3.3.1
Jab tum haath se do decimal numbers add karte ho, har column mein ek digit likhte ho aur kabhi kabhi 1 ko agle column mein carry karte ho. Binary addition bilkul waisi hi hai, lekin har column mein sirf bits hoti hain (0 ya 1). Ek half adder do bits add karta hai. Ek full adder teen bits add karta hai (do inputs + ek carry jo andar aa rahi hai). Ye woh building blocks hain jo, ek saath chain karke, ek CPU ko kisi bhi length ke numbers add karne dete hain.
KYA chahiye hame: ek circuit jo, input bits dene par, unka arithmetic sum binary mein output kare.
Dikkat yeh hai: 1 + 1 = 1 0 2 do outputs chahiye:
Sum (current column ka bit)Carry (woh bit jo agle column mein jaati hai)
KAISE hum logic dhundhte hain: truth table banao, phir Boolean expression padho.
Ek combinational circuit jo do single bits A B Sum bit aur ek Carry bit produce karta hai. Isse "half" isliye kaha jaata hai kyunki yeh carry-in ko accept nahi kar sakta kisi pichle stage se.
Har input combination aur arithmetic result list karo:
A B A + B Carry C
Sum S
0
0
0
0
0
0
1
1
0
1
1
0
1
0
1
1
1
2 = 1 0 2
1
0
Yeh step kyun? Table bas "ones ginlo aur binary mein likho" hai. Abhi koi Boolean guessing nahi — pure arithmetic truth.
Ab columns padho:
S A B alag hon → woh hai XOR .C dono 1 hon → woh hai AND .
Real multi-bit addition mein, har column ko uske daayein column se ek carry bhi milti hai. Half adder ise ignore karta hai, isliye woh incomplete hai. Ek full adder teen inputs handle karta hai: A B C in
Ek combinational circuit jo teen bits add karta hai — A B C in Sum S Carry-out C o u t
A B C in sum(dec)
C o u t S
0
0
0
0
0
0
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
2
1
0
1
0
0
1
0
1
1
0
1
2
1
0
1
1
0
2
1
0
1
1
1
3
1
1
Yeh step kyun? Phir se, teenon inputs mein 1s gino aur count ko 2-bit binary ( C o u t S )
Sum S S = 1 odd ho (count 1 ya 3 wali rows). "Odd number of 1s" teeno ka XOR hai:
S = A ⊕ B ⊕ C in Kyun? XOR output ko har baar flip karta hai jab ek input 1 ho; odd count use 1 par chhodta hai.
Carry C o u t C o u t = 1 kam se kam do inputs 1 hon. Minterms group karo:
C o u t = A B + B C in + A C in majority function hai — teenon inputs mein se majority jo value rakhti hai woh output karo.
Doosri form equivalent kyun hai (algebra ka steel-man):
( A ⊕ B ) C in = ( A B ˉ + A ˉ B ) C in A B A = B = 1 A B A , B A ⊕ B = 1 C in = 1 A B + A C in + B C in
Intuition Reuse karo, reinvent mat karoEk full adder = HA1 jo A , B S 1 = A ⊕ B C 1 = A B HA2 S 1 C in S = S 1 ⊕ C in C 2 = S 1 C in C o u t = C 1 + C 2 = A B + ( A ⊕ B ) C in
Worked example 1) Half adder se1 + 1 A = 1 , B = 1 S = 1 ⊕ 1 = 0 C = 1 ⋅ 1 = 1 ( C , S ) = ( 1 , 0 ) = 1 0 2 = 2 Yeh step kyun? Barabar bits ka XOR 0 hota hai (woh alag nahi hain); AND 1 hai (dono true hain). Result sahi se decimal 2 ke barabar hai.
