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Early computing devices were more complicated than the ancient [[abacus]] date to the seventeenth century. French mathematician [[Blaise Pascal]] designed a gear-based device that could add and subtract, selling around 50 models. The [[stepped reckoner]] was invented by [[Gottfried Leibniz]] by 1676, which could also divide and multiply. Due to the limitations of contemporary fabrication and design flaws, Leibniz' reckoner was not very functional, but similar devices ([[Leibniz wheel]]) remained in use into the 1970s.{{sfn|Blum|2011|p=13–14}} In the 19th century, Englishman [[Charles Babbage]] invented the [[difference engine]], a mechanical device to calculate [[polynomial]]s for astronomical purposes.{{sfn|Blum|2011|p=14}} Babbage also designed a general-purpose computer that was never built. Much of the design was incorporated into the earliest computers: [[punch card]]s for input and output, [[memory (computing)|memory]], an arithmetic unit analogous to [[central processing unit]]s, and even a primitive [[programming language]] similar to [[assembly language]].{{sfn|Blum|2011|p=15}}
In 1936, [[Alan Turing]] developed the concept of the [[universal Turing machine]] to model any type of computer,
{{sfn|Blum|2011|pp=21, 23}} The universal Turing machine was a type of [[stored-program computer]] capable of mimicking the operations of any [[Turing machine]] (computer model) based on the [[software]] instructions passed to it. The storage of [[computer program]]s is key to the operation of modern computers and is the connection between computer hardware and software.{{sfn|Blum|2011|p=25}} Even prior to this, in the mid-19th century mathematician [[George Boole]] invented [[Boolean algebra]]—a system of logic where each [[proposition]] is either true or false. Boolean algebra is now the basis of the [[circuit (computer science)|circuit]]s that model the [[transistor]]s and other components of [[integrated circuit]]s that make up modern computer hardware.{{sfn|Blum|2011|pp=34-35}} In 1945, Turing finished the design for a computer (the [[Automatic Computing Engine]]) that was never built.{{sfn|Blum|2011|pp=71-72}} [[File:Von Neumann Architecture.svg|thumb|Von Neumann architecture scheme]]
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