3.1.2
Computers binary (0s aur 1s) mein sochte hain, lekin lambi binary strings jaise 11010110 ko humans ke liye padhna, likhna, aur bina galti copy karna bahut mushkil hai. Hexadecimal (base 16) aur octal (base 8) binary ke liye shorthand hain. Inke exist karne ki puri wajah: 16 = 2 4 8 = 2 3 4 bits pack karta hai aur har octal digit exactly 3 bits pack karta hai. Bits ko neat chunks mein group karne se hum ek 8-bit byte ko 8 characters se sirf 2 hex digits mein shrink kar sakte hain.
Intuition "Sirf decimal use karo" idea ko steel-man karnaDecimal (base 10) humein natural lagta hai, toh binary ko seedha decimal mein convert kyun na karein? Kyunki 10 power nahi hai. Decimal aur binary ke beech koi clean, digit-by-digit mapping nahi hai — tumhe poori division arithmetic karni padti hai. Hex aur octal bit boundaries ke saath perfectly align hote hain, isliye conversion sirf binary string ko groups mein chop karna hai. Yahi inka killer feature hai.
b b b d n d n − 1 … d 1 d 0 0 ≤ d i < b value = ∑ i d i ⋅ b i i position (place value) hai. b base ya radix hai.
System
Base b
Digits used
Binary
2
0,1
Octal
8
0–7
Decimal
10
0–9
Hex
16
0–9, A,B,C,D,E,F
Definition Hex letter digitsKyunki 9 ke baad humare paas numeric symbols khatam ho jaate hain, hex letters use karta hai: A = 10 , B = 11 , C = 12 , D = 13 , E = 14 , F = 15
Bas definition mein plug karo. Har digit ko uske place value b i
Intuition Baar baar base se divide kyun karte hain?Jab tum N b remainder exactly last digit d 0 i ≥ 1 d i b i b d 0
4 7 10 47 ÷ 16 = 2 r 15 ( = F ) 2 ÷ 16 = 0 r 2 2 , F ⇒ 2 F 16
Bottom→top kyun? Pehla remainder sabse chhota place value hai (1 6 0
1101011 0 2 3 011 2 010 6 110 11 ko 011 se pad karo)
⇒ 32 6 8
Pad kyun? 8 3 left mein zeros se padding karne se value nahi badlti.
Worked example Hex → octal (binary se hoke jaao)2F 16 0010 1111 00 101111 → 0 010 1111 → 5 101 7 111 0 10111 1 2 5 7 8
Binary se hoke kyun? Hex aur octal ek doosre ke saath align nahi hote (unke group sizes 4 aur 3 alag hain), lekin dono binary ke saath align hote hain — isliye binary universal middle-man hai.
Definition Real hardware/code mein bases kaise mark hote hain
Subscript: 2F 16 10 1 2 1 7 8
Hex prefix: 0x2F (C, Python) ya $2F (assembly) ya 2Fh.
Octal prefix: 0o17 (Python) ya leading 0 jaise 017 (old C).
Binary prefix: 0b101.
Common mistake Bits ko LEFT se group karnaKyun sahi lagta hai: Hum English left-to-right padhte hain, isliye left se grouping shuru karte hain.
Kyun galat hai: Place values right se badhti hain (2 0 , 2 1 , … Fix: Hamesha right se group karo; leftmost group ko leading zeros se pad karo.
Common mistake Yeh bhool jaana kiA F Kyun sahi lagta hai: Maths mein letters usually unknowns ke liye hote hain.
Fix: Hex mein, A F 10 15 0xFF = 15 ⋅ 16 + 15 = 255
Common mistake Hex ko octal ke saath seedha "digit swapping" se mix karnaKyun sahi lagta hai: Dono sirf "grouped binary" hain, toh lagta hai groups swap kar sakte hain.
Fix: 4 3 binary ke through jaao.
Common mistake Yeh sochna ki10 Fix: 1 0 2 = 2 1 0 8 = 8 1 0 16 = 16 Kisi bhi base b 10 b
Recall Feynman: 12-saal ke bacche ko samjhaoSocho binary ek bahut lamba word hai jo sirf I aur O letters se bana hai, aur use miscopy karna aasaan hai. Hex ek secret code ki tarah hai jahan har 4 letters ko ek single nickname milta hai (0–9, phir A–F). Octal bhi yahi karta hai lekin 3 letters per nickname ke saath. Toh ek giant binary word ek short, tidy code ban jaata hai. Decode karne ke liye, tum bas har nickname ko uske chhote bits mein wapas spell karo — koi maths nahi chahiye. Ten (decimal) yeh trick nahi kar sakta kyunki 10
Mnemonic Group sizes yaad karo"Hex ke 4 pair hain, Octal ke 3 pair hain." (Hex = 2 4 → 4 = 2 3 → 3 F ifteen sabse bada hex digit hai (F = 15
Ek hex digit exactly 4 bits se kyun map hota hai? Kyunki 16 = 2 4 0 15
Ek octal digit exactly 3 bits se kyun map hota hai? Kyunki 8 = 2 3
Hex digit F 15
10 1 2 b 10 Kisi bhi base b 10 b 1 0 2 = 2 1 0 16 = 16
Binary ko hex mein group karte waqt, kis side se group karte hain aur kyun? Right se, kyunki place values (2 0 , 2 1 , …
2F 16 2 ⋅ 16 + 15 = 47
1101011 0 2 1101 0110 = D6 16
1101011 0 2 011 010 110 = 32 6 8
Hex↔octal ko binary ke through kyun convert karte hain? 4-bit aur 3-bit groups ek doosre ke saath align nahi hote, lekin dono binary ke saath align hote hain, jo common middle-man hai.
Decimal→base b Pehla remainder sabse chhota place value (b 0
0xFF decimal mein kya equal hai?255 15 ⋅ 16 + 15
16 equals 2^4 packs 4 bits
8 equals 2^3 packs 3 bits
