In Mathematical, the method of Héron or Babylonian method is an effective method of extraction of square Racine. It bears the name of the mathematician Héron of Alexandria but certain former calculations seem to prove that the method is older.

Principle

To determine the square root of the number has, one chooses a number $x\_0$ rather near to √A, in general the whole Partie √A, then one builds a continuation defined by recurrence by

$x\_\; \{n+1\}\; =\; \backslash \; frac\; \{x\_n+\; \backslash \; frac\; \{has\}\; \{x\_n\}\}\; \{2\}$

The continuation thus obtained is a decreasing continuation starting from the second term, converging towards √A.

Convergence in is quadratic: the difference between each term and the limit √A evolve/move like the square of the preceding variation

$x\_\; \{n+1\}\; -\; \backslash \; sqrt\; \{has\}\; =\; (x\_\; \{N\}\; -\; \backslash \; sqrt\; \{has\})\; (x\_\; \{N\}\; -\; \backslash \; sqrt\; \{has\})\; \backslash \; frac\; \{1\}\; \{2\; x\_n\}$

i.e. the number of exact decimals doubles with each iteration.

If the first term of the continuation is a rational Integer or , all the successive terms will be rational numbers, which makes it possible to approach an irrational number such as √2 by a continuation of rational S.

The algorithm requires with each stage to make a division, which itself requires a all the more long succession of operations as the required precision is important (one supposes that one does not have calculating machine, without what the algorithm would be useless). Nevertheless, algorithm is robust, it supports well some approximations (and even some errors, the effect will be to delay obtaining the result but will not empéchera to obtain it), which makes it possible to be satisfied with divisions (not too) false, at least at the beginning.

Geometrical motivation

The current mathematical presentation does not allow the description of the geometrical principle. In the Greek mathematicians, to extract the square root of has is to find a square whose surface is A. By taking an arbitrary rectangle on side X and of the same surface, it is necessary that the other side has as a length A/X. But this rectangle is not square (in general). To make it right-angled , it is enough to take a rectangle of which the length is the arithmetic Mean on the two preceding sides is

$\backslash \; frac\; \{X+A/X\}\; \{2\}$

and whose surface remains A. By reiterating the process infinitely, one gradually transforms the rectangle into of the same square surface

Generalization of the method

A similar method exists to extract the nth root of a number (see Calculation algorithm of the nth root).

The method of Héron is a particular case of the Méthode of Newton. Indeed, in the method of Newton, it is a question of finding one zero of a function F by using the following recurrence:

$x\_\; \{n+1\}\; =\; x\_n\; -\; \backslash \; frac\; \{F\; (x\_n)\}\{f\text{'}\; (x\_n)\}$

By taking

$f\; (X)\; =\; x^2\; -\; has\; \backslash ,$

the recurrence becomes

$x\_\; \{n+1\}\; =\; x\_n\; -\; \backslash \; frac\; \{x\_n^2-A\}\; \{2x\_n\}\; =\; \backslash \; frac\; \{x\_n^2+A\}\; \{2x\_n\}\; =\; \backslash \; frac\; \{x\_n\; +\; \backslash \; frac\; \{has\}\; \{x\_n\}\}\; \{2\}$

See too

• square Root of two
• Calculation algorithm of the nth root
• Method of Newton
• Method of drop by drop the

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