At the root of the square root
Essential in geometry as in analysis, square roots nevertheless took many centuries to establish themselves as full-fledged numbers in mathematics. The difficulty in evaluating them did not help their acceptance! In fact, one often has to calculate approximate values. For this, what better than a good algorithm? You undoubtedly know Heron's method or the digit-by-digit extraction method, but this feature will finally help you understand "why and how it works".
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Extracting square roots: the main algorithms
Calculating square roots "with a quill," as people said in the 18th century, is a challenging art with many facets. Let's explore some famous methods and see why they work.
Lightning calculators
How do lightning calculators use root-extraction algorithms so quickly?
Newton's method
Do you know how your calculator—or your smartphone's calculator—computes the square root of a number a > 0?
Roots and continued fractions
There are other ways to represent real numbers besides our decimal notation! One of the best known is to write a number as a continued fraction. The resulting representations, even when truncated, very quickly provide "good" approximations of the original number.
Square matrices and transformations of the plane
The notion of a square root can be extended to very general sets. One may then obtain more than two square roots—even infinitely many! Matrices provide one example. Orientation-preserving similarities, viewed through the lens of complex numbers, lead us back to less startling results.
Numbers or not?
Irrational numbers had a troubled beginning. Introducing roots of integers that are not perfect powers raised many questions about the nature and essence of numbers. Is a radical incommensurable with unity and its subdivisions a number?