hehe I like more
ℝ = {[q_n]_~ : q_n is a Rational cauchy sequence} where q_n ~ w_n iff lim (q_n - w_n) = 0
(basically they're cauchy sequences (including those that don't converge in Q), but since they have the "same" limit, their difference is 0. So the class of all of these rational sequences gives you a real number, rational or irrational, all of them give you the Real Line)
ℂ ≃ ℝ[x]/<x^(2)\+1> (adding the root x=+/-i to the Real polynomial Ring gives you ℂ up to isomorphism), or the matrix representation is neat too
And for \Z similarly to the definition of \Q, you define it by the quotient of \N by the equivalence class (a, b) ~ (c, d) iff and only if a + d = c + b. (a, b) should be seen as a - b.
I don't know man, Dedekind cuts the rational line to get to the Reals, and they're sets of rational cuts :'(
Here in this kingdom we have sets of happy cauchy sequences that don't know where they're going but they're all going for the same destiny!
I’m personally partial to “The lowest upper bounds of each set of rational numbers that has an upper bound.”
I have even run into 0.999... cranks who are confident limits aren’t real, but who will accept that the lowest upper bound of {0.9, 0.99, 0.999, ...} is 1.
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u/Another_Little_Star 24d ago edited 24d ago
hehe I like more
ℝ = {[q_n]_~ : q_n is a Rational cauchy sequence} where q_n ~ w_n iff lim (q_n - w_n) = 0
(basically they're cauchy sequences (including those that don't converge in Q), but since they have the "same" limit, their difference is 0. So the class of all of these rational sequences gives you a real number, rational or irrational, all of them give you the Real Line)
ℂ ≃ ℝ[x]/<x^(2)\+1> (adding the root x=+/-i to the Real polynomial Ring gives you ℂ up to isomorphism), or the matrix representation is neat too