Are uncomputable numbers a subset of real numbers?

Question

I know that Chaitin's constant is an uncomputable real number, but I'm curious as to if there are any proven examples of non-real uncomputable numbers? Could uncomputable numbers be complex?

The Wikipedia page on types of numbers says that computable numbers are real numbers. An answer on Quora says that computable numbers are complex numbers. Given that neither of these are really necessarily cited, I'm curious if there exists any mathematical definition of the computable numbers and whether that helps define the uncomputable numbers?

Thanks so much!

Answer 1

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.

We don't really care what the definition of a computable number is. There are lots of numbers that are clearly computable, like all the rationals. You can compute them based on the field axioms. As long as you have only countably many constants, there are only countably many computable numbers because there are only countably many finite strings of characters to compute a number. As there are uncountably many reals, most of them are uncomputable. The negative definition is what makes it hard to specify an uncomputable number. If you can define it, usually you can compute it.

Answer 2

A computable number which can be computed to any number of digits desired by a Turing machine, according to Wolfram Math World. From this definition, which talks of digits, I would infer that computable and uncomputable numbers are a subset of real numbers.