In Computation
Computer arithmetic cannot directly operate on real numbers, but only on a finite subset of rational numbers, limited by the number of bits used to store them, whether as floating-point numbers or arbitrary precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas for them (such as, or) rather than their rational or decimal approximation; however, it is not in general possible to determine whether two such expressions are equal (the constant problem).
A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.
Read more about this topic: Real Number
Famous quotes containing the word computation:
“I suppose that Paderewski can play superbly, if not quite at his best, while his thoughts wander to the other end of the world, or possibly busy themselves with a computation of the receipts as he gazes out across the auditorium. I know a great actor, a master technician, can let his thoughts play truant from the scene ...”
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