Is every real number representable as an exact expression?

Question

I want to know: does every real number have an "exact" form or not?

By an exact expression I mean kind of like a closed form involving a finite number of elementary functions. For example, $e^{\pi\sqrt{163}}+\sqrt{1/\mathrm{arccot}(\sqrt[4]{17})}$.

I'm sorry for the vague question, but I hope someone can answer (possibly after making additional assumptions, if necessary, to make the question more concrete).

No. Uncountabley many real numbers. Countabley many exact expressions. — Eric, May 20 '23 at 03:42
@Eric: Makes sense, thanks! And thank you for the link, morrowmh. Intresting stuff that's really hard for me to understand right now. — John Smith, May 20 '23 at 03:44
And I'm guessing there is no example of a concrete real number that we can say definitely doesn't have an exact form? — John Smith, May 20 '23 at 03:46
You brought up elementary functions... have you considered the use of non-elementary functions? — JMoravitz, May 20 '23 at 03:50
@JMoravitz: I thought it would be interesting to know the answer first for elementary functions, but since the answer is negative, it'd be nice to learn from you what happens in the non-elementary case? — John Smith, May 20 '23 at 03:52
Your last comment is verging on paradoxical. What would it mean to have a concrete real number that does not have a concrete exact form? — Lee Mosher, May 20 '23 at 04:03
Your question is not crystal clear for me. However, irrational numbers can have infinite digits behind the decimal and since we can't even know exactly all the digits, it may not be possible to find a function that represents it. So, the answer would be no. — NoChance, May 20 '23 at 04:27
I mean, you used $e$ and $\pi$. Those were symbols introduced when we discovered a number that didn't have an "exact form" at the time. We can always do that again. — JonathanZ, May 20 '23 at 04:27
Useful reading: What is a closed form number? by Timothy Y. Chow (1999). Also, see the MSE question What is the reason that equations such as $\tan x = 2x$ can only be solved with the help of algorithms? — Dave L. Renfro, May 20 '23 at 09:30
Suppose you have a supper-pupper constant. You intrioduce symbol Ц for that constant. Now Ц is exact expression of your constant. And, of course, any constant is an elementary function. — Anixx, May 20 '23 at 20:33

IV_ · Accepted Answer · 2023-05-20T20:05:38.933

1.) Elementary numbers

The elementary functions were defined by Liouville ([Ritt 1948]). Ritt ([Ritt 1948] p. 60) defined the elementary numbers and asked the problem whether some simple transcendental equations have solutions that are elementary numbers.

Lin and Chow proved, assuming Schanuel's conjecture is true, that no irreducible algebraic equation of both $z$ and $e^z$ have solutions except $0$ that are elementary numbers or explicit elementary numbers respectively.

And Chow, assuming Schanuel's conjecture is true, and Khovanskii proved that the algebraic equations that aren't solvable in radicals cannot be solved by explicit elementary numbers.

Therefore, there are real numbers that aren't elementary numbers or explicit elementary numbers respectively.

Which kinds of equations of elementary functions can have elementary solutions?

Polynomials with degree $5$ solvable in elementary functions?
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2.) Closed-form numbers

If we ask for solutions that are closed-form numbers, we have to ask for solutions in closed form, e.g. in terms of Special functions.

If we ask for closed-form numbers, we have to define which set of functions we allow.

We know that there are equations whose solutions cannot be represented in terms of certain known Special functions, e.g. Lambert W, Generalized Lambert W or HyperLambert W.

What are the closed-form inverses of $x \sinh(x), x \cosh(x), x \tanh(x), x\ \text{sech}(x), x \coth(x), x\ \text{csch}(x)$?

What are the closed-form inverses of $x+\sinh(x)$, $x+\cosh(x)$?
$\ $

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

[Khovanskii 2014] Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014

[Khovanskii 2019] Khovanskii, A.: One dimensional topological Galois theory. 2019

[Khovanskii 2021] Topological Galois Theory - Slides 2021, University Toronto

Khovanskii's publications

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

[Ritt 1948] Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. Columbia University Press, New York, 1948

score 3 · Answer 2 · answered May 20 '23 at 14:53

(In addition to IV_'s answer) There are only countably many forms of mathematical expression comprising finitely many symbols, such as the examples you gave. However, as Cantor showed with his classic diagonal argument, the real numbers are uncountable. Therefore, there are real numbers for which no finite symbolic expression exists.

Is every real number representable as an exact expression?

2 Answers2