Other people have covered why this doesn't cover all irrationals, so I'll just instead share a cool math fact about all this: Mathematicians split the irrationals into two groups: the algebraics and the transcendentals. Algebraics are irrational numbers that are the solutions of polynomials (with integer coefficients), like `sqrt(2)` is the solution to `x² - 2 = 0` and `sqrt(2) + sqrt(3)` is the solution to `x⁴ - 10x² + 1= 0`. Every other irrational, like π or e or `ln(6)`, is transcendental. Now the cool part: algebraics are countable! To see this, you can encode any polynomial as an array of integer coefficients, starting with constant. So `x² - 2` becomes `[-2, 0, 1]`, and `x⁴ - 10x² + 1` becomes `[1, 0, -10, 0, 1]`. So "the set of all polynomials" is the same size as "the set of all arrays of integers". Using your shower approach, we can show the set of all arrays of integers are countable, meaning the set of all polynomials is countable. And then, because each polynomial has at most `len-1` roots, the set of all roots — aka the set of all algebraic numbers — is countable too. Incidentally, if you're a fan of gremlin languages, J has the `p.` operator, which automatically finds roots of polynomials. ``` p. _2 0 1 +-+----------------+ |1|1.41421 _1.41421| +-+----------------+ ```