False Position Method - Arithmetic and Algebra

Arithmetic and Algebra

In problems involving arithmetic or algebra, the false position method or regula falsi is used to refer to basic trial and error methods of solving problems by substituting test values for the unknown quantities. This is sometimes also referred to as "guess and check". Versions of this method predate the advent of algebra and the use of equations.

For simple false position, the method of solving what we would now write as ax = b 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 = x′ · b ÷ b′. This technique is found in cuneiform tablets from ancient Babylonian mathematics, and possibly in papyri from ancient Egyptian mathematics.

Likewise, double false position arose in late antiquity as a purely arithmetical algorithm. It was used mostly to solve what are now called affine linear problems by using a pair of test inputs and the corresponding pair of outputs. This algorithm would be memorized and carried out by rote. The oldest surviving document demonstrating knowledge and proficiency in the technique is the Indian mathematical text Vaishali Ganit (c. 3rd century BC). 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 quadratic polynomial. A more typical example is this "joint purchase" problem:

Now an item is purchased jointly; everyone contributes 8, 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.

Between the 9th and 10th centuries, the Egyptian Muslim mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭāʾayn). The oldest surviving writing on double false position from the Middle East is that of Qusta ibn Luqa (10th century), a Christian Arab mathematician from Baalbek, Lebanon. He justified the technique by a formal, Euclidean-style geometric proof. Within the tradition of medieval Muslim mathematics, double false position was known as hisāb al-khaṭāʾayn ("reckoning by two errors"). It was used for centuries, especially in the Maghreb, to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance), as well as purely recreational problems. The algorithm was often memorized with the aid of mnemonics, such as a verse attributed to Ibn al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna, all three being mathematicians of Moroccan origin.

Leonardo of Pisa (Fibonacci) devoted Chapter 13 of his book Liber Abaci (AD 1202) to explaining and demonstrating the uses of double false position, terming the method regulis elchatayn after the al-khaṭāʾayn method that he had learned from Arab sources.