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Square Root Algorithm A simple approximation method is used by calculators to find square roots. The basic steps to finding y ϭ ͙x are first to guess the value of y, calculate y2, and then find r ϭ x Ϫ y2. Then if the magnitude of r is small enough, return y as the answer. Otherwise, increase or decrease y (depending on whether r is positive or negative) and repeat the process. The number of intermediate calculations required can be reduced by avoiding finding y2 and x Ϫ y2 for each value of y.

For example, some advanced scientific calculators allow the user to input numbers using up to twelve decimal places, and return results in this same format, but fifteen-digit numbers are actually used during calculation. Reverse Polish Notation and Algebraic Logic System The Polish logician Jan Lukasiewicz demonstrated a way of writing mathematical expressions unambiguously without using parentheses in 1951. For example, given the expression (2 ϩ 3) ϫ (7 Ϫ 1), each operator can be written before the corresponding operands: ϫ ϩ 2 3 Ϫ 7 1.