A computable number is a number that can be calculated by a finite computer program. All the numbers you have ever heard of like 3 , √2, π, e, etc. are computable. Some numbers (like π) are represented by an infinite string of nonrepeating digits. It would take an infinite amount of time to write all these digits down.
What things are not computable?
A non-computable is a problem for which there is no algorithm that can be used to solve it. Most famous example of a non-computablity (or undecidability
Are irrational numbers computable?
The proof of this fact is actually quite easy if you know the difference between countable and uncountable sets. We first show that the set of all possible computer programs is a countable set. ... In fact, since only countably many irrational numbers can be computed, “ most” irrational numbers are not computable !
Do non-computable numbers exist?
Not only do non-computable numbers exist , but in fact they are vastly more abundant than computable numbers. Many, many real numbers are simply infinite sequences of seemingly random digits, with no pattern or special property. ... As one such example, consider a number whose part before the decimal point is 0.
What are non-computable numbers?
Chaitin’s constant is an example (actually a family of examples) of a non-computable number. It represents the probability that a randomly-generated program (in a certain model) will halt . It can be calculated approximately, but there is (provably) no algorithm for calculating it with arbitrary precision.
What makes a problem Undecidable?
In computability theory, an undecidable problem is a type of computational problem
Are undecidable problems solvable?
The corresponding informal problem is that of deciding whether a given number is in the set. A decision problem A is called decidable or effectively solvable if A is a recursive set and undecidable otherwise .
Is Rayo’s number the biggest number?
Rayo’s number is a large number named after Mexican associate professor Agustín Rayo (born 1973) which has been claimed to be the largest (named) number.
How do you prove a number is normal?
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/ b .
How do you prove a number is computable?
A real number is computable if and only if there is a computable Dedekind cut D corresponding to it . The function D is unique for each computable number (although of course two different programs may provide the same function). A complex number is called computable if its real and imaginary parts are computable.
What is the Omega number?
The omega numbers simply reference how many carbons away from the methyl end of the fatty acid chain that the first carbon-carbon double bond appears . If the double bond is three carbons away, it’s called an omega-3 fatty acid.
Are all algebraic numbers computable?
The set of algebraic numbers is countable (enumerable), and therefore its Lebesgue measure as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). ... All algebraic numbers are computable and therefore definable and arithmetical.
What is Uncomputable?
Uncomputable: function that cannot be computed by any Turing machine . ● Extra important now to differentiate functions vs. programs/Turing machines. Theorem 9.5: Uncomputable Functions. There exists a function that is not computable by any Turing machine.
What is an undecidable problem example?
Examples – These are few important Undecidable Problems: Whether a CFG generates all the strings or not ? As a CFG generates infinite strings, we can’t ever reach up to the last string and hence it is Undecidable. ... Since we cannot determine all the strings of any CFG, we can predict that two CFG are equal or not.
How do you know if a problem is undecidable?
In computability theory, an undecidable problem is a type of computational problem
Can humans solve the halting problem?
Humans can’t solve the halting problem even for restricted cases where computers can, just imagine trying to analyze an otherwise trivial Turing machine that was larger than you could read in your lifetime. ... Every case a computer can solve the halting problem a human can as well, it may just take longer.
