- "Computable" means that there is a (finite ) algorithm that can (in principle) calculate ALL the digits of the number.
In the case of an integer N , the program "print(N)" already does the job, since the input of N is finite.
So, N clearly must be computable. 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..
How many computable numbers are there?
The computable numbers are an infinite set.
We have provided an injective function g that maps every computable number to a single natural number: a G\xf6del number.
Any set with such a function is countable, and therefore computable numbers are countable.Dec 1, 2021.
Is Pi Uncomputable?
Definitions.
Archimedes' constant (pi), along with other well-known numbers such as Pythagoras' constant (√2) and the golden ratio (φ) are all examples of a type of real number which we say is computable, despite also being irrational (real numbers which cannot be constructed from fractions of integers).Jul 11, 2019.
Is Pi Uncomputable?
Definitions.
Archimedes' constant (pi), along with other well-known numbers such as Pythagoras' constant (√2) and the golden ratio (φ) are all examples of a type of real number which we say is computable, despite also being irrational (real numbers which cannot be constructed from fractions of integers)..
What is an example of a computable number?
The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers..
What is the computational number theory?
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to .
What is the concept of on computable numbers?
Undoubtedly the most famous theoretical paper in the history of computing, "On Computable Numbers" is a mathematical description an imaginary computing device designed to replicate the mathematical "states of mind" and symbol-manipulating abilities of a human computer..
What is the meaning of computability?
Computability is the ability to solve a problem in an effective manner.
It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science.
The computability of a problem is closely linked to the existence of an algorithm to solve the problem..
What makes a number computable?
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.
They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals..
What numbers are not computable?
However, the set of all irrational numbers is uncountable, so there must be some irrational number whose decimal expansion is not computable In fact, since only countably many irrational numbers can be computed, “most” irrational numbers are not computable.
- Alan Turing's 1936 paper "On Computable Numbers" effectively founded computer science because it introduced the concept of a theoretical machine, now known as a Turing machine, that could perform any computation that could be described in a mathematical formula.
- All algebraic numbers are computable and therefore definable and arithmetical.
For real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic. - If you have a non-computable number, there is NO fixed-size algorithm that can generate the number to an arbitrary precision.
You can have an algorithm that generates the number to 1,000,000 digits.
But if you want 1,000,001 digits of precision, you can't use the algorithm. - The computable numbers are an infinite set.
We have provided an injective function g that maps every computable number to a single natural number: a G\xf6del number.
Any set with such a function is countable, and therefore computable numbers are countable.Dec 1, 2021
