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2. Positional Numbering System Part 2 (Conversion Decimal to Octal & Hexadecimal)

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Computer Organization & Architecture

1. Positional Numbering System(Conversion Decimal to Octal & Hexadecimal) :

I. Decimal to Octal :-
◼ Conversion from Decimal Number to Octal Number ( )10 = ( )8 .
◼ Phele video main maine bataya ki Octal ka base 8 hota hai ( )8.
◼ Decimal to octal main convert krne k liye ham sabse phele Decimal ko convert krnge Binary main uske baad jaise Binary main convert ho jayega phir Binary ko convert krnge Octal main.
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◼ Let's take a example.
    (117)10 = (?)8
◼ Phele ke video maine btaya ki ham Binary main kaise convert krtey hai.

Toh sabse phele Rough work krnge lekin isko ham jarur krnge kyoki isiki help se ham question ko solve kr payenge.
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◼ 128 limit k bahar hai matlab 117 se bada h toh ye eliminate ho gya (128 > 117).
◼ Ab dekh rhe hai ki 64 117 se chota h matlab limit k ander hai to 'In' matlab 1 ho gya (64 < 117).
◼ Phir 32 ki baari (64 + 32) krnge toh 96 ata hai who bhi 117 se chota h matlab limit k ander hai toh ye bhi 'In' matlab 1 ho gya.
◼ Phir 16 ki baari (64 + 32 + 16) krnge toh 112 ata hai who bhi 117 se chota h matlab limit k ander hai toh ye bhi 'In' matlab 1 ho gya.
◼ Phir 8 ki baari (64 + 32 + 16 + 8) krnge toh 120 ata hai lekin who 117 se bada h matlab limit k bahar hai toh ye 'Out' matlab 0 ho gya.
◼ Phir 4 ki baari (64 + 32 + 16 + 4) krnge toh 116 ata hai who bhi 117 se chota h matlab limit k ander hai toh ye bhi 'In' matlab 1 ho gya.
◼ Phir 2 ki baari (64 + 32 + 16 + 4 + 2) krnge toh 118 ata hai lekin who 117 se bada h matlab limit k bahar hai toh ye 'Out' matlab 0 ho gya.
◼ Phir 1 ki baari (64 + 32 + 16 + 4 + 1) krnge toh 117 ata hai to ab who 117 ke barabar h matlab limit k equal hai toh ye bhi 'In' matlab 1 ho gya.
◼ Toh finally 117 ka binary bana.
    (117)10 = (1110101)2
◼ Abhi toh Decimal se Binary me convert kiye hai ab Binary ko Octal me convert krnge.
◼ Joh binary number aya h usko ham 3-3 bit me divide krnge sbhi bits ko jha se starting point dikhaya hai wha se.
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   001    110    101
◼ Ab humne isko 3-3 bits me divide kr diya hai lekin last me jb 1 bit baccha toh usme ham 3 bits me dikhane k liye two zero(00) extra lga diya kyoki isse value me koi ferk ni hoga lekin hamesha zero's ko aage add karenge baad m nahi.
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◼ Ham isko 3-3 bits main kyo divide kr rhe hai ye batata hu main, ab jaise octal start hota hai 0 se aur end hota hai 7 par.
   ( 0, 1, 2, 3, 4, 5, 6, 7 ) isme last digit 7 hai .
◼ Ye joh table banaya hai usme se number liya hai toh dikh rha hai ki 1, 2 aur 4 ko add krne par aata hai 7 joh ki octal ka last digit hai agar 8 add krtey hai toh 7 se bada ho jayega jisse Octal number nahi rahega toh table m kevel 1,2 aur 4 number tak lenge isliye ham octal me 3 bits me number ko divide krtey hai.
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◼ Divide krne k bad ab isko ham Octal me convert krnge.
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◼ Jha se start from this side likha hai wha se ab ham start krnge.
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◼ Jaise ki maine betaya tha ki agr zero(0) hai toh who out hai aur one(1) hai oh who in hai.
    1 0 1 = 4 + 1 = 5
◼ Ab 1 1 0 ki bari hai
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    1 1 0 = 4 + 2 = 6
◼ Ab last hai 0 0 1
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    0 0 1 = 1
◼ Toh Octal main kya bna.
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◼ Toh hame (117)10 = (165)8 conversion mil gaya Decimal to Octal lekin ye sirf rough work ke liye hai kuki exam main aise marks ni milenge.
◼ Lekin rough main ye kam krne ke baad ab exam main marks lene ke liye ye kya kam karenge dekhte h.
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◼ Sabse phele is table ko bena lenge jisme left side main 8-8 likh denge kyoki Octal main convert kerna hai or top par 117 likh denge.
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◼ Hamara Octal Number hai 165 jaise image main dikh rha h waise likh denge.
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◼ ((8 * 1)+6) = 14 isliye humne 14 likha.
◼ ((8 *14)+5) =117 isko check kiya ki joh 117 hai who mera 117 aya ki ni agar aya toh answer thik wrna galat toh mera answer thik aya hai.
◼ Finally humari table mil gai or easy bhi hai kuki multiplication or addition division se easy hota hai.
◼ Finally ye table banana k bad hum apna result likh denge (117)10 = (165)8

