

2⁴ 2³ 2² 2¹ 2⁰

16 8 4 2 1

1 1 0 1 0

1*16

+1*8

+0*4

+1*2

+0*1 Conversion of binary representation to decimal

 Covert (11010)₂ to decimal

Octal Number System

 The base is 8

 The digits are 0-7

 Each position represents a power of the base 8.

 For example:- decimal

equivalent to the octal

number 421 is 273

Converting Decimal to Octal

(1988)

= (?)

8 1988 Rem

8 248 4

8 31 0

8 3 7

0 3

(642)

= (?)

Converting Octal to Decimal

(642)

= (?)

6 x 8

= 6 x 64 = 384 + 4 x 8

= 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2

Converting Octal to Decimal

Hexadecimal Number System

 The base is 16

 digit symbols in base 16 are 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F

 Each position represents

a power of the base 2.

222 13 0 16 3567 16 222 16 13

32 16 0

36 62 13

32 48

47 14 32

15

Converting Decimal to Hexadecimal

(DEF)

= (?)

D x 16

= 13 x 256 = 3328 + E x 16

= 14 x 16 = 224 + F x 16º = 15 x 1 = 15

Sum = (3567)

Converting Hexadecimal to Decimal

Shortcut Method for Converting a Binary Number to its Equivalent

Octal Number

Shortcut Method for Converting an Octal Number to Its Equivalent

Binary Number

Shortcut Method for Converting a Binary Number to its Equivalent

Hexadecimal Number

Shortcut Method for Converting a Hexadecimal Number to its

Equivalent Binary Number

Q.1- (EF)

+ (27)

= (?)

Q.2- (C2D51)

+ (27655)

= (?)

Some Problems

Q.1- (EF)

+ (27)

= (?)

(EF)

= E x 16

+ F x 16º

= 14 x 16 + 15 x 1

= 224 + 15 = (239)

(27)

= 2 x 8

+ 7x8º = 16 + 7 = 23 (EF)

+ (27)

= (239)

+ (23)

= (262)

= (523)

Q.2- (C2D51) + (27655) = (?)

Some Problems

Thank You