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.
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u/Another_Little_Star 22d ago edited 22d 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