Let script B sign (x) denote the Banach algebra of all bounded linear operators on a Banach space x. We show that script B sign(x) is finite if and only if no proper, complemented subspace of x is isomorphic to x, and we show that script B sign(x) is properly infinite if and only if x contains a complemented subspace isomorphic to x ⊕ x. We apply these characterizations to find Banach spaces x₁, x₂ and x₃ such that script B sign(x₁) is finite, script B sign(x₂) is infinite, but not properly infinite, and script B sign(x₃) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces η₁ and η₂ such that script B sign(η₁ and script B sign(η₂) are infinite without being properly infinite, script B sign(η₁) has a continued bisection of the identity, and script B sign(η₂) has no continued bisection of the identity. Finally, we exhibit a unital C*-algebra which is finite and has a continued bisection of the identity.