In mathematics, the regula falsi, method of false position, or false position method refers to a family of algorithms used to solve linear equations and smooth nonlinear equations for a single unknown value. In its oldest known examples found in cuneiform and hieroglyphic writings, the method replaces simple trial and error with proportional correction of an initial guess. In modern usage, the method relies on linear interpolation based on two different guesses.

Two basic types of false position method can be distinguished historically, simple false position and double false position.

Simple false position is aimed at solving problems involving direct proportion and can be thought of as an early algorithm for division. Such problems can be written algebraically in the form: determine x such that

if a and b are known. The method begins by using a test input value x′, and finding the corresponding output value b′ by multiplication: ax′ = b′. The correct answer is then found by proportional adjustment, x = ⁠b/ b′⁠ x′.

As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation x + ⁠x/4⁠ = 15. This is solved by false position. First, guess that x = 4 to obtain, on the left, 4 + ⁠4/4⁠ = 5. This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply x (currently set to 4) by 3 and substitute again to get 12 + ⁠12/4⁠ = 15, verifying that the solution is x = 12.

Double false position is aimed at solving more difficult problems that can be written algebraically in the form: determine x such that

if it is known that

Double false position is mathematically equivalent to linear interpolation. By using a pair of test inputs and the corresponding pair of outputs, the result of this algorithm given by,