Abstract
Mathematicians have developed many methods for solving polynomial equations, ranging from the basic formula giving the solutions of quadratic equations already known to ancient Greeks to very clever numerical algorithms developed in our contemporary computer age. In this short note I shall describe the history of one of the more interesting methods, Newton’s method, from its introduction in the seventeenth century up to the beautiful and unexpected results obtained in the twenty-first century, showing how a mathematical theory can be still well alive and kicking five centuries after it was born.
Talk given in the Congress “Matematica e Cultura 2012” on March 23, 2013
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Notes
- 1.
The use of the word “root” to denote a solution of an equation (see, e.g.,[1, p.163]) goes back to the Arabic mathematician al-Khwārizmī, that in 830 AD wrote Al-jabr, the first modern algebra book. He thought of the variable x as the hidden source of the equation, as the root is the hidden source of a plant; and finding the solution as akin to bringing to light this source, that is to extracting the root from the soil.
- 2.
In a sense, this means that to solve the general quadratic equation is enough to be able to solve the very specific quadratic equation \( {y}^2=m \).
- 3.
And most unpractical applications too; for the remaining cases usually knowing the exact value of the square root is irrelevant, it suffices to know that it exists (and when it does).
- 4.
Botanical metaphors abound in this subject…
- 5.
If the roots are real, the only real bad seed is the middle point of the segment delimited by the two roots. If the polynomial is real but the roots are complex (necessarily conjugated), the axis of the segment connecting the two roots is the real axis; in other words, all real points are bad seeds, consistently with the fact that this polynomial has no real roots.
- 6.
This seemingly self-evident statement actually is a deep theorem of plane geometry known as Jordan curve theorem; see, e.g., [8] for a proof.
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Abate, M. (2015). À LA RECHERCHE DES RACINES PERDUES (In Search of Lost Roots). In: Emmer, M. (eds) Imagine Math 3. Springer, Cham. https://doi.org/10.1007/978-3-319-01231-5_18
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