Are there any examples of non-computable real numbers?

Is this true, that if we can describe any (real) number somehow, then it is computable?

For example, $$\pi$$ is computable although it is irrational, i.e. endless decimal fraction. It was just a luck, that there are some simple periodic formulas to calcualte $$\pi$$. If it wasn't than we were unable to calculate $$\pi$$ ans it was non-computable.

If so, that we can't provide any examples of non-computable numbers? Is that right?

The only thing that we can say is that these numbers are exist in many, but we can't point to any of them. Right?

• I can't be more precise since I don't know :) I agree that probably there can be some desriptions of non-computable numbers -- this is what my question about -- please provide some for me to feel...
– Dims
Aug 8, 2013 at 13:43
• Roll a ten sided die numbered 0-9 an infinite number of times. May 20, 2019 at 3:13

I haven't thought this through, but it seems to me that if you let $BB$ be the Busy Beaver function, then $$\sum_{i=1}^\infty 2^{-BB(i)}=2^{-1}+2^{-6}+2^{-21}+2^{-107}+\ ... \ \approx 0.515625476837158203125000000000006$$ should be a noncomputable real number, since if you were able to compute it with sufficient precision you would be able to solve the halting problem.

• (+1) I like this answer because it explicitly provides a great many initial digits of an uncomputable real number and has a very simple easy-to-understand definition. The current state of knowledge about the max-shifts function is that $BB(5) \ge 47176870$, so the present example is actually 0.51562547683715820312500000000000616297582203915472977912941627176741932192527428924222476780414581298828125000..., where there are at least $\lfloor 47176870 \ log_{10}2\rfloor - 107 = 14201545$ zeros after the $...8125$ and before the next positive digit. May 31, 2014 at 13:52
• Every finite sequence of digits is the start of the decimal expansion of an uncountable family of non-computable numbers
– MJD
May 31, 2014 at 18:50
• That's true, of course; but it's beside the point, which is that this particular uncomputable number (viz., $\sum_{i=1}^\infty 2^{-BB(i)}$) has these particular digits. Jun 1, 2014 at 4:29
• "Sufficient" is not a subjective term; it is a mathematical term of art with a specific technical meaning.
– MJD
Jan 27, 2015 at 5:18
• While it's true that "sufficient" is a technical term and not necessarily incorrect, I believe "arbitrary" is more appropriate here. In order for a number to be computable, it needs to be able to be specified with arbitrary precision. "Sufficient" implies some threshold that you must cross. "Arbitrary" implies that that threshold is infinite.
– Tim
Feb 17, 2017 at 18:40

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.

• Yes. You can obviously change the probability with which a random program halts by changing the programming language or computational model you are working in. But each language has its own associated $\Omega$.
– MJD
Aug 8, 2013 at 13:52
• An $\Omega$ number does represent something, but not (as the WIkipedia article also says) "the probability that a randomly generated program will halt", because (1) there is not any canonical probability measure on the set of programs, and (2) the non-canonical measure used to generate a particular $\Omega$ number may not be a probability measure. I realize I am being picky about this, but it is a common point of confusion in that area. Aug 8, 2013 at 19:23
• @MJD: An $\Omega$ number, relative to a universal prefix-free function $U$, is exactly the measure of the set of infinite binary sequences $f \in 2^\omega$ such that there is a finite prefix of $\sigma$ of $f$ on which $U$ will halt. The measure is the usual fair-coin probability measure on $2^\omega$, which is fixed and independent of $U$. Aug 8, 2013 at 20:31
• The key points: (1) The measure is on $2^\omega$, not on $2^{<\omega}$ or on "programs". (2) There is no useful probability measure on $2^{<\omega}$ or any countable discrete set. (3) It is not even clear what a "program" for a prefix-free universal function $U$ should be, since every string is a legal input for $U$. Is a "program" then an arbitrary string (in which case the set of programs is not prefix-free, and the infinite sum of $2^{-|p|}$ over all programs $p$ diverges) or is a "program" just an input on which $U$ halts (in which case the probability a "random program" halts is 1). Aug 8, 2013 at 20:35
• Reminder: Anyone can edit Wikipedia and make it better... May 29, 2014 at 3:05

Any language can be turned into a number, by setting the $i^{th}$ decimal to 1 if the $i^{th}$ word is in the language, and to 0 otherwise. So we can build for instance the number $H$, which describes the halting problem and is therefore uncomputable.

• Would this example be more or less equivalent to the busy beaver example given by MJD?
– k_g
Dec 18, 2014 at 3:03
• Yes it is more or less the same idea up to encoding, which is basically the reason why we can exhibit uncomputable real numbers, because a real number can encode an undecidable language. Dec 18, 2014 at 13:28
• Love this answer. Jan 19, 2015 at 14:07
• This is hilarious and yet at the same time useless to the reader. Oct 20, 2017 at 7:25

I usually consider the function $$f(n)=1$$ if the $$n^{th}$$ computer program halts with input $$0$$ and $$f(n)=0$$ otherwise. The real number $$r=\sum_{n\in \mathbb{N}} f(n)\times 2^{-n}$$ must be non-computable, otherwise you would be able to solve the halting problem.

Note: in order to accept this argument you must accept that there is a bijection between natural numbers and computer programs, which can be easily done by assigning values to the symbols used to write programs.

Examples of uncomputable numbers, like the one clearly presented by Luis Fonsera, seem to assume that the halting of any particular Turing machine can be determined (by some other Turing machine ?). Is this always true? It not then for some values of n, Fonsera's function f(n) might not be determined and so the uncomputable number r is undetermined. An example might be a Turing machine that searches for violations of the Goldbach conjecture. Another could be a Turing machine, T, that searches through ratios of two rational numbers to see if it can find one that is squareroot of 2. We know that this programme wil never halt but I am at a loss to see how a Turing machine could come to show this just by examining the programme of machine T. Are some mathematical truths like the irrationality of squareroot 2 uncomputable?

Most finite strings of digits have high Kolmogorov complexity (Li & Vitanyi, 1997). This means they can not be produced by a Turing machine much smaller than the length of the string itself. There is a counting argument for this, which (overly simplified) says that there are less strings with length n-1 than there are with length n. If this is also the case for infinite strings (which I assume, but don't remember) it follows that most real numbers are uncomputable.

Li, Ming; Vitányi, Paul (1997). An Introduction to Kolmogorov Complexity and Its Applications. Springer.

• You can show that there are uncountably many uncomputable real numbers more directly - there are only countably many Turing machines, so there are only countably many computable numbers, so there are uncountably many remaining real numbers that aren’t computable. May 24 at 15:19