[ajkjk]

I am very confused why you are continually talking about rationals as if they are not real. every real number is also a rational number, in the usual conception of things, are they not? Perhaps you are distinguishing the two? like regarding 1.000 as an equivalence classes of cauchy sequences is not the same as 1.000 as the equivalence class of a/a?

because when I picture 1.000 I am clearly imagining a real number. Likewise if I imagine pi, as defined any way you like.

[nilkn]

My language was sloppy, but I'll admit I thought it was pretty obvious that we were talking about defining the rest of the reals starting from the rationals -- obvious enough that it didn't need clarification. I can't edit my prior comment, but you may imagine it has been amended in the obvious way with that clarity made explicit rather than implicit and reply to it again if you're interested in continuing the conversation.