Fibonacci actually lists several different methods for constructing Egyptian fraction representations (, chapter II.7). He includes the greedy method as a last resort for situations when several simpler methods fail; see Egyptian fraction for a more detailed listing of these methods. As Salzer (1948) details, the greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, earliest and most notably by ; see for instance and . A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative dates back to .