Boolean algebra
/ BOO-lee-an /
Boolean algebra is the math of true and false — a tidy algebra in which every quantity is either 1 (true) or 0 (false), and the operations are AND, OR, and NOT. It is the grammar that lets us write down, simplify, and reason about what gates do, the same way ordinary algebra lets us manipulate numbers. Named after George Boole, who in the 1800s realized logic could be done with symbols and rules long before there were any electronics to run it on. When you write a condition like 'ready AND (not busy)', you are already doing Boolean algebra.
Its variables hold only 0 or 1. AND is written like multiplication (A times B, often just AB) and gives 1 only when both are 1. OR is written like addition (A + B) and gives 1 when either is 1. NOT is an overbar or a prime (A' means 'not A'). From a few laws you can rewrite expressions: A AND 1 = A, A OR 0 = A, A AND A' = 0, A OR A' = 1, and crucially De Morgan's laws, which say not(A AND B) = (not A) OR (not B), and not(A OR B) = (not A) AND (not B). A small worked step: ready AND (not busy) OR (ready AND busy) simplifies, by factoring out ready, to just ready — so the second clause was never needed. That kind of simplification removes gates from a real circuit.
Why it matters: Boolean algebra is the bridge between a specification in words and a pile of physical gates. Design a logic function by writing its truth table, read off a Boolean expression, simplify it algebraically (or with a Karnaugh map), and the simplified expression maps directly to fewer, faster gates. The honest point is that simpler-looking algebra usually means a smaller, cheaper, lower-power circuit — but 'simplest expression' and 'fewest transistors' do not always coincide, because real gates like NAND are cheaper than the AND-then-NOT they replace.
Verify De Morgan with a truth table. Take not(A AND B). For (A,B) = (0,0),(0,1),(1,0),(1,1), A AND B is 0,0,0,1, so its NOT is 1,1,1,0. Now (not A) OR (not B): 1,1,1,0. They match in every row — the two expressions are the same gate, which is exactly why NAND can stand in for OR-of-inverts.
De Morgan's laws let you trade ANDs for ORs (and back) by flipping the inputs and output — a designer's everyday move.
Boolean algebra is not the same as ordinary arithmetic, even though we borrow + and times. In Boolean algebra 1 OR 1 = 1, not 2 — there is no carrying, because the only values that exist are 0 and 1.