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I have a question that I think is quite weird and I can't find an answer.

Is there any real number that I can't find using only $+,-,\times,\div,$ limits and radicals?

For example, using some series, I can find $\pi$ or $e$. But is there any number that I can't find using only those?

Sorry if I made any mistakes, I don't know how to ask that question or which tags to put.

Community
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alper akyuz
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  • These are called uncomputable numbers. They exist: however they cannot be computed, so we cannot give you any example. – Crostul Feb 20 '18 at 12:07
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    The real numbers can be constructed as limits of Cauchy-series with elements in ℚ. – P. Siehr Feb 20 '18 at 12:07
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    What about the Chaitin's constant, Crostul? An uncomputable number which is precisely defined (https://en.wikipedia.org/wiki/Chaitin's_constant) – rafa11111 Feb 20 '18 at 12:14
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    What does "allowing limits" mean? – Joppy Feb 20 '18 at 12:17
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    There are only countably many predicates you could write down in the language of mathematics. So there are many real numbers which are not definable. – Tim kinsella Feb 20 '18 at 14:32
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    @Timkinsella: Please read this post for a proper analysis of that notion. If you do not do it properly, your claim that "there are many real numbers which are not definable" would either be ill-defined or false. – user21820 Feb 20 '18 at 14:56
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    "Most" real numbers require an infinite amount of information to specify. So the observable universe is too small to hold the description of just one real number which is "typical" in this sense. Such a number will be a sequence of rationals, taking the limit, but the universe is too small to hold a sequence of rationals, no matter how few atoms you use to write each fraction (and the numerators and denominators will grow huge since the typical number is irrational). – Jeppe Stig Nielsen Feb 20 '18 at 15:50
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    Can you say more clearly what you mean by "find"? Pi and e are numbers that match a description, like "the ratio of a circle's circumference to its diameter" or "the amount a bank account with a dollar in it will increase to in a year if 100% interest is compounded continuously", and then we can come up with algorithms that give us approximations to this description. There are descriptions of numbers that we know exist that cannot be found in this way because there is no algorithm that gives us an approximation. Is that what you're after? – Eric Lippert Feb 20 '18 at 17:10
  • @Timkinsella is correct, there are only a countably infinite number of possible expressions but there are an uncountably infinite number of reals. Which means that an uncountable infinite number of reals cannot be 'found' or described by finite expressions. And BTW, the size of the physical Universe is irrelevant to abstract math. – David R Tribble Feb 20 '18 at 18:45
  • @DavidRTribble: Not true: see this post. The misconception here is concluding too much -- the argument only implies that any ZFC universe of sets does not contain a bijection between its set of real numbers and its set of possible expressions (written in a countable language). As a misconception, IMO it's rather similar similar to Skolem's paradox. –  Feb 20 '18 at 19:15
  • This is interesting: http://wiki.c2.com/?UnknowableNumbers – Jake Feb 20 '18 at 19:41
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    I voted to close as unclear. While the question might be interesting, you need to specify a lot of things in order to get a simple answer. For start, there are only countably many definitions of numbers as a definition has to be finite and there are only countably many (English) texts. However, the reals are uncountable. So you either have to allow different definitions of your numbers, or you have to restrict what you look for. – yo' Feb 20 '18 at 19:50
  • This got closed before I could write an answer, but I believe the concept of a dense set is the formal version of what you are seeking. The rational numbers are a dense subset of the reals, and it's trivial to show that rational numbers can be formed via division (one of your operators). – Cort Ammon Feb 20 '18 at 21:39
  • @user21820 I intended that as an informal metamathematical statement. It seems to me both undeniably true and also, by its nature, impossible to formalise (though of course it has some obvious formal approximants). I'm of course aware of pitfalls like Richard's paradox, but it seems to me these are just examples of pushing an informal notion further than its built to go. – Tim kinsella Feb 21 '18 at 04:08
  • @DavidRTribble: If you do not have a proper grasp of logic, please refrain from reinforcing the false misconception that many people hold. First read the post I linked to very carefully. Once you fully understand that post, you will know that what I and Hurkyl said in our earlier comments are correct. – user21820 Feb 21 '18 at 04:18
  • @Timkinsella: I was aware that your statement could have been informal, but it is a too frequent mistake by beginners in logic, so I strongly feel it is important to get it right. Also, have you read and understood the linked post? Richard's paradox is only a miniscule part of the real issue here. – user21820 Feb 21 '18 at 04:22
  • I'm nominating this question to be re-opened. The OP did not ask a question tagged (logic), he/she asked a question tagged (real-analysis), her/his question was answered by @OveAhlman and accepted, and that answer received 19 upvotes by the community. The question had an ambiguity which allowed for other interpretations (regarding computable numbers), and that is often, perhaps usually, a good reason for closing. But in this case the community has handled the ambiguity graciously (for the most part), and I think there might be further interesting answers out there. – Lee Mosher Feb 22 '18 at 13:15