II Conversion from Decimal Number to Hexadecimal Number ( )10 = ( )16 :-
◼ Ye bhi wahi tarika hai jaise octal me use kiya tha bus yha ham log decimal ko hexadecimal main convert kernge.
◼ Decimal ko Hexadecimal main convert kerne k liye sabse phele hum Decimal ko Binary me convert kernge jaise Binary main convert ho jayega uske baad hum Hexadecimal me convert kr denge.
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◼ Lekin poora process same rahega bas waha 3-3 bit m divide kar re the lekinyaha hum 4-4 bits m divide karenge.
◼ Let's take a example
    (117)10 = (?)16
◼ Toh sabse phele Rough work krnge lekin isko ham jarur krnge kuki isiki help se ham question ko solve kr payenge lekin rough main hi krenge.

<----aap chahe to ye part skip kar sakte h kyoki same humne uper dekh liya h ---->

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◼ 128 limit k bahar hai matlab 117 se bada h toh ye eliminate ho gya (128 > 117).
◼ Ab dekh rhe hai ki 64 117 se chota h matlab limit k ander hai to 'In' matlab 1 ho gya (64 < 117).
◼ Phir 32 ki baari (64 + 32) krnge toh 96 ata hai who bhi 117 se chota h matlab limit k ander hai toh ye bhi 'In' matlab 1 ho gya.
◼ Phir 16 ki baari (64 + 32 + 16) krnge toh 112 ata hai who bhi 117 se chota h matlab limit k ander hai toh ye bhi 'In' matlab 1 ho gya.
◼ Phir 8 ki baari (64 + 32 + 16 + 8) krnge toh 120 ata hai lekin who 117 se bada h matlab limit k bahar hai toh ye 'Out' matlab 0 ho gya.
◼ Phir 4 ki baari (64 + 32 + 16 + 4) krnge toh 116 ata hai who bhi 117 se chota h matlab limit k ander hai toh ye bhi 'In' matlab 1 ho gya.
◼ Phir 2 ki baari (64 + 32 + 16 + 4 + 2) krnge toh 118 ata hai lekin who 117 se bada h matlab limit k bahar hai toh ye 'Out' matlab 0 ho gya.
◼ Phir 1 ki baari (64 + 32 + 16 + 4 + 1) krnge toh 117 ata hai to ab who 117 ke barabar h matlab limit k equal hai toh ye bhi 'In' matlab 1 ho gya.
◼ Toh finally 117 ka binary bana.
    (117)10 = (1110101)2

<----------------yaha par skip part ka end h -------------------->

◼ Abhi toh Decimal se Binary me convert kiye hai ab Binary ko Hexadecimal me convert krnge.
◼ Joh binary number aya h usko ham 4-4 bit me divide krnge sbhi bits ko jha se starting point dikhaya hai wha se.
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   0111    0101
◼ Isko hum 4-4 bits main divide kr rhe hai lekin dekh rhe hai ki last m 3 bits hai toh isko 4 bits banana hoga toh iske liye hume last 3 bits m ek zero (0) bit add kr denge.
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◼ Ab main ye betata hu ki isko hum 4-4 bits m kyo divide krtey hai, jaise ki humne dekha hexdecimal start hota hai 0 se or end hota hai 15 par.
◼ ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ) isme last digit hai 15 but maine ye betaya ki jaise isme double digit ata hai toh numbers ki jagah characters aa jata hai.
◼ Isme hum 1, 2, 4, 8 ko add kartey hai toh 15 ata hai jo ki last digit hai agar 16 bhi add kr dete hai toh 15 se greater ho jata hai toh 16 ni add krtey hai isiliye sirf 4 bits lete h.
◼ Ab 1, 2, 4, 8 lete hai isliye hum 4 bits use krtey hai Hexadecimal main.
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◼ Jaise Binary main convert kar liya uske baad hum isko 4-4 bits m divide kr denge.
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◼ Jha se strating point likha hai wha se start kr denge.
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◼ Jaise ki maine betaya tha ki agar zero (0) hai toh out ho jayega aur agar one (1) hai toh In ho jayega or jo In h sirf wahi aad hoga.
    0 1 0 1 = 4 + 1 = 5
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◼ Ab isme
    0 1 1 1 = 4 + 2 + 1 = 7
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◼ Toh ab hume (117)10 = (75)16 conversion mil gaya lekin ye sirf rough work hai kyoki exam main aise krenge toh marks ni milega.
◼ Lekin rough main ye krne k baad ab exam main marks lene k liye ye kam krnege.
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◼ Sabse phele ek uper dikhaya hua table bana lenge jisme left side main 16-16 likh denge kyoki Hexadecimal main convert krna hai or top 117 likh denge.
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◼ ((16 * 7)+5) = 117
◼ Finally hume humari table mil gai or easy bhi hai kuki multiplication or addition division se easy hota hai.
◼ Finally ye table benane k baad hum apna result likh denge (117)10 = (75)16

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