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!
Best Answer
Let's completely forget all notions of number entirely.
It's not hard to show this operation (which I will call $\oplus$ rather than $\bullet$) has nice algebraic properties: it's commutative, associative, and it respects the ordering on line segments.
Alongside $\oplus$, we could pick some segment to be a unit, and use another geometric construction to define an operation $\otimes$.
These two operations on line segments have very good algebraic properties; without using any notion of number, we've still managed to construct a system in which we can do arithmetic, algebra, and analysis — in fact, once we add in a notion of direction, these operations satisfy the complete ordered field axioms.
What is the length of a line segment? The line segment itself. Addition is literally defined as "extending a length".
I believe this overall description is a relatively accurate description of actual history — e.g. that Greek geometers did algebra with line segments, considering the segments themselves as a notion of quantity; e.g. see Euclid's elements, book 2. The real number system came into being precisely as numeric representations of these geometric objects
In fact, AFAIK, for a long time mathematicians considered numbers quantifying length and numbers quantifying area as completely different kinds of numbers, rather than the point of view today where both kinds of quantities are described by the same number system, but possibly with units attached.
The perspective that treats real numbers as being more fundamental than geometric notions is, I think, mainly an artifact of how math is taught in modern times.