Is it possible to "produce" a non-real number without the square root operation?

Question

I am trying to prove that various expressions are real valued functions. Is it possible to state that, because no square roots (or variants such as quartic roots etc) are in that function, it is a real valued function?

Answer 1

"Function" is such an incredibly broad notion that it is wholly implausible to give a general answer to this sort of question that isn't something trivial like "Applying a real-valued function always gives a real number".

More meaningful questions need to restrict the scope; for example, one might ask what sort of numbers you can produce by plugging real numbers into the operations of addition, subtraction, multiplication, and division. (where by those words I refer to the usual arithmetic operations) (and, of course, I mean that you only plug in numbers that are in the domain of the operation)

And, in fact, you can only get real numbers by doing so.

Incidentally, assuming they satisfy the "usual laws of arithmetic" (i.e. the field axioms), the sorts of functions you can produce by using these four arithmetic operations, are called rational functions.

That said, depending on the fine details of exactly what you mean by your question, this particular fact might be just another trivial one, of the form "applying a real-valued function always gives a real number".

This question is more interesting when you ask questions like "what values can I get from rational functions if I only start with rational numbers and $\sqrt{2}$?" (answer: the numbers that can be written in the form $a + b \sqrt{2}$, where $a,b$ are rational numbers)

Answer 2

Well, $\sin^{-1} 2 = 1.570796327-1.316957897i$ (for a usual branch.) I don't know if this counts as using square root. The series for $\sin^{-1}x$ is found by integrating the series for $1/\sqrt{1-x^2},$ so there's a square root floating around.