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The mathematical foundations of computation developed from practical methods for representing and comparing quantities. Early practices such as one-to-one correspondence, measurement against standards, tallying, and geometric constructions gradually led to abstract numbers and formal numeral systems. The development of positional Hindu-Arabic numerals and mathematical notation made it possible to perform and record operations such as addition, subtraction, multiplication, division, square roots, and logarithms systematically. Mathematical statements could then be communicated, verified, formalized, and proven, as illustrated by methods such as Euclid's algorithm for finding greatest common divisors. Computation later became closely connected with algorithms, logic, and symbolic representation. Mechanical devices such as the abacus, the Antikythera mechanism, and Babbage's proposed Analytical Engine demonstrated how mathematical procedures could be automated, while Ada Lovelace's work illustrated the idea of a general program operating on data. In the twentieth century, Boolean algebra, switching circuits, truth tables, universal logic gates, and Turing's model of computation provided theoretical foundations for digital computers. These ideas, combined with binary representation and electronic technology, enabled programmable, stored-program, personal, and supercomputers.
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