Are the computable numbers complete?
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 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 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 every rational number computable?
If irrational and trancendental numbers like √2, π, and e are computable, one begins to wonder if there are any uncomputable numbers. It turns out that almost every number is uncomputable.
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 Rayo’s number the biggest number?
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 is not computable?
A non-computable is a problem for which there is no algorithm that can be used to solve it. An example of a non-computable is the halting problem.
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.
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 bigger than a Googolplexianth?
Graham’s number is bigger than the googolplex. It’s so big, the Universe does not contain enough stuff on which to write its digits: it’s literally too big to write. But this number is finite, it’s also an whole number, and despite it being so mind-bogglingly huge we know it is divisible by 3 and ends in a 7.
How many zeros are in a Googolplexianth number?
Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes; that is, a 1 followed by a googol zeroes.
What is computable problem?
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.