The carry-lookahead adder in advance — using only its own two input digits — whether it will make a carry or merely pass one along . Because that decision needs no waiting, a single machine can look at all positions at once and announce every carry simultaneously instead of one-at-a-time.
This page assumes you have seen nothing . We build every letter, sign, and picture the parent note uses, in an order where each brick rests on the one below it. If a symbol appears in the parent, it appears here first — defined in plain words, anchored to a picture, and justified.
A bit is a single digit that is either 0 1 off = 0 , on = 1 .
We write multi-bit numbers most-significant on the left , like 1011 0 1 i A a i
Intuition Why index from the right?Position i 2 i from position i i + 1 — from small weight to the next-bigger weight — so counting from the right makes "the next position up" mean "+ 1
a i b i a i i b i i i subscript : a label, not a multiplication. a 2 a
Everything in the parent note is built from exactly three operations on bits. We meet each with its picture — a truth table , which simply lists the answer for every possible input.
Definition AND — written as a producta b a i b i a AND b 1 only when both are 1 . In this topic it is written like multiplication: a i b i a i b i 1 ⋅ 1 = 1 0
a b a b
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Definition OR — written as a suma + b a OR b 1 at least one is 1 a + b not ordinary arithmetic (1 + 1 1 2
a b a + b
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a ⊕ b ⊕ exclusive OR : 1 different , 0 same . It answers the question "do these disagree?"
a b a ⊕ b
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Intuition Why XOR is the natural "sum digit"When you add two bits, the result digit is: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 exactly the XOR table. So XOR = "the sum digit, ignoring any carry." That is why the parent's sum formula uses ⊕ + Boolean Algebra for the full ruleset these three obey.
c i 1 + 1 = 2 1 + 1 + 1 = 3 carry . c i coming into position i c i + 1 i sends out to position i + 1 c 0 0
Intuition Why the carry is the villainThe sum digit s i = a i ⊕ b i ⊕ c i c i only thing that makes each column wait is that c i c i without waiting . Everything in Ripple-Carry Adder vs CLA is a fight over how fast the carries appear.
A Full Adder is the one-column machine. Given a i b i c i
Now read every symbol you already own:
s i 1 a i b i 1 ( a i ⊕ b i ) c i
Intuition Reading the carry-out in wordsA column sends a carry when either it is guaranteed to on its own (a i b i or it received one and is willing to forward it (( a i ⊕ b i ) c i generate and propagate .
g i p i g i = a i b i ( ==generate== ) p i = a i ⊕ b i ( ==propagate== ) g i = 1 p i = 1 both depend only on a i , b i — they need no carry, so they are ready instantly.
Intuition The picture: a lock and a pipePicture each column as a small gate on a water pipe. Generate = the column has its own faucet: water appears regardless of upstream. Propagate = the column's valve is open, so upstream water flows straight through. If a column neither generates nor propagates (g i = 0 , p i = 0 0 absorbs the carry — nothing comes out. This is why c i + 1 = g i + p i c i
The parent's general formula uses two shorthand symbols. Both are just "do this to a list."
∑ ∑ k = 0 i − 1 X k means X 0 + X 1 + ⋯ + X i − 1 k = 0 k 0 i − 1 k i − 1 +
∏ ∏ j = k + 1 i p j means p k + 1 p k + 2 ⋯ p i
Intuition Why we need them hereThe lookahead carry c i + 1 ∑ ∏ k bookkeeping , not new maths.
A real logic gate takes a tiny but nonzero time to react to its inputs. We count that as one gate delay . A signal passing through an AND then an OR costs 2 gate delays . See Propagation Delay & Fan-in .
O ( n ) n O ( n ) n O ( n ) constant delay — a fixed number of gate levels no matter how big n
XOR written as circled plus
Generate g_i and Propagate p_i
What is a bit, in one lamp-word? A single digit, either 0 (off) or 1 (on).
What does the subscript in a i A position label — "bit number i
When is a i b i Only when both a i b i
When is a i + b i When at least one of them is 1.
When is a i ⊕ b i When the two bits differ (disagree).
Why is XOR the sum digit of two bits? Its table matches 0 + 0 = 0 , 0 + 1 = 1 , 1 + 0 = 1 , 1 + 1 = 0
What is c i The carry coming into position i c i + 1
What does generate g i = a i b i This column makes a carry no matter what comes in.
What does propagate p i = a i ⊕ b i This column forwards whatever carry arrives.
Why can g i , p i They use only a i , b i
What does ∏ j = k + 1 i p j AND together p k + 1 p i
What does O ( n ) The delay grows in proportion to the number of bits.
Carry-lookahead adder — the parent topic these foundations feed.Full Adder — the one-column machine whose equations we unpack.Ripple-Carry Adder — the slow baseline whose carry-wait we diagnose.Boolean Algebra — the rulebook for AND, OR, XOR used throughout.Propagation Delay & Fan-in — where "gate delay" and O ( n ) 
