Waitamoment… The rationals are not a subset of the irrationals. No rational number is irrational, by definition. The rationals and the irrationals together form the real numbers.
And there are a number of other interesting subsets of the complex numbers, not all of which fall into this hierarchy. For example, you can have the Gaussian integers, which are complex but with integer coefficients, or the set of complex numbers with rational coefficients (what’s that one called?), both of which are rings and thus have algebraic properties corresponding to those we’re familiar with.
To the best of my recollection (and this was just from seeing them in a Whittimore lecture series):
Let K be a number field (a finite extension field of Q. For each finite prime of K (yes, there’s a way to recover primes in number fields) let K[sub]v[/sub] be the completion of K at v and o[sub]v[/sub] be its valuation ring. Now, A[sub]K[/sub] is a subset of the direct product over all primes (finite and infinite). If we denote an element of the direct product by (a[sub]v[/sub]), it is in A[sub]K[/sub] iff a[sub]v[/sub] is in o[sub]v[/sub] for all but finitely many of the finite primes v. A[sub]K[/sub] inherits addition and multiplication and K imbeds in the obvious way along the “diagonal”.
I remember that they’re useful because they satisfy certain universality properties, so evaluating algebraic groups on them or taking Adèlic representations of algebras can be fruitful. I, personally, hate numbers and have as little to do with the Adèles as possible.