eli5: How were square roots calculated in the first place, before calculators?
In: Mathematics
First, think about what a square root actually is: the length of a side of a square that has the given area.
Let’s say you want to find the square root of 10005. Draw a square to help visualize this, and assume that that square’s area is 10005. You know that the square root of 10000 is 100, so let’s draw that square inside the first one with the upper left corners of the two squares on top of each other. You now have a square that you know has an area of 10000, and the square surrounding it shares the same area, plus a strip at the bottom and a strip at the right. Those two strips are the same size: 100 (the length of the sides of the inner square) times some unknown additional width, which we’ll call x. (There’s also a small square at the bottom right that’s x by x, but let’s ignore it for the moment.) That extra area must be equal to the difference of the areas of the two squares: 10005-10000, or 5. So let’s assume that the two strips together equal 5. Since we have two, let’s say each is 2.5. Now we have an area, 2.5, and the length of one of the sides, 100, so it’s easy to find the length of the other side: 2.5 ÷ 100, or 0.025. Well, the length of that additional unknown side is how much bigger the square of area 10005 is than the square of area 10000. So that means the square’s sides are 100.025, and that’s what the square root would be. Except we ignored that x by x sized square at the bottom right, so this is just an estimate, but it’s very close.
If it’s not close enough, though, you can repeat this process with the new 100.025-sided square we just made. It’s actually a little bigger than 10005, so you’ll have to deal with subtracting strips along the edge instead of adding them, but that shouldn’t be too big a deal. Then you can continue repeating the process as many times as you like. (It’s important to note that square roots that are not whole numbers are definitively irrational numbers, so you will never find the exact value. Estimates are the best you can hope for.)
[Matt Parker has a video about this](https://youtu.be/Bwt5EZEb1Ns), and I stole his example values. His drawings may help.
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