Most real numbers can never be calculated
, they’re uncomputable, which suggests that mathematics is full of things that we can’t know, that we can’t calculate. This is related to something famous called Gödel’s incompleteness theorem from 1931, five years before Turing.
What makes a number computable?
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.
Are all real numbers 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. Every computable number is arithmetical.
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
How do you know if a function is computable?
The function f such that f(n) = 1 if there is a sequence of at least n consecutive fives in the decimal expansion of π, and
f(n) = 0 otherwise
, is computable.
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
.
What is the highest computable number?
In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. … The inverse of this bijection is an injection into the natural numbers of the computable numbers, proving that
they are countable
.
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
.
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.
Is the empty set computable?
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.
What is meant by a verifier for a language?
A verifier for a language L is
a TM V
with the following properties: ● V is a decider (that is, V halts on all inputs.) ● For any string w ∈ Σ*, the following is true: w ∈ L iff ∃c ∈ Σ*.
What does it mean for a problem to be 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
What is 5% as a normal number?
| chart for: Percentage conversions | Percent equals Decimal No equals Fraction equals | 5% 0.05 1 ⁄ 20 | 7.5% 0.075 3 ⁄ 40 | 10% 0.1 1 ⁄ 10 |
|---|
Is zero a normal number?
Zero does not have a positive or negative value. However,
zero is considered a whole number
, which in turn makes it an integer, but not necessarily a natural number. … They have to be positive, whole numbers. Zero is not positive or negative.
