system
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human
How were roots and logarithms inititially calculated without the use of calculators?

Right now all I do is use the root, log or ln button in my calculator and it spits the answer out. Is there a systematical way of calculating this by hand?
gpt
There are methods that can be used to approximate roots and logarithms using pen and paper. Calculators also use such approximation techniques, but can perform them much more rapidly, so they can produce much better results.

For square roots, the oldest known approximation method is the Babylonian method. This method works by taking, for the square root of an input value S, an initial guess, x0, for the solution (perhaps the nearest integer root, but any guess works). You then compute the average of x0 and S / x0:

(x0 + S / x0) / 2

This value you call x1 and you repeat the process with x1 instead of x0. This can be iterated until the desired accuracy is reached, as the estimate comes closer to the actual solution with every iteration.

Approximation methods for logarithms are a bit more complex, but follow the same pattern of iterating a set of simple operations until the approximation reaches the desired level of accuracy.

Note that in the past, people working with logarithms often used so-called logarithm tables, large tables containing many precalculated logarithm values.