5 Answers5

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Allowing limits is very strong.

A real number $y$ has an integer part and a decimal part $a.x_1x_2x_3x_4...$ We can construct this real number as a limit in the following way:

Let $y_1 = a, y_2=a.x_1, y_3 =a.x_1x_2 , y_4 = a.x_1x_2x_3 , y_5=...$ clearly each $y_i$ is a rational number and $\lim_{n\to\infty} y_i = y$.

Ove Ahlman
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    Although you're not just allowing limits; you're allowing arbitrary sequence formation as well. –  Feb 20 '18 at 13:09
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    Perhaps it's worthwhile saying that the sequences $x_i$ and $y_i$ are definable in terms of $y$ alone, namely by induction $y_1 = a = \lfloor y \rfloor$, $x_i = \lfloor (y-y_{i}) / 10^{i} \rfloor$, $y_{i+1} = y_{i} + x_i \cdot 10^{-i}$. The OP can tell us whether he allows mathematical induction and whatever other bits of logic might or might not be implicit in the question (sequences clearly are implicit in the question, given the examples of $\pi$ and $e$). More to the point, the OP can tell us whether we may use the "greatest integer" function $\lfloor \cdot \rfloor$. – Lee Mosher Feb 20 '18 at 14:45
  • @LeeMosher One can define a sequence converging to $e$ in a way that, depending on your definitions, uses only the arithmetic operations (in particular, the series definition of $e$ is hypergeometric). The floor function in your formula isn't the biggest problem; the biggest problem is the symbol "$y$", so you're implicitly allowing arbitrary real constants, which makes the problem trivial. – Jack M Feb 21 '18 at 12:27
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Maybe Chaitin's constant is somthing you are looking for.

It is non-computable, so we cannot approximate it to arbitrary precision, even with stronger tools than the ones allowed by you.

M. Winter
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    To be more precise, there are many different Chaitin's constants (it depends on the encoding in the definition), and in some of them a few digits can be computed: http://mathworld.wolfram.com/ChaitinsConstant.html [Sorry if my first version of this comment sounded rude - I accidentally hit enter before I finished it] – David Szwer Feb 20 '18 at 13:00
  • Generally, any non-computable number should be counted as a "real number that [one] can't find using only $+,-,\times,\div,$ limits and radicals" (starting from 1, say). And since the real numbers are a lot more than the number of programs/algorithms/expressions/whatever one can write as a finite sequence of a finite number of characters, there are a lot of non-computable numbers. – md2perpe Aug 04 '21 at 10:26
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Every real number $x$ can be written in the form: $$x=n+\sum_{k=0}^\infty x_k\left(\frac{1}{10}\right)^k,\ x_k\in\{0,1,\dots,9\}$$ and we are used to call $n$ in integer part and $x_k$ its decimal digits. So the answer to you question is yes.

Moreover, you can write any real number $x$ in the form: $$x=n+\sum_{k=1}^\infty b_k\left(\frac{1}{2}\right)^k,\ b_k\in\{0,1\}$$ where $b_k$ are its binary digits.

So, you only need two digits and the operations you've mentioned.

Vassilis Markos
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Any real number can be expressed as a limit of rational numbers, and rational numbers can be expressed by dividing two whole numbers.

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It depends upon what you mean by "number that I can't find", but there are an infinity of real numbers that you can't define in any way using what you are describing.

Consider the set of numbers you can "define". The definition of such a number is some kind of text you can encode as a binary number, so this set is a countably infinite set.

Because real numbers are uncountable, there is an infinity of real numbers you can't define.

Karlotcha Hoa
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