Are All Real Numbers Computable? - Fixanswer

Other examples of non-computable numbers are known:

the Chaitin’s con- stant Ω [2]; the real number such that its n-th digits equals 1 if a given universal TM halts for input n, and 0 otherwise

(see[3]); the real number whose digits ex- press the solutions of the busy beaver problem.

What makes a number computable?

A real number is computable

if and only if the set of natural numbers it represents (when written in binary and viewed as a characteristic function) is computable

.

Are rational numbers computable?

It turns out that almost every number is uncomputable. To understand this we first introduce the concept of a set being countable. A set is called countable if it can be put in one-to-one coorespondence with the integers. For instance,

rational numbers are countable

.

Are computable real numbers countable?

While the set of real numbers is uncountable,

the set of computable numbers is only countable

and thus almost all real numbers are not computable. That the computable numbers are at most countable intuitively comes from the fact that they are produced by Turing machines, of which there are only countably many.

What things are not computable?

(Undecidable simply means non-computable in the context of a decision problem, whose answer (or output) is either “true” or “false”). Non-Computable Problems – A non-computable is

a problem for which there is no algorithm that can be used to solve it

.

What is the meaning of computable?

Definition of computable

:

capable of being computed

.

Are all rational numbers constructible?

All rational numbers are constructible

, and all constructible numbers are algebraic numbers (Courant and Robbins 1996, p. 133). If a cubic equation with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins 1996, p. 136).

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).

Is the empty set computable?

Examples and non-examples

The empty set is computable

. The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbers less than a given natural number is computable.

Is Pi a computable number?

Yes, π is computable

. There are a few equivalent definitions of computable, but the most useful one here is the one you have given above: a real number r is computable if there exists an algorithm to find its n th digit.

What is loader’s number?

Loader’s number is essentially

a busy beaver number for the calculus of constructions

, which is possible to compute since all coc programs terminate. In particular, loader. c defines a function called D .

Are there real numbers that are Uncomputable not generated by a Turing machine )?

You can show that there are uncountably many uncomputable real numbers more directly – there are only countably many Turing machines, so there are only countably many computable numbers, so

there are uncountably many remaining real numbers that aren’t computable

.

Is Pi a normal number?

Irrationality. In the 18th century, the Swiss mathematician Johann Lambert proved that

π is an irrational number

. This means that it is impossible to express π as a fraction of two integers. As a consequence, π has an infinite number of digits and does not end in an infinitely repeating pattern of digits.

What does the Church Turing thesis state?

The Church-Turing thesis (formerly commonly known simply as Church’s thesis) says that

any real-world computation can be translated into an equivalent computation involving a Turing machine

.

What is a cut in math?

A Dedekind cut is

a partition of the rational numbers into two sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element

. The set B may or may not have a smallest element among the rationals.

How do you know if something is computable?

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Are all solvable problems computable?

A mathematical problem is computable if it can be solved in principle by a computing device. Some common synonyms for “computable” are “solvable”, “decidable”, and “recursive”.

Hilbert believed that all mathematical problems were solvable, but in the 1930’s Gödel, Turing, and Church showed that this is not the case

.

What makes something undecidable?

In computability theory, an undecidable problem is a type of computational problem that

requires a yes/no answer, but where there cannot possibly be any computer program that always gives the correct answer

; that is, any possible program would sometimes give the wrong answer or run forever without giving any answer.

Is every function computable?

I’d like to share a simple proof I’ve discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function!

What does it mean to be effectively computable?

is effectively computable

if there is an effective procedure or algorithm that correctly calculates f

. An effective procedure is one that meets the following specifications.

Is Ackermann function computable?

The Ackermann function is the simplest example of a well-defined total function which is

computable but not primitive recursive

, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Dötzel 1991).

Are all algebraic numbers constructible?

Not all algebraic numbers are constructible

. For example, the roots of a simple third degree polynomial equation x3 – 2 = 0 are not constructible. (It was proved by Gauss that to be constructible an algebraic number needs to be a root of an integer polynomial of degree which is a power of 2 and no less.)

Are complex numbers constructible?

—

A complex number is constructible if and only if it is algebraic and the field generated by its conjugates is a finite extension of Q whose degree is a power of 2

. The remaining part is usually proved using Galois theory.

How do you prove a number is constructible?

Definition. A real number r ∈ R is called constructible

if there is a finite sequence of compass-and-straightedge constructions that, when performed in order, will always create a point P with at least one coördinate equal to r

.

Are all irrational numbers transcendental?

Transcendental numbers are irrational, but

not all irrational numbers are transcendental

. For example, x
²
– 2 = 0 has the solutions x = ±Square root of√2; thus, Square root of√2, an irrational number, is an algebraic number and not transcendental.

What is the difference between transcendent and transcendental?

A transcendental idea is applied immanently when it is applied only to an object within the limits of experience. It is applied transcendently when it is applied to an object beyond the limits of experience or to an object falsely believed to be adequate with, and to correspond to, it.

How do you know if a number is transcendental?

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument

. For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as 5π, π-3/√2, (√π-√3)
⁸
, and
⁴
√π
⁵
+7 are transcendental as well.

Are the sets 0 and ∅ empty sets?

the set {0} is not an empty set

.. it has only one element which is 0 . But the set {Ø} is an empty set .

Are all finite sets decidable?

Every finite set is decidable

since we can always “hard-code” a Turing machine to accept a given finite set: fixing a1,…,an, just write a program which on input k checks whether k=a1, whether k=a2,… , whether k=an, and outputs “YES” if the answer to one of these questions is YES and outputs “NO” otherwise.

Is natural numbers decidable?

In fact one can restrict oneself to the concept of a decidable set of natural numbers, since the more general case can be reduced to this case by enumerating the objects under consideration.

A set M of natural numbers is said to be decidable if there exists a general recursive function f such that M={n:f(n)=0}

.

What is a Turing machine in theory of computation?

A Turing machine is

a computational model, like Finite Automata (FA), Pushdown automata (PDA), which works on unrestricted grammar

. The Turing machine is the most powerful computation model when compared with FA and PDA. Formally, a Turing machine M can be defined as follows − M = (Q, X, ∑, δ, q0, B, F)

How long is Rayo’s number?

Rayo’s number: The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with

less than a googol (10100) symbols

.

Is Rayo’s number finite?

Definition. The definition of Rayo’s number is a variation on the definition:

The smallest number bigger than any finite number

named by an expression in the language of first-order set theory with a googol symbols or less.

What’s bigger than Rayo’s number?

Note that, in this new theory, Rayo’s number can now be described very briefly, in terms of this new constant! So

H(1, 10
¹⁰⁰
)

will be much larger than Rayo’s number.