# Extracting square roots by hand **URL:** \ **Category:** In My Humble Opinion\ **Created:** [August 11, 2017, 2:17pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332 "2017-08-11T14:17:15Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![DCnDC](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dcndc/32/2842_2.png) [@DCnDC](https://boards.straightdope.com/u/DCnDC)\ **Post date:** [August 11, 2017, 4:21pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/21 "2017-08-11T16:21:38Z")
I’m certain it was taught to us. It was a _lot_ more complicated than pushing the “square root” button on the calculator, so it didn’t stick. ---
**Author:** ![Channing\_Idaho\_Banks](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/channing_idaho_banks/32/250_2.png) [@Channing\_Idaho\_Banks](https://boards.straightdope.com/u/Channing_Idaho_Banks)\ **Post date:** [August 11, 2017, 6:08pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/22 "2017-08-11T18:08:03Z")
I thought this was going to be a gardening thread. ---
**Author:** ![EinsteinsHund](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/einsteinshund/32/8400_2.png) [@EinsteinsHund](https://boards.straightdope.com/u/EinsteinsHund)\ **Post date:** [August 11, 2017, 6:35pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/23 "2017-08-11T18:35:09Z")
I learned it, maybe in 8th or 9th grade and remember that it was a similar algorithm like long division, but I never really used it since calculators were already a common item by then, and so I forgot the details. ---
**Author:** ![That\_Don\_Guy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/that_don_guy/32/538_2.png) [@That\_Don\_Guy](https://boards.straightdope.com/u/That_Don_Guy)\ **Post date:** [August 11, 2017, 7:15pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/24 "2017-08-11T19:15:56Z")
I learned what turned out to be Newton-Raphson (for square roots, start with a guess, replace it with the average of the guess and the original number divided by the guess, and keep repeating) in elementary school, and discovered the “one digit at a time” method on my own. The latter is: 1. Divide the number into groups of two digits; if it is not an integer, form groups on both sides of the decimal point. 2. The first digit of the answer is the square root of the largest square \<= the first group; subtract the square, then append the two digits in the next group. 3. The next digit is the largest digit where that digit multiplied by (the number formed by doubling the digits found so far and appending that digit to it) \<= the number formed by the groups so far; repeat (3) as often as necessary. ---
**Author:** ![suranyi](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@suranyi](https://boards.straightdope.com/u/suranyi)\ **Post date:** [August 11, 2017, 7:30pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/25 "2017-08-11T19:30:05Z")
I learned it, but I never used it beyond those few weeks in class, and I certainly don’t remember how to do it now. ---
**Author:** ![What\_Exit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/what_exit/32/10652_2.png) [@What\_Exit](https://boards.straightdope.com/u/What_Exit)\ **Post date:** [August 11, 2017, 7:43pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/26 "2017-08-11T19:43:00Z")
I learned but have since forgotten it. Same with Logs. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [August 11, 2017, 8:13pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/27 "2017-08-11T20:13:04Z")
Going full circle, I didn’t learn it until college, when I took a “Theory of Math” course in preparation for learning to be a teacher. The professor was, I think, a fine mathematician, but a horrible teacher–he was the sort to teach with his back to the class, muttering while he wrote equations on the board. I’m pretty decent at math (nowhere near good enough to make a living in a math profession, but enough to impress other non-mathematicians), and I could follow what he was saying if I concentrated; plenty of other folks in the class were completely lost. I don’t remember how to find square roots. I do remember, while he scratched out instructions on the chalkboard, one girl turning around, catching her friend’s eye, and pantomiming shooting herself in the temple. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [August 11, 2017, 8:24pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/28 "2017-08-11T20:24:09Z")
> [@That\_Don\_Guy](#): > > I learned what turned out to be Newton-Raphson (for square roots, start with a guess, replace it with the average of the guess and the original number divided by the guess, and keep repeating) in elementary school, and discovered the “one digit at a time” method on my own. > > The latter is: > > 1. Divide the number into groups of two digits; if it is not an integer, form groups on both sides of the decimal point. > 2. The first digit of the answer is the square root of the largest square \<= the first group; subtract the square, then append the two digits in the next group. > 3. The next digit is the largest digit where that digit multiplied by (the number formed by doubling the digits found so far and appending that digit to it) \<= the number formed by the groups so far; repeat (3) as often as necessary. Yeah, that’s the method I learned (and provided the links for above). I can do it other ways, too, but that way works well if I have paper for it. ---
