Number System… Monika Gope Lecturer IICT, KUET. Inside The Computer 6/3/2014 2.

Number System…

Monika Gope

LecturerIICT,

KUET

Inside The Computer

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Decimal to Binary

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3Step 1

Technique 1:

Decimal to Binary

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4Step 2

Technique 1:

Decimal to Binary

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5Steps

Technique 1:

Decimal to Binary

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6Steps

Technique 1:

Decimal to Binary

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7Step 1

Technique 2:

Decimal to Binary

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8Step 2

Technique 2:

Decimal to Binary

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9Step 3

Technique 2:

Decimal to Binary

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10Step 4

Technique 2:

Decimal to Binary

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11Step 5

Technique 2:

Decimal to Binary

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12Step 6

Technique 2:

Decimal to Binary

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13Step 7

Technique 2:

Decimal to Binary

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14Step 8

Technique 2:

Decimal to Binary

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Step 9

Technique 2:

Binary to Decimal

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16Step 1

Technique 1:

Binary to Decimal

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17Step 2

Technique 1:

Binary to Decimal

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18Step 3

Technique 1:

Binary to Decimal

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19Step 4

Technique 1:

Binary to Decimal

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Step 5

Technique 1:

Binary to Decimal

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21Step 1

Technique 2:

Binary to Decimal

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22Step 2

Technique 2:

Binary to Decimal

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23Step 3

Technique 2:

Binary to Decimal

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24Step 4

Technique 2:

Binary to Decimal

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25Step 5

Technique 2:

Binary to Decimal

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26Step 6

Technique 2:

Binary to Decimal

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27Step 7

Technique 2:

Binary to Decimal

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28Step 9

Technique 2:

Binary to Decimal- Decimal to Binary

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• (123)10 = 1111011

• (11100011)2 = 227

• 150 = 10010110

• 10100010 = 162

Summary of Decimal Number System

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3805001000

)10*3()10*8()10*5()10*1()1583( 012310

• A positional Number System• Has 10 Symbols or Digits (0,1,2,3,4, 5,6,7,8 ,9). Hence its base is 10.•The maximum value of single digit is 9.• Each position of a digit represent a specific power of the base 10.

Summary of Decimal Number System

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In positional number systems, there are only few symbols, called digits, and these symbols represent values, depending on the position , they occupy in the number.

The value of each digit is determined by three steps◦ The digit itself◦ The position of the digit in the number◦ The base of the number systems

Examples◦ Decimal number systems◦ Binary number systems◦ Octal number systems◦ Hexadecimal number systems.

Summary of Binary Number System• As we know that decimal system the base

is equal to 10. It means that there are 10 digits in decimal system i.e. 0,1,2,3,…,9.• Binary number system is same as decimal

system, except that the base is 2, instead of 10.• In Binary System there are only two digits

(0,1) , which can be used.• Each position in a binary system represent

the power of base (2)

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Summary of Binary Number System

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)21()20()21()20()21()10101( 012342

= 16 + 0 + 4 + 0 + 1 = 21

Octal Number System• In Octal Number System, the base is 8.• So, there are eight digits: 0,1,2,3,4,5,6 and 7.• Each position in an octal number represents a

power of base 8.• Since there are only 8 digits, so 3 bits are

sufficient to represent any octal number in binary,

• Example

= 1024 + 0 + 40 + 7 = 1071

)87()85()80()82()2057( 01238

)82( 3

Hexadecimal Number System• In hexadecimal number system, the base is 16. • So, there are 16 digits or symbols in hexadecimal

number system.• First 10 digits are 0,1,2,3,4,5,6,7,8,9.• The remaining six digits are denoted by the

symbols A, B, C, D, E and F, representing the decimal values 10, 11,12, 13, 14 and 15, respectively.• Each position in the hexadecimal system

represents a power of the base (16)• Example:

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

)16()16()161()1( 01216 FAAF

Converting From One Number System To Another• Any number in one number system can be

represented in other number system.

• In computer system, the input and output is mostly in decimal number system.

• So, computer takes input in decimal and convert it into binary than process the input and converts again in decimal to produce output. Which is understandable to user.

Converting From Decimal to Another BaseFour Steps• Divide the decimal number to be

converted by the value of new base.• Record the remainder from Step 1 as the

right most digit of the new base number.• Divide the quotient of the previous divide

by the new base.• Record the remainder from step 3 as the

next digit (to the left) of the new base. • Repeat Step 3 and 4 until the quotient

becomes zero.

Converting From Decimal To Another Base• Example

2 42 remainder 21 0

10 1 5 0 2 1 1 0 0 1 Hence 42 = 101010

Converting to Decimal from Another BaseThree steps.

1. Determine the column value of each digit (this depends on the position of the digit and the base of the number).

2. Multiply the obtain column values by the digits in the corresponding columns.

3. Sum the products calculated in step 2. The total is equivalent value in decimal.

Converting to Decimal from Another BaseExample • Step 1: Determine the values.

12345

• Step 2: Multiply Column values by corresponding column digits(1x16) + (1x8) + (0x4) + (0x2) + (1x1)

• Step 3: Sum the product 16 + 8 + 0 + 0 + 1 = 25

2)11001(

162

82

42

22

12

4

3

2

1

0

Decimal to Octal

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41Step 9

Octal to Decimal

• The given number is 2018

2018 = (2 * 82) + (0 * 81) + (1 * 80)

= 2 * 64 + 0 * 8 + 1 * 1

= 128+0+1

= 129

The equivalent decimal number for 2018 is 129

2018 = 129

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Decimal to Octal - Octal to Decimal1. The given number is (532) 10 = ( ?) 8

2. The given number is (532) 8 = ( ?) 10

Answer:

1. 1024, 2. 346

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Hexadecimal Numbering systems• Base: 16• Digits: 0, 1, 2, 3, 4, 5, 6,

7,8,9,A,B,C,D,E,F

• Hexadecimal number: 1F416

powers of : 164 163 162 161 160

decimal value: 65536 4096 256 16 1

Hexadecimal number: 1 F 4

Hexadecimal Numbering systems

Four-bit Group Decimal DigitHexadecimal Digit

0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F

Hexa to Decimal ConversionTo convert to base 10, beginning with the

rightmost digit multiply each nth digit by 16(n-1), and add all of

the results together.Ex: 1F416

positional powers of 16: 163 162 161 160

decimal positional value: 4096 256 16 1Hexadecimal number: 1 F 4

256 + 240 + 4 = 50010

Converting From a Base Other Than 10 to a Base Other than 10Two Steps• Convert The original number into decimal number system• Convert the decimal number obtained in step 1 to the new number.

Shortcut Method of converting Binary number system to Octal Number System• Step 1: Divide the digits into group of

three starting from the right

• Step 2: Convert each group of three binary digits to one octal digit using the method of binary to decimal conversion.

Shortcut Method of converting Binary number system to Octal Number System

Shortcut Method of converting Octal Number System to Binary number system

Shortcut Method of converting Binary number system to Hexadecimal Number System• Step 1: Divide the digits into group of

four starting from the right

• Step 2: Convert each group of four binary digits to one Hexadecimal number.

Shortcut Method of converting Binary number system to Hexadecimal Number System

Shortcut Method of converting Hexadecimal Number System to Binary number system• Step 1: Convert Each Hexadecimal

digit to a 4 digit binary number.

• Step 2: Combine all the resulting binary groups into a single binary number

Shortcut Method of converting Hexadecimal Number System to Binary number system

Fractional Numbers

Formation of Fractional Numbers in Binary Number System

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Anyone who has never made a mistake has never tried anything

new.Einstein