In mathematics, the regula falsi, method of false position, or false position method is 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 historical types 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
a x = b , {\displaystyle ax=b,}
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
f ( x ) = a x + c = 0 , {\displaystyle f(x)=ax+c=0,}
if it is known that
f ( x 1 ) = b 1 ; f ( x 2 ) = b 2 . {\displaystyle {\begin{aligned}f(x_{1})&=b_{1};\\f(x_{2})&=b_{2}.\end{aligned}}}
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,
x = b 1 x 2 − b 2 x 1 b 1 − b 2 , {\displaystyle x={\frac {b_{1}x_{2}-b_{2}x_{1}}{b_{1}-b_{2}}},}
would be memorized and carried out by rote. Indeed, the rule as given by Robert Recorde in his Ground of Artes (c. 1542) is:
For an affine linear function,
f ( x ) = a x + c , {\displaystyle f(x)=ax+c,}
double false position provides the exact solution, while for a nonlinear function f it provides an approximation that can be successively improved by iteration.
History The simple false position technique is found in cuneiform tablets from ancient Babylonian mathematics, and in papyri from ancient Egyptian mathematics. Double false position arose in late antiquity as a purely arithmetical algorithm. In the ancient Chinese mathematical text called The Nine Chapters on the Mathematical Art (九章算術), dated from 200 BC to AD 100, most of Chapter 7 was devoted to the algorithm. There, the procedure was justified by concrete arithmetical arguments, then applied creatively to a wide variety of story problems, including one involving what we would call secant lines on a conic section. A more typical example is this "joint purchase" problem involving an "excess and deficit" condition:
Now an item is purchased jointly; everyone contributes 8 [coins], the excess is 3; everyone contributes 7, the deficit is 4. Tell: The number of people, the item price, what is each? Answer: 7 people, item price 53.
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