Table of Contents
Are most numbers Uncomputable?
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.
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 .
Does there exist a non-computable calculation?
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.
Are the real numbers computable?
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.
Are most numbers normal?
Abstract. A real number is called normal if every block of digits in its expansion occurs with the same frequency. A famous result of Borel is that almost every number is normal.
What is the most normal number?
Universally, given the prevalence of binary based computers, 0 and 1 are probably the most common numbers.
Are Uncomputable numbers countable?
The set of all finite strings in a given language is obviously countable as it is the countable union of finite sets. Because R∖Computable=Uncomputable, the uncomputables are uncountable.
What does Uncomputable mean?
Not computable
Filters. Not computable; that cannot be computed. adjective.
Are there any uncomputable numbers?
If irrational and trancendental numbers like √2, π, and eare computable, one begins to wonder if there are any uncomputable numbers. It turns out that almost every number is uncomputable. To understand this we first introduce the concept of a set being countable.
Is a real number always definable?
It is always definable, but in different models of Z F C it wil have different values, sometimes they will be computable (e.g. if G C H holds) and sometimes they could be non-computable (as above). So this gives us a definition of a real number which is not provably computable and not provably uncomputable!
Are the real numbers countable or uncountable?
If we attempt to do the same thing with the real numbers we will find that it is impossible; the real numbers are uncountable. Mathematicians would say that the rational numbers form a set of measure zero along the real axis. Computable numbers are numbers that can be calculated by a finite computer program. All rational numbers are computable.
How do you prove a number is not normal to all bases?
Just define a nested sequence of intervals around the real. That is a computable sequence because the real in their intersection is computable. Numbers which are not normal to all bases also fail a test. This requires a little more thought, but isn’t too bad.