I've come to realize that the more I study some math subjects the more I question some results or ideas that seemed trivial or obvious to me. My question is about the real numbers and their geometric interpretation as a line.
I get that real numbers being complete and ordered are naturally visualized as a number line. Now, consider two positive numbers $a$ and $b$ and find them in the number line. We define the binary operation $a•b$ like this: "take a compass and open it from O to $b$, now draw a circle with such compass with $a$ as its center. The point on the right where the circle intersects the line is $a•b$".
This, of course, corresponds to $a+b$ but the reason is not that obvious to me. Intuitively we use addition when we "add" something so if I'm "adding" a line segment with length $a$ to another line segment with length $b$ the length of the resulting line must be $a+b$. But with a formal definition of addition (via Dedekind cuts for example) that property of addition doesn't seem that clear to me. I guess my question is: why must addition correspond to $a•b$? Why, if I have a line segment of length 3/2 and other one of length 5/3 can I be sure that if I arrange them so that one starts exactly when the other one ends, the length of the resulting line will be 19/6?
Please forgive me if this question is too obvious or trivial. Thank you!