How to define length geometrically

Question

My boss claims that distance in $\Bbb{R}^n$ between two vectors, $a=[a_1, a_2, \dots a_n]$ and $b=[b_1,b_2,\dots b_n]$ is defined as:

$$d = \sqrt{\sum (b_i-a_i)^2}$$

I claim that no, under a geometric treatment one only needs to define length in one dimension. From there, one can apply Pythagoras theorem to get derive the Euclidean distance in $\Bbb{R}^2$ and inductively from there to $\Bbb{R}^n$. My question is two-fold:

1. Is it true that Euclidean distance in $\Bbb{R}^n$ must be defined and can't be derived once we define it in $\Bbb{R}^1$ and just apply Pythagoras theorem repeatedly (and the definition of $\Bbb{R}^n$ as the space spanned by orthogonal vectors)?
2. In modern geometry, how is distance/ length in $\Bbb{R}^1$ defined? I understand everything is built on Euclid's five postulates. The first postulate talks about being able to draw a line segment between two points. But I see nothing defining length in $\Bbb{R}^1$.

Also, is there a textbook on Geometry that covers these concepts?

Answer 1

Mathematically we can identify lengths with the norms and distances with metrics. What you've noticed is that given some norm $\|\cdot \|$ then we can define a metric as $d(x,y):=\|y-x||$, called the induced metric.

The Kuratowski embedding theorem ensures that ever metric space can be embedded into some Banach space so we can always think of a metric as being induced by some norm. So that means if we're giving a a means by which to measure length we can define distances and every means of defining distances can be related to some means of measuring length.

I first encountered these concepts in linear algebra and analysis. Norms are often related to inner products and so show up naturally in linear algebra. Metric spaces are fundamental to all considerations in analysis and so norms are too.

Answer 2

I assume that when you and your boss are talking about distances in $\mathbb{R}^n$ you mean the ordinary distance we use most of the time, not the fact that there are other metrics on that space - that is, other consistent ways to define "distance".

I think this is (roughly) the answer to (2):

If you have a good set of axioms for synthetic geometry (that is, geometry from geometric postulates like Euclid's, modernized to fill gaps, as Hilbert did) then if you choose a segment and decree that it has length $1$ you can prove that allows you to introduce coordinates on a line such that the absolute value of the difference defines the distance you want. The Pythagorean theorem (which is in fact equivalent to the parallel postulate) allows you to introduce the usual coordinate system in the plane (once you choose axes). The "distance formula" follows, in all finite dimensions.

So if you are starting from geometry, you introduce coordinates and the Pythagorean theorem tells you how to calculate distances as real numbers. (Euclid didn't do that. For the Greeks the square of the hypotenuse was a literal square with the hypotenuse as one side.) If you start from a coordinate system then with the usual algebraic definition of lines you can prove Euclid's postulates, including the parallel postulate and the Pythagorean theorem.