1.1.2
Intuition Badi idea (WHY yeh exist karta hai)Computers binary (base-2 ) mein sochte hain, lekin binary humans ke liye painful hai: number 11111111 human-readable bhi ho aur binary mein trivially convertible bhi. Trick yeh hai: ek aisa base chuno jo power of 2 ho.
8 = 2 3 octal (base-8): ek octal digit = exactly 3 bits .16 = 2 4 hexadecimal (base-16): ek hex digit = exactly 4 bits .
Kyunki base ek power of 2 hai, binary se/binary mein conversion sirf bits ko group karna hai — koi division nahi, koi arithmetic nahi. Yahi poora reason hai in bases ke use hone ka.
Definition Positional number systemsEk base-b d n d n − 1 … d 1 d 0 value = ∑ i = 0 n d i ⋅ b i 0 ≤ d i < b
Binary (b = 2 { 0 , 1 } Octal (b = 8 { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } Hexadecimal (b = 16 { 0 , … , 9 , A , B , C , D , E , F }
Intuition WHY ek hex digit = 4 bitsEk 4-bit binary number 0000 1111 0 15 one-to-one ek hex digit pe map hota hai, groups ke beech koi carrying nahi . Same logic: 3 bits ka range 0 7
1101011 0 2 → Step A — right se 4s mein group karo: 1101 0110.
Right se kyun? Kyunki place value leftward badhti hai; rightmost bit 2 0 Step B — har group convert karo: 110 1 2 = 8 + 4 + 1 = 13 = D 011 0 2 = 4 + 2 = 6 Answer: D 6 16 Decimal se check karo: D 6 = 13 ⋅ 16 + 6 = 208 + 6 = 214 1101011 0 2 = 128 + 64 + 16 + 4 + 2 = 214
Worked example 2) Same binary1101011 0 2 → Step A — right se 3s mein group karo: 11 010 110 → left pad karo → 011 010 110.
Pad kyun karo? Leftmost group ko 3 bits chahiye; missing high bits zeros hain.
Step B: 011 = 3 010 = 2 110 = 6 Answer: 32 6 8 Check: 32 6 8 = 3 ⋅ 64 + 2 ⋅ 8 + 6 = 192 + 16 + 6 = 214
Worked example 3) Decimal→ 70 0 10 least-significant pehle ):
700 ÷ 16 = 43 r 12 ( C ) 43 ÷ 16 = 2 r 11 ( B ) 2 ÷ 16 = 0 r 2 bottom-to-top padho: 2B C 16 Bottom-to-top kyun? Pehla remainder d 0 Verify karo: 2 ⋅ 256 + 11 ⋅ 16 + 12 = 512 + 176 + 12 = 700
→ 0x 2 B C 2 = 0010 B = 1011 C = 1100 0010 1011 110 0 2 101011110 0 2 Math kyun nahi chahiye? Har hex digit independently apne 4-bit block mein expand hota hai.
left se group karna
Kyun sahi lagta hai: Hum left-to-right padhte hain, toh left se grouping shuru karte hain.
Kyun galat hai: Place value rightmost bit (2 0 Fix: Hamesha right se group karo; missing high bits ko 0
Common mistake Division remainders ko top-to-bottom padhnaKyun sahi lagta hai: Hum naturally remainders ki list usi order mein padhte hain jis order mein likhi.
Kyun galat hai: Pehla remainder units digit hai, aakhri most significant hai. Top-to-bottom number ko reverse kar deta hai.
Fix: Remainders bottom-to-top padho (last computed = leftmost digit).
Common mistake Hex letters ko unki alphabet position samajhnaKyun sahi lagta hai: "A pehla letter hai, toh A = 1."
Kyun galat hai: Hex mein, letters 9 ke baad count karte rehte hain: A = 10 Fix: A F = 10 15
Common mistake Base label bhool jaana10 likhna ambiguous hai: 1 0 2 = 2 1 0 8 = 8 1 0 16 = 16 Fix: prefixes use karo — 0b1010 (binary), 0o12 (octal), 0x0A (hex).
Recall Feynman: 12-saal ke bacche ko explain karoImagine karo binary ek lamba zip-code hai sirf 0s aur 1s ka — super accurate lekin padhna thaka dene wala. Hexadecimal jaisa hai jaise un 4 0/1 lights ko pack karo ek symbol mein (0–9 phir A–F). Kyunki 16 3 lights ek waqt mein chops karta hai. Computers inhe isliye use karte hain kyunki yeh binary ke under same hain, lekin human eyes ke liye kaafi aasaan hain.
Mnemonic Grouping sizes yaad rakho"Oct‑3, Hex‑4." Octal 3 bits group karta hai; hex 4 bits group karta hai. (Octal = 8 = 2 3 2 4
Memory addresses aur colors: 0xFF0000 (red), addresses jaise 0x7FFF — compact, byte-aligned (1 byte = 2 hex digits).Bit masks aur flags: 0x0F clearly matlab "low 4 bits set."File permissions (Unix): chmod 755 octal use karta hai kyunki permissions 3 bits ke groups mein aati hain (rwx).Debugging: Memory dumps hex mein dikhaye jaate hain taaki har byte ek tidy 2-digit pair ho.
Binary represent karne ke liye hexadecimal ko decimal ke upar kyun prefer kiya jaata hai? Kyunki 16 = 2 4
Ek octal digit kitne bits represent karta hai, aur kyun? 3 bits, kyunki 8 = 2 3
Hex digits A se F ki value kya hai? A=10, B=11, C=12, D=13, E=14, F=15.
1101 011 0 2 D 6
70 0 10 2 B C
Binary ko hex mein group karte waqt, kis side se group karte hain, aur kyun? Right se, kyunki place value rightmost bit (2 0
Decimal→base-b division se remainders padhte waqt, kaun sa order? Bottom-to-top (last remainder most significant digit hai).
Base se repeated division digits kyun extract karta hai? N mod b d 0
Prefixes 0b, 0o, 0x ka matlab kya hai? Krama se Binary, octal, hexadecimal literals.
0 x 2 B C 2=0010, B=1011, C=1100 → 101011110 0 2
Binary number system — underlying base jise hex/octal abbreviate karta hai.Positional notation and place value — master equation N = ∑ d i b i Bytes and memory addressing — kyun 2 hex digits = 1 byte.Bitwise operations and masks — hex masks ko readable banata hai.Unix file permissions (chmod) — octal practice mein.ASCII and character encoding — code points hex mein dikhaye jaate hain.
Human unreadable long strings
Place value equation N=sum di b^i
Group bits, no arithmetic
Divide-remainder for decimal
