Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
Alan Turing memorial statue in Sackville Park Image credit: User:Lmno |
Alan Mathison Turing, OBE (June 23, 1912 – June 7, 1954), was an English mathematician, logician, and cryptographer.
Turing is often considered to be the father of modern computer science. Turing provided an influential formalisation of the concept of the algorithm and computation with the Turing machine, formulating the now widely accepted "Turing" version of the Church–Turing thesis, namely that any practical computing model has either the equivalent or a subset of the capabilities of a Turing machine. With the Turing test, he made a significant and characteristically provocative contribution to the debate regarding artificial intelligence: whether it will ever be possible to say that a machine is conscious and can think. He later worked at the National Physical Laboratory, creating one of the first designs for a stored-program computer, although it was never actually built. In 1947 he moved to the University of Manchester to work, largely on software, on the Manchester Mark I then emerging as one of the world's earliest true computers.
During World War II, Turing worked at Bletchley Park, Britain's codebreaking centre, and was for a time head of Hut 8, the section responsible for German Naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine which could find settings for the Enigma machine.
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This logic diagram of a full adder shows how logic gates can be used in a digital circuit to add two binary inputs (i.e., two input bits), along with a carry-input bit (typically the result of a previous addition), resulting in a final "sum" bit and a carry-output bit. This particular circuit is implemented with two XOR gates, two AND gates and one OR gate, although equivalent circuits may be composed of only NAND gates or certain combinations of other gates. To illustrate its operation, consider the inputs A = 1 and B = 1 with C_{in} = 0; this means we are adding 1 and 1, and so should get the number 2. The output of the first XOR gate (upper-left) is 0, since the two inputs do not differ (1 XOR 1 = 0). The second XOR gate acts on this result and the carry-input bit, 0, resulting in S = 0 (0 XOR 0 = 0). Meanwhile, the first AND gate (in the middle) acts on the output of the first gate, 0, and the carry-input bit, 0, resulting in 0 (0 AND 0 = 0); and the second AND gate (immediately below the other one) acts on the two original input bits, 1 and 1, resulting in 1 (1 AND 1 = 1). Finally, the OR gate at the lower-right corner acts on the outputs of the two AND gates and results in the carry-output bit C_{out} = 1 (0 OR 1 = 1). This means the final answer is "0-carry-1", or "10", which is the binary representation of the number 2. Multiple-bit adders (i.e., circuits that can add inputs of 4-bit length, 8-bit length, or any other desired length) can be implemented by chaining together simpler 1-bit adders such as this one. Adders are examples of the kinds of simple digital circuits that are combined in sophisticated ways inside computer CPUs to perform all of the functions necessary to operate a digital computer. The fact that simple electronic switches could implement logical operations—and thus simple arithmetic, as shown here—was realized by Charles Sanders Peirce in 1886, building on the mathematical work of Gottfried Wilhelm Leibniz and George Boole, after whom Boolean algebra was named. The first modern electronic logic gates were produced in the 1920s, leading ultimately to the first digital, general-purpose (i.e., programmable) computers in the 1940s.
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