eli5: How were square roots calculated in the first place, before calculators?

Question

eli5: How were square roots calculated in the first place, before calculators?

In: Mathematics

Answer 1

There isn’t really an ELI5, since you don’t learn this until Calc 2 at most schools, but a common way to calculate square roots, exponential functions, or log functions is with a Taylor series expansion. A Taylor series takes a function that’s difficult to calculate by hand and expresses it as the infinite sum of simpler functions. You keep adding together smaller and smaller numbers until they are close enough to the exact number for your practical use.

This article does a decent job showing you how: https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Series_and_Expansions/Taylor_Expansion_II

Quick edit: many modern calculators actually do something like this for finding square roots and logs. They can calculate 8 decimal places in less than a second.