Boolean algebra turned true and false into circuit math
George Boole built an algebra whose variables are truth values, not ordinary numbers. Decades later Claude Shannon noticed the same rules describe switching circuits—so the logic of and, or, and not became the language of digital electronics.
Boolean algebra differs from school algebra in two sharp ways. Variables take the truth values true and false—usually written 1 and 0—and the basic operations are conjunction, disjunction, and negation rather than the four arithmetic operations. The subject formalizes logical calculation the way elementary algebra formalizes numerical calculation. It appears in set theory, statistics, programming languages, and especially digital design. Classical propositional sentences translate into Boolean expressions, though quantifiers from first-order logic sit outside its reach.
Boole introduced the system in The Mathematical Analysis of Logic (1847) and expanded it in An Investigation of the Laws of Thought (1854). Huntington credits Henry M. Sheffer with suggesting the name "Boolean algebra" in 1913, while Charles Sanders Peirce titled an 1880 chapter with a near spelling of the phrase. Leibniz's earlier algebra of concepts matches, in deductive strength, the set-theoretic Boolean framework and foreshadowed the binary theme. Late nineteenth-century work by Jevons, Schröder, Huntington and others shaped the modern abstract structure; in 1936 M. H. Stone proved each Boolean algebra matches some field of sets under isomorphism.
Shannon, studying switching circuits in the 1930s, cast switching algebra as the two-element Boolean algebra and made gate design algebraic. Today VLSI tools often represent Boolean functions with reduced ordered binary decision diagrams for synthesis and verification. Satisfiability—whether some assignment makes a Boolean formula true—was the first problem shown to be NP-complete. Arithmetic mod 2 mirrors XOR and AND; De Morgan's laws let you rebuild conjunction from disjunction and negation, or the reverse. Laws familiar from ordinary algebra coexist with Boolean-only identities such as absorption and idempotence that fail if you plug in ordinary integers.
Source: Boolean algebra
