Fundamentals of Computer and programming in C

(CMP 104 )

NUMBER SYSTEM

NUMBER SYSTEM

NON-POSITIONAL

SYSTEMNON-POSITIONAL

SYSTEMPOSITIONAL

SYSTEMPOSITIONAL

SYSTEM

New Symbolic representation for every number.Decimal Roman

1,2,3,4,5,6,7,8,9

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

10 X

11 XI

12 XII

--- ----

New numbers are formed using digits: 0-9 and a decimal point.

10 1 Dec. 1/10 1/100

3 5 . 2 5

3×10 5×1 2×1/10 5×1/100

30 5 .20 .05

35 25

+ +

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POSITIONAL NUMBER SYSTEMDecimal (BASE 10) Number System uses 10 symbols 0,1,2,3,4,5,6,7,8,9 called digits.

Integer numbers Integers are whole numbersExamples 1, 2, -3, 50, 675, -560, …..

Decimalnumbers

Real numbersNumbers that has fractions like 687. 345, -49.56, …

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NUMBER SYSTEM

• In computer real numbers are referred to as floating point numbers.

• Floating point numbers are represented as<Integer part> <Radix Point> <Fractional part>

34568 . 56735

34568.56735

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Decimal Number System

In decimal number system the value of a digit is determined by digit × 10 position . In integer numbers the position is defined as 0,1,2,3,4,5,… starting from the rightmost position and moving one position at a time towards left.

Position 4 3 2 1 010 position 10 4 10 3 10 2 10 1 10 0

Position value 10000 1000 100 10 1Digits 7 2 1 3 4Digit Value 7×10 4 2×10 3 1×10 2 3×10 1 4×10 0

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Decimal Number System

Digit Value 7×10 4 2×10 3 1×10 2 3×10 1 4×10 0

Digit Value 70000 2000 100 30 4Integer Number

70000 + 2000 + 100 + 30 + 4

72134

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Decimal Number System

In floating point numbers the position is defined as 0,1,2,3,4,5,… starting from the radix point and moving one position at a time towards left, and -1,-2,-3, … starting from the radix point and moving towards right one position at a time.

436.85 = 4 × 100 + 3 × 10 + 6 × 1 . 8 × 0 .1 + 5 × 0.01

Position

Place Value

Digits

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Data Representation for Computers

• Computers store numeric (numbers) as well non-numeric (text, images and others) data in binary representation (binary number system).

• Binary number system is a two digits (0 and 1), also referred to as bits, so it is a base 2 system.

• Binary number system is also a positional number system. In this system, the position definition is same as in decimal number system.

• In binary number system the value of a digit is determined by digit × 2 position .

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Binary Number System

Position 4 3 2 1 02 position 24 2 3 2 2 2 1 2 0

Position value 16 8 4 2 1Binary Digits 1 1 1 0 1Digit Value 1×2 4 1×2 3 1×2 2 0×2 1 1×2 0

1 × 16 1×8 1×4 0×2 1×116 8 4 0 1

+ = (29)10(11101)2

Value = digit × 2position

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Binary Number System

Position

Place Value

Digits

5 75

Floating Point Number (101.11)2= (5.75)10

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Can we make new number systems?10 2 3 4 5 6 7 8 …. --- 16

0123456789

101112131415

01

1011

100101110111

10001001101010111100110111101111

012

101112202122

100101102110111112120

0 1 2 3

10 11 12 13 20 21 22 23 30 31 32 33

01234

1011121314202122232430

012345

10111213141520212223

0123456

101112131415162021

01234567

1011121314151617

0123456789ABCDEF

BaseBase

Relationship

Decimal

BinaryBinary

OctalOctal

Hexadecimal

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Decimal, Binary, Octal and Hexdecimal

(2057)8 = 2 x 83+0x82+5x81+7x80 = 2 x 512+ 0 x 64 + 5 x 8 + 7 x 1= 1024+0+40+7= (1071)10

