Are there real numbers that can only be expressed via a complex expression? Someone recently told me certain real numbers could only be expressed in closed form via an expression involving complex numbers.
Is this true?  If so do these numbers have a name?
What is a simple example?
 A: This phenomenon occurs prominently in  the (historically interesting) casus irreducibilis of third degree polynomial equations with three real roots. Take the equation
$$x^3-2x^2-6x+5=0$$
as an example.
A: To answer my final part about a nice, minimal example:
$$\sqrt[3]{1+i \sqrt{7}}+\sqrt[3]{1-i \sqrt{7}}$$
Which $\approx$ 2.6016791318831542525
This value was adapted from one of the roots of the polynomial posted in the accepted answer, then back solved for the generating cubic to ensure it is a real casus irreducibilis.  (Previous version using $\sqrt{5}$ did not have a rational cubic.)
The generating cubic is simply $x^3 - 6x - 2 = 0$.
As noted by a @Steven Stadnicki , this is only irreducible under the radicals.  Using Euler's formula we can find a trigonometric representation of this number.
$$2 \sqrt{2} \cos{\left(\frac{\tan^{-1}{\left(\sqrt{7}\right)}}{3}\right)}$$
A: $2$ things to be noted:


*

*The set of real numbers $(\mathbb{R})$ is a subset of the set of complex numbers $(\mathbb{C})$. Any real number is a complex number but the converse is not true.

*If any number (real or not) can be expressed involving certain non-zero complex expressions, then it is a complex number, by definition.
If this is not what you intended to ask, then please tell me.
