Are all real numbers computable?

Real numbers used in any explicit way in traditional mathematics are always computable in this sense. But as Turing pointed out, the overwhelming majority of all possible real numbers are not computable. For certainly there can be no more computable real numbers than there are possible Turing machines.

What did Turing prove?

Turing showed, by means of his universal machine, that mathematics was also undecidable. To demonstrate this, Turing came up with the concept of “computable numbers,” which are numbers defined by some definite rule, and thus calculable on the universal machine.

What did Alan Turing prove in his paper on computable numbers with an application to the Entscheidungsproblem?

Turing’s proof is a proof by Alan Turing, first published in January 1937 with the title “On Computable Numbers, with an Application to the Entscheidungsproblem.” It was the second proof (after Church’s theorem) of the conjecture that some purely mathematical yes–no questions can never be answered by computation; more …

Who coined the term Turing machine?

Alan Turing
Turing machines, first described by Alan Turing in Turing 1936–7, are simple abstract computational devices intended to help investigate the extent and limitations of what can be computed. Turing’s ‘automatic machines’, as he termed them in 1936, were specifically devised for the computing of real numbers.

Are computable numbers transcendental?

Yes, every incomputable number is transcendental, or, differently said, every algebraic number is computable. (Because it is possible to compute an arbitrary close rational approximation to every algebraic number).

Where is Turing machine now?

A working reconstruction of one of the most famous wartime machines is now on display at The National Museum of Computing. With Colossus, it is widely regarded as having shortened the war, saved countless lives and was one of the early milestones on the road to our digital world.

Is quantum computer Turing machine?

Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine.