The Man of Numbers: Fibonacci's Arithmetic Revolution by Devlin Keith

Author:Devlin, Keith [Devlin, Keith]
Language: eng
Format: mobi, epub
Publisher: Bloomsbury Publishing Plc
Published: 2011-07-04T16:00:00+00:00

Find a number such that if it be raised to the third power, and the result added to twice the same number raised to the second power, and if that result be then increased by ten times the number, the answer is twenty.

Leonardo solved the equation by approximation. There is no record of the exact method he used, but there were a number of techniques available at the time. It is likely his argument went something like this: The first step is to observe that the unknown number must be between 1 and 2. For if it were 1, the answer to the computation would be less than 20, and if it were 2 the answer would exceed 20. So a reasonable initial guess is x = 1.5. The idea now is to iteratively refine this guess, gradually homing in on the correct answer. The key step is deciding how to improve each approximation.

If x solves the equation x3 + 2x2 + 10x = 20, then x(x2 + 2x + 10) = 20, and so

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