Worked example 2) Full adder ke saathA = 1 , B = 1 , C in = 1 S = 1 ⊕ 1 ⊕ 1 = 1 C o u t = ( 1 ⋅ 1 ) + ( 1 ⋅ 1 ) + ( 1 ⋅ 1 ) = 1 ( C o u t , S ) = ( 1 , 1 ) = 1 1 2 = 3 Yeh step kyun? Teen ones → binary 11
Worked example 3) FAs chain karkeA = 01 1 2 , B = 00 1 2 1 + 1 + 0 → S 0 = 0 , C = 1 1 + 0 + 1 → S 1 = 0 , C = 1 0 + 0 + 1 → S 2 = 1 , C = 0 = 10 0 2 = 4 = 3 + 1 Yeh step kyun? Har column ka C o u t C in ripple-carry adder hai.
Common mistake "Half adder ki carryA + B Kyun sahi lagta hai: "add" OR jaisa lagta hai, aur OR of ( 1 , 1 ) Trap: OR of ( 0 , 1 ) koi carry nahi aati. Carry tab 1 honi chahiye jab dono 1 hon.
Fix: Carry AND hai (A ⋅ B
Common mistake "Full adder SumA + B + C in Kyun sahi lagta hai: "sum" ka matlab hi add karna hai, aur + Trap: Boolean algebra mein, + Fix: Sum parity hai → teeno ka XOR : A ⊕ B ⊕ C in
Common mistake Kaunsa output kaunsa hai, yeh confuse karna.Fix mnemonic: Sum woh hai jo tum is column mein See karte ho; Carry woh hai jo tum agle column mein Cart karte ho.
Recall Feynman: ek 12-saal ke bacche ko samjhaoSocho tum 1 + 1 add kar rahe ho sirf 0 aur 1 buttons ke saath. Answer hai "do," lekin tum "2" nahi likh sakte — tumhare paas sirf 0 aur 1 hain! Toh tum yahan ek 0 likhte ho aur ek chota 1 agli jagah carry kar dete ho, jaise normal addition mein carry karte hain. Half adder ek choti machine hai jo do buttons ke liye yeh karti hai. Lekin kabhi kabhi ek carry daayein se aati hai, toh tum actually teen cheezein add kar rahe ho. Full adder woh badi machine hai jo teeno handle kar sakti hai. Full adders ki ek row laga do aur woh bade numbers add kar sakti hai, ek column ek baar — aise computers add karte hain!
Mnemonic Equations yaad karo"Sum = XOR, Carry = MAJORity."
Half: S = A ⊕ B C = A B S = C o u t = Ise yahan dekho (Sum), agale darwaze par Cart karo (Carry).
Answers dhak do aur dono truth tables scratch se rebuild karo, phir equations.
Kisi bhi binary adder cell ke do outputs kya hain? Sum (is column ka bit) aur Carry (agle column ko pass hone wala bit).
Half adder Sum equation? S = A ⊕ B
Half adder Carry equation? C = A ⋅ B
Ise "half" adder kyun kehte hain? Yeh sirf do bits add karta hai aur pichle stage se carry-in accept nahi kar sakta.
Full adder ke kitne inputs hote hain? Teen: A B C in
Full adder Sum equation? S = A ⊕ B ⊕ C in
Full adder Carry-out equation? C o u t = A B + A C in + B C in
Do half adders se full adder — final carry? C o u t = A B + ( A ⊕ B ) C in
Common mistake: kya half-adder carry OR hai? Nahi — yeh AND hai; carry sirf tab jab DONO bits 1 hon.
Full adders chain karne par kaun sa multi-bit adder milta hai? Ek ripple-carry adder (har C o u t C in
A = 1 , B = 1 , C in = 1 ( C o u t , S ) ( 1 , 1 ) = 3
XOR gate — woh parity behavior jo Sum output ko power deta haiAND gate aur OR gate — carry logic banate hainRipple-carry adder — full adders chain kiye hueCarry-lookahead adder — tez carry, ripple delay se bachta haiTruth tables and minterms — yahan use ki gayi derivation methodTwo's complement subtraction — adders subtraction ke liye reuse hote hainCombinational circuits — parent family (koi memory nahi, output = f(inputs))
because 1+1 equals 10 base 2
CPU adds any-length numbers
