Why do we call $\sqrt{-1}$ imaginary and $-1$ real? Looking at the numbers I can see in nature only positive real numbers. Because many problems couldn't be solved using only positive real numbers a new number set, called "negative numbers" was introduced. Later on, for similar reasons, complex numbers were introduced and the term of "imaginary". 

Can anyone please explain to me why $-1$ is "more real" than $i$? For me both seem to be a convention, a name given to the result of a function or a set of functions to the positive real numbers.

Disclaimer: I'm not a mathematician, I'm just trying to better understand some basic concepts and conventions in math
In mathematics, every modifier, such as "continuous, real, rational, ...", is reassigned meaning, lest there be unnecessary confusion such as "your 'continuous" is different than mine" that wastes time. So, if you come across a terminology in some math book, find its definition within the book; do not take a dictionary and check the meaning of the term... 

I am not sure if you are asking about the origin of the reason why the word "imaginary" was chosen. If not, then leave the "secular" meaning of the word alone. If you do this, then it is crystal clear that "imaginary number" is no more than a word used to name the relatives of $\sqrt{-1}$. Note also that "positive" and "negative" in math have nothing to do with any sentiments. 

Logically, you can call the real numbers "imaginary" and the imaginary numbers "real". Then in your book, the theorem "the square of every real number is $\geq 0$" becomes "the square of every imaginary number is $\geq 0$". Noticed? The name thing is rather superficial :), which in math serves as a mnemonic device and a shortcut of reference.
Whenever you have got a question about why we call something so.... refer to the definition.
According to Wikipedia:

A real number  is a value that represents a quantity along a line.
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit $i$, which is defined by its property $i^2 = −1$.

There is no problem in understanding that why $-1$ is a real number as you can easily construct a line and represent $-1$ on that line. The problem lies in the definition of imaginary numbers. It is a real number times $i$.  Why $i$ ?? Because imaginary numbers are  defined that way and changing the definition is not so good I think.  :) :) :)