Exactly as I have written above, a 1D vector is the difference between 2 points on a straight line.
In most mathematical handbooks it is now traditional to define first the real numbers deriving them from integer numbers, via rational numbers, without any geometric interpretations, then to define the straight line and the mapping between the real numbers and the straight line. Then usually the vectors are derived from tuples of real numbers.
I believe that this methodology is very wrong because it occults the real meaning and usefulness of vectors, real numbers, complex numbers and of many other important mathematical entities. This sequence of the derivations of the main mathematical objects leads to many confusions and mistakes and it also does not correspond with the historical development of mathematics, where the real numbers, that is the "measures", as they were initially called, were indeed obtained since the earliest times as quotients of differences between points, i.e. as quotients of vectors (e.g. the quotient between a measured length and a standard length, e.g. a foot), and not as some limits of rational sequences.
It is perfectly possible and much clearer in my opinion, to start from axioms of the affine spaces (i.e. the spaces of points) and to derive the real numbers (and everything else in geometry) from that.
The result that the real numbers correspond to limits of rational sequences is very important in practice, but it is not the motivation for the introduction of real numbers. Why would you believe a priori that the limits of rational number sequences are of any interest for you?
The points are interesting, because they model your environment. Then the vectors and then the real numbers, complex numbers, matrices etc. become interesting to be able to model the geometric transformations of the points.
In most mathematical handbooks it is now traditional to define first the real numbers deriving them from integer numbers, via rational numbers, without any geometric interpretations, then to define the straight line and the mapping between the real numbers and the straight line. Then usually the vectors are derived from tuples of real numbers.
I believe that this methodology is very wrong because it occults the real meaning and usefulness of vectors, real numbers, complex numbers and of many other important mathematical entities. This sequence of the derivations of the main mathematical objects leads to many confusions and mistakes and it also does not correspond with the historical development of mathematics, where the real numbers, that is the "measures", as they were initially called, were indeed obtained since the earliest times as quotients of differences between points, i.e. as quotients of vectors (e.g. the quotient between a measured length and a standard length, e.g. a foot), and not as some limits of rational sequences.
It is perfectly possible and much clearer in my opinion, to start from axioms of the affine spaces (i.e. the spaces of points) and to derive the real numbers (and everything else in geometry) from that.
The result that the real numbers correspond to limits of rational sequences is very important in practice, but it is not the motivation for the introduction of real numbers. Why would you believe a priori that the limits of rational number sequences are of any interest for you?
The points are interesting, because they model your environment. Then the vectors and then the real numbers, complex numbers, matrices etc. become interesting to be able to model the geometric transformations of the points.