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7. Positional Numbering System Part 7 (Conversion Binary to Decimal, Octal & Hexadecimal with Fixed or Radix Point)

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Computer Organization & Architecture

1. Conversion From Binary to Decimal, Octal and Hexadecimal with Fixed or Radix point

1.Binary to Decimal

◼ Isme hum convert karnge base 2 ko base 10 main ()2 = ()10
◼ Let's take an example
    (1101101.101)2 = (?)10
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◼ Hum dekh rhe hai ki yha par do number hai ek point ke left side main or dusra point ke right side main jaisa image main diya hai toh sabse phele point ke left side wale ko convert karnge.
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◼ Toh left side wale number ko decimal mai convert karnge toh usko likh liya hai jaisa image main likha hai or isko hum convert jaisa table lekar kartey the waise hi karnge.
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◼ Phir joh table betaya tha waise hi likh lenge jaisa image main likha hai or niche joh image hai woh 2 ki power ki values hai.
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◼ Jha se strat from here likha hai wha se start kar denge likhna jaise image main likha hai.
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◼ Yha par 2 ki power ki value hai woh likh liya hai jaisa image main likha hai.
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◼ Ab kya kar rhe hai ki sabhi number ko joh multiply diya hai usko multiply kar diya hai jaisa image main kiya hai.
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◼ Sab ko add karne ke baad total 109 aaya hai jaisa image main kiya hai to hamara point ke left side wala convert ho gaya.
◼ Ab point ke right side wale ko convert karnge.
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◼ Ye point ke right side main number hai toh usko likh liya hai jaisa image main likha hai or ise bhi waise hi convert karnge jaisa table betaya tha waise hi hoga bas isme 2 ki power -1 se chelga jaise 2-1 , 2-2 , 2-3 , 2-4... infinity.
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◼ Left side wale number ko convert karte time humne 20 se start kiya tha or 2 ki power ko badhaate gaye the.
◼ To ab point ke right side m 2 ki power -1 se start hogi or negative number ki taraf badhaati jaygi.
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◼ Hum dekh rahe hai ki isme joh 2 ki power minus main hai to uski value nikalne ke liye jitni power h utni baar 5 ko likh kar multiply kar denge jaisa image main kiya hai.
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◼ Ab kya kr rhe hai ki table main jaise solve krtey the waise hi solve kr denge jaisa image main likha hai.
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◼ Ab jha se start from here likha hai wha se start kar denge likhna jaise image main likha hai.
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◼ Phir ab 2 ki power ki joh value hai use likh denge jaisa image main kiya hai.
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◼ Ab total nikal denge jaisa image main kiya hai.
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◼ Jaisa hi point ke dono side ka result aa jayega phir dono ko add kar denge jaisa image main kiya hai.
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◼ Finally humara result aa gaya toh jaisa image main likha hai waise hi result likh denge.
    (1101101.101)2 = (109.625)10

2.Binary to Octal

◼ Isme hum convert karnge base 2 ko base 8 main ()2 = ()8
◼ Let's take an example
    (101101.10011101)2 = (?)8
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◼ Isme hum dekh rhe hai ki 2 number hai ek point k left side main or ek point k right side main toh sabse phele convert kartey hai point k left side wale number ko.
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◼ Lekin jab hum Binary se Octal m convert karte h to Binary bits ko hum 3-3 ke pair m divide karte h.
◼ To jo bits point ke left side m h unhe hum jaha se start from here wala arrow h waha se divide karna start karnge or point ke right side wali bits ko jaha arrow h waha se divide karna start karge.
◼ To sabse pehle point k left side wali bits ko divide karnge.
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◼ Ab isko hum 3-3 bits main divide kar denge jaisa image main kiya hai.
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◼ 3-3 bits m divide karne ke baad table ka 4 2 1 lete h octal main convert karne k liye fir sabhi group ko ek ek karke table use karke convert karnge jaisa image main kiya hai.
◼ To humara left side wali bits Octal m convert ho gayi lekin abhi right side wali bits bachi h to usse convert karte h.
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◼ Ab 3-3 bits main divide karnge jha se start from here likha hai wha se jaisa image main kiya hai.
◼ Lekin last m sirf 2 hi bits h to use 3 bits karne k liye ‘0 1’ ke aage ek 0 laga denge jaisa image m kiya h.
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◼ To ab right side wali bits bhi Octal m convert ho gayi.
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◼ Ab point k left or right dono sides convert hone k baad dono sides mila kar likh denge.
◼ Finally hum result aise likh denge jaise image m likha h.
    (101101.10011101)2 = (55.472)8

3.Binary to Hexadecimal

◼ Isme hum convert karnge base 2 ko base 16 main ()2 = ()16
◼ Let's take an example
    (1101110111.10110011011)2 = (?)16
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◼ Isme bhi hum dekh rhe hai ki 2 number hai ek point k left side main or ek point k right side main toh sabse phele convert kartey hai point k left side wale number ko.
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◼ Lekin jab hum Binary se Hexadecimal m convert karte h to Binary bits ko hum 4-4 ke pair m divide karte h.
◼ To jo bits point ke left side m h unhe hum jaha se start from here wala arrow h waha se divide karna start karnge or point ke right side wali bits ko jaha arrow h waha se divide karna start karge.
◼ To sabse pehle point k left side wali bits ko divide karnge.
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◼ Ab 4-4 bits main divide ho gaya lekin dekh rhe hai ki last wale group m sirf 2 hi bits h to use bhi 4 bits m karne ke liye uske aage 2 zero’s laga denge jaisa image main kiya hai.
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◼ 4-4 bits m divide karne ke baad table ka 8 4 2 1 lete h Hexadecimal main convert karne k liye fir sabhi group ko ek ek karke table use karke convert karnge jaisa image main kiya hai.
◼ To humara left side wali bits Hexadecimal m convert ho gayi lekin abhi right side wali bits bachi h to usse convert karte h.
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◼ Ab 4-4 bits main divide karnge jha se start from here likha hai wha se jaisa image main kiya hai.
◼ Lekin last m sirf 3 hi bits h to use 4 bits karne k liye ‘0 1 1’ ke aage ek 0 laga denge jaisa image m kiya h.
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◼ Hexadecimal main convert krne k liye table 8 4 2 1 lenge ek ek karke sabhi groups ko convert karnge jaisa image main kiya hai.
◼ Jha 11 aaya hai waha B isliye liya hai kyoki Hexadecimal main double digit number aate hi usko hum ek Character de dete hai jaisa image main kiya hai.
◼ 10 ko ‘A’ , 11 ko ‘B’ , 12 ko ‘C’ , 13 ko ‘D’ , 14 ko ‘E’ or last 15 ko ‘F’ se denote karte h.
◼ To ab right side wali bits bhi Octal m convert ho gayi.
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◼ Ab point k left or right dono sides convert hone k baad dono sides mila kar likh denge.
◼ Finally hum result aise likh denge jaise image m likha h.
    (1101110111.10110011011)2 = (377.B36)16

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