What Are Computable Numbers?

space → 400 newline → 401 etc. Using this mapping, we can assign a unique integer to any text file. Since finite computer programs can always be represented as a finite text file, the number of computable numbers is countably infinite.

What is the highest computable number?

program by Ralph Loader that came in first place for the Bignum Bakeoff contest, whose objective was to write a C program (in 512 characters or less) that generates the largest possible output on a theoretical machine with infinite memory. It is among the largest computable numbers ever devised.

Do non-computable numbers exist?

Other examples of non-computable numbers are known: the Chaitin’s con- stant Ω ; the real number such that its n-th digits equals 1 if a given universal TM halts for input n, and 0 otherwise (see); the real number whose digits ex- press the solutions of the busy beaver problem.

Why are computable numbers countable?

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. … This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals.

Are imaginary numbers computable?

The real and imaginary parts of a complex number are computable from the complex number, so the computable complex numbers are those with computable real and imaginary parts. It doesn’t matter whether you look at computable numbers as reals or complex numbers.

What problems are not computable?

(Undecidable simply means non-computable in the context of a decision problem, whose answer (or output) is either “true” or “false”). 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) is the Halting Problem.

What is the meaning of computable?

: capable of being computed.

Is Googolplex bigger than infinity?

Almost inevitably, at this point someone proffers an even bigger number, “googolplex.” It is true that the word “googolplex” was coined to mean a one followed by a googol zeros. … True enough, but there is nothing as large as infinity either: infinity is not a number. It denotes endlessness.

Are all algebraic numbers computable?

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.

Is Infinity calculable?

Unlike indefiniteness, infinity doesn’t have a limit that is beyond what is actually calculable; rather, infinity is without limit.

Is EA normal number?

A number is said to be absolutely normal if it is normal in all integer bases greater than or equal to 2. … It is widely believed that the (computable) numbers √2, π, and e are normal, but a proof remains elusive.

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).

Is Uncomputable a word?

Not computable; that cannot be computed.

What do you mean by universal Turing machine?

In computer science, a universal Turing machine (UTM) is a Turing machine that simulates an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simulated as well as the input to that machine from its own tape.

Which one of the following is the common synonym of computable?

Some common synonyms for “computable” are “solvable”, “decidable”, and “recursive”.

What is Turing computable function define recursive function?

A language is called computable (synonyms: recursive, decidable) if there is a computable function f such that for each word w over the alphabet, f( w ) = 1 if the word is in the language and f( w ) = 0 if the word is not in the language.

Is the equivalence problem computable?

Example: The equivalence problem is not partially computable.

Are all decision problems computable?

If this decision problem were decidable then the function that yields the answer of the function problem is computable. Every decision problem can be converted into the function problem of computing the characteristic function of the set associated to the decision problem.

What is the concept of Undecidability?

: not capable of being decided : not decidable …

Is Pi irrational or transcendental?

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of π’s transcendence. in which the nth digit after the decimal point is 1 if n is equal to k! (k factorial) for some k and 0 otherwise.

Are real numbers part of complex numbers?

Real numbers are to be considered as special cases of complex numbers; they’re just the numbers x + yi when y is 0, that is, they’re the numbers on the real axis. For instance, the real number 2 is 2 + 0i. The numbers on the imaginary axis are sometimes called purely imaginary numbers.

Are imaginary numbers irrational?

Imaginary Numbers Have Applications

If the number line is expanded to become a number plane, some numbers that are neither rational nor irrational can be plotted. These are “imaginary numbers” which are defined as multiples of the square root of -1.