Can the “Babylon Method” be used Exponents with Decimal Arguments

approximation

I was reading about the methods that the Ancient Babylonian Civilization used for approximately calculating square roots:

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  • Could this method be used for approximately calculating any "root" – for instance, could we use this method to approximately calculate the cube of some number "S"?

  • Could this method be used for approximately calculating an exponent with a "decimal argument"? For instance, the square root of "S" can be written as S^0.5 – Could we use this method for approximately calculating S^0.3?

Although there are now more modern ways to approximate these calculations, I am interested in learning about the limitations of these ancient methods that existed far before calculators and computers!

Thanks!

Source:

Best Answer

Yes. The Baylonian method can indeed be generalized easily to $$x_n=\frac{1}{k}\left((k-1)x_{n-1}+\frac{S}{x_{n-1}^{k-1}}\right)$$ to compute any $k-th$ root (although the convergence is very slow). Once computed the $k-th$ root you can then compute the $l-th$ power if you want to calculate $S^{l/k}$

Remark: It's possible to prove that this recursion converges, yet it is quite a mess.

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