(1AF)16 = 1 x 162+Ax161+Fx160

= 1 x 256+ 10 x 16 + 15 x 1= 256+160+15= (431)10

(1101)2 = 1 x 23+1x22+0x21+1x20 = 1 x 8+ 1 x 4 + 0 x 2 + 1 x 1= 8+4+0+1= (13)10

Binary to decimal

Octal to decimal

Hexadecimal to decimal

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More examples

Ternary (base-3) numbersQuaternary (base-4) numbersQuinary (base-5) numbers

Mayan number (base-20) system

(211)3 = 2 x 32 + 1 x 31 + 1 x 30 =18 + 3+1 = (22)10

(211)4 = 2 x 42 + 1 x 41 + 1 x 40 =32 + 4+1 = (37)10

(211)5 = 2 x 52 + 1 x 51 + 1 x 50 =50 + 5+1 = (56)10

Senary (base-6 ) numbers?? (base-7) numbersTridecimal or Tredecimal (base-13) numbers

Ex. (211)6 = (?)10 (211)7 = (?)10

(211)13 = (?)10

(211)20= (?)1012/04/23

From Decimal to Another Base1. Divide the decimal

number by the new base.2. Record the remainder as

the right most digit.3. Divide the quotient of the

previous divide by the new base.

4. Record the remainder as the next digit.

5. Repeat step 3& 4 until the quotient becomes 0 in step 3.

ExampleConvert (25)10=()2

(25)10=(11001)2

Number/Base

Quotient Reminder

25/2 12 1

12/2 6 0

6/2 3 0

3/2 1 1

1/2 0 1

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From Decimal to Another BaseConvert (42)10=()22 42 Remainder

21 0

10 1

5 0

2 1

1 0

0 1

Convert (952)10=()8

Convert (42)10=(101010)2

8 952 Remainder

119 0

14 7

1 6

0 1

Convert (952)10=(1670)8

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From Decimal to Another BaseConvert (428)10=()1616 428 Remainder

26 12 C

1 10 A

0 1

Convert (100)10=()5

Convert (428)10=(1AC)16

5 100 Remainder

20 0

4 0

0 4

Convert (100)10=(400)5

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From Decimal to Another BaseConvert (100)10=()44 100 Remainder

25 0

6 1

1 2

0 1

Convert (1715)10=()12

Convert (100)10=(1210)4

12

1715 Remainder

142 11 B

11 10 A

0 11 B

Convert (1715)10=(BAB)12

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Converting from a base other than to a base other than 10

1. Convert the original number to a decimal number.2. Convert that decimal number to the new base.

Convert (545)6 to () 4

(545)6 = 5 x 62+4 x 61+ 5 x 60 = 5 x 36 + 4 x 6 + 5 x 1 = 180+24+5= (209)10

4 209 Remainder

52 1

13 0

3 1

0 3

545)6 = (209)10=(3101) 4

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Converting form a base other than to a base other than 10

Convert (101110)2 to () 8

(101110)2 = 1 x 25+0 x 24+1 x 23 +1 x 22+1 x 21+0 x 20 = (46)10

8 46 Remainder

5 6

0 5(101110)2 = (46)10=(56)8

(11010011)2 = 1 x 27+1 x 26+0 x 25 +1 x 24+0 x 23 +0 x 22+1 x 21+1 x 20 = (211)10

Convert (11010011)2 to () 16

16

211 Remainder

13 3 3

0 13 D

(11010011)2 = (211)10=D3) 16

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Shortcut methods

Binary to Octal1.Start from the rightmost position, make groups of three binary digits.2.Convert each group into octal digit

Convert (101110)2 to () 8 101 110

(5 6)8Octal to Binary1.Convert each octal to three digit binary.2.Combine them in a single binary number (5 6)8

(101 110)212/04/23

Shortcut methods

Convert (562) 8 to ()2

5 6 2

010110101

Convert (6751) 8 to ()2

6

101111110

7 5 1

00112/04/23

Shortcut methods Binary to Hexadecimal conversion1.Starting from the right most position make groups of 4 binary digits2.Convert each group its hexadecimal equivalent digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Convert (10 1110 0000 1000)2 to () 16

(0)

1110 10 0000 1000

(2) (8)(14)

(2E08) 16

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Shortcut methods

Convert (1EBA2F ) 16

(B)

00100001 1110 1011

(1) (2)(E)

Hexadecimal to Binary conversion1.Convert each hexadecimal digit 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F into 4 binary digit.

1111

(F)

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Floating Point

368.65

368.65 x 10-1 = 36.865

368.65 x 10-2 = 3.6865368.65 x 10-3 = .36865

368.65 x 101 = 3686.5

368.65 x 102 = 36865.

.36865 x 103 36865. x 10-2

Mantissa Exponent12/04/23