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

You used a format that looks like long division but isn’t, to get the answer one digit at a time. Using (a+b)^2 = a^2 + 2ab + b^2, but backwards, guessing each new b as you went.

1 . 4 1 4

__________________________________________________

1 | 2 . 00 00 00

1 . 00 = 1*1

. ——

1 . 00

4

. 96 = 2(4 *) 4

——

. 04 00

1

. 02 81 = 28(1 *)1

——–

. 01 19 00

4

. 01 12 96 = 282(4 *)4

———–

. 00 06 04 00

, etc. At each layer you take what you’ve got, multiply by 2, stick on your guess for the next digit, then multiply by that guess, shifted over to the right 2 more places; find the guess that gets you closest to the current remainder without going over. Subtract, put that guess digit on the end, repeat.

–Dave, cube roots can be done this way too, but not fourth roots or higher