# Church Turing thesis [closed]

What is the exact statement of the Church Turing thesis?

Is it fair to say anything computable in the physical world can be computed by a Turing machine? If so, how does a Turing machine handle continuous variable computations (for example, computing solutions to differential equations or storing/computing any real number)? It will take infinitely many tape squares to specify anything continuous. If the input itself is infinite, what does it even mean to compute based on the infinite input?

## closed as unclear what you're asking by dkaeae, Evil, David Richerby, Apass.Jack, Discrete lizard♦Jul 21 at 16:00

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• In a nutshell, the Church-Turing thesis addresses pencil-and-paper methods. How do you represent real numbers on paper? – dkaeae Jun 21 at 7:35
• The Church–Turing thesis is not a formal statement. It doesn't have an exact statement. – Yuval Filmus Jun 21 at 11:02
• Have you consulted Wikipedia? – Yuval Filmus Jun 21 at 11:02
• – Discrete lizard Jul 18 at 9:17
• Possible duplicate of Is there any data structure that can't be represented or described inside a computer?. Although that question and answer does not mention Church-Turing thesis, it does address the vague questions raise here. – Apass.Jack Jul 20 at 2:31

Neither physical computers nor the Turing machine model can exactly represent the value of a continuous quantity or a non-rational transcendental real number. They can, for example, only approximate $$\pi$$ by calculating or storing its value to some finite (but maybe very large) number of decimal places.