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"We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter R"

"The geometric line can, as soon as one chooses two distinct points on it, be made into a commutative ring, with the two points as respectively 0 and 1"

1 - What is this choice of the points 0,1 for? Sure they are the two unit elements of the ring R, but are they used to generated the other elements in R ? because from these 0,1 it seems he constructs other elements of R as: 1+1, 1+1+1...

2 - But if this is the case, are the integers the only elements of R ?

then he "promotes" the ring structure of R as an algebra over the rationals because it's required for the element 1+1, 1+1+1 to have an inverse.

3 - so.. is because of this Q, given all we have is 0,1, that it become possible to include the rationals in R?

4 - what about the irrationals then? Is \pi included in R? if so how to obtain \pi from 0,1 and Q?

My general understanding is that the real numbers are include in R, which moreover, in order to include the nonzero and nilpotent elements, has to loose its field structure in favor of a ring structure (with the subring of the rational). In others words my four questions above seem all "bad questions" but I really don't understand his comment about what the points 0,1 are for, how they give a ring structure to the geometric line or if/how they are used to generate others elements in R. thanks much ;)