**Author:** ![don\_t\_ask](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@don\_t\_ask](https://boards.straightdope.com/u/don_t_ask)\ **Post date:** [August 11, 2017, 10:17pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/29 "2017-08-11T22:17:29Z")
I didn’t learn it at school. One evening, back in the 60s, my father, who was a surveyor, showed me how to do it. I can’t recall what prompted the lesson but I was really impressed. And although I don’t think I ever actually had cause to use it I still recall exactly how to do it and how the algorithm works. ---
**Author:** ![Dendarii\_Dame](https://avatars.discourse-cdn.com/v4/letter/d/bbce88/32.png) [@Dendarii\_Dame](https://boards.straightdope.com/u/Dendarii_Dame)\ **Post date:** [August 11, 2017, 11:52pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/30 "2017-08-11T23:52:35Z")
My sister taught me to do it by interpolating between the squares. Nobody else ever taught me at all. Including you, OP. And we’re married. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [August 11, 2017, 11:55pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/31 "2017-08-11T23:55:51Z")
> [@Dendarii\_Dame](#): > > My sister taught me to do it by interpolating between the squares. Nobody else ever taught me at all. > > Including you, OP. > > And we’re married. We’ll talk… ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [August 11, 2017, 11:56pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/32 "2017-08-11T23:56:25Z")
I didn’t learn it in school (except that I knew how to approximate them with a Taylor expansion once I learned calculus). My wife learned the method that looks a bit like long division which she taught me later. ---
**Author:** ![jtur88](https://avatars.discourse-cdn.com/v4/letter/j/e9c0ed/32.png) [@jtur88](https://boards.straightdope.com/u/jtur88)\ **Post date:** [August 12, 2017, 12:17am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/33 "2017-08-12T00:17:05Z")
Yes, and I still remember how. ---
**Author:** ![Johanna](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johanna/32/8318_2.png) [@Johanna](https://boards.straightdope.com/u/Johanna)\ **Post date:** [August 12, 2017, 12:36am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/34 "2017-08-12T00:36:39Z")
I answered yes, but only because I taught myself to do it. At least school gave me the idea of it. ---
**Author:** ![HelenGawn](https://avatars.discourse-cdn.com/v4/letter/h/a88e4f/32.png) [@HelenGawn](https://boards.straightdope.com/u/HelenGawn)\ **Post date:** [August 12, 2017, 3:32am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/35 "2017-08-12T03:32:56Z")
Aside: My dad and I were on a road trip from California to Texas, which is of course 2000 miles of very boring scenery. We spent about 500 of those miles trying to work out how to do this. We finally got it, and the next morning did not remember how. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 12, 2017, 4:08am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/36 "2017-08-12T04:08:14Z")
Yes, but only for fun. I rediscovered Newton’s method, though I didn’t know it at the time: x’ = (x + c/x)/2 The derivation: f(x) = x[sup]2[/sup] - c f’(x) = 2x x’ = x - f(x)/f’(x) x’ = x - (x[sup]2[/sup] - c) / 2x x’ = 2x[sup]2[/sup]/2x - (x[sup]2[/sup] - c) / 2x x’ = (x[sup]2[/sup] + c) / 2x x’ = (x + c/x) / 2 ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [August 12, 2017, 8:57am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/37 "2017-08-12T08:57:36Z")
Yes. It was the last step in the math section at my Montessori school, which I reached near the end fourth grade. I don’t think they necessarily expect you to get that far in curriculum, as you move at your own pace. It’s just there to give kids more of a challenge without getting into higher math that would require an additional system created to teach the Montessori way. I do find it weird that most people were taught Newton’s method, though. It’s so tedious. I was taught [this algorithm](https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Decimal_.28base_10.29). The examples show how easy it is, without using variables as the description does. It’s just slightly more involved than long division. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [August 12, 2017, 12:41pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/38 "2017-08-12T12:41:48Z")
They tried to teach me the bastardized “long division” form in grade school. I love Math and it was clear that this was a really stupid method. Learned Newton’s Method in late high school. Okay, now _that_ I can use. The convergence rate is great so it is much more efficient than one might think. Throw in a shortcut or two and it just works. But I was also taught how to use a slide rule in school. YMMV. ---
**Author:** ![Tim\_R.Mortiss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim_r.mortiss/32/142_2.png) [@Tim\_R.Mortiss](https://boards.straightdope.com/u/Tim_R.Mortiss)\ **Post date:** [August 12, 2017, 9:36pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/39 "2017-08-12T21:36:28Z")
Yes, I learned it, but not in school. My dad used to drill us kids in math problems at the dinner table every night. I was a sophomore in college before I learned anything in a math class I hadn’t already learned over spaghetti and meatballs at home. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [August 12, 2017, 9:47pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/40 "2017-08-12T21:47:02Z")
I learned the [Newton–Raphson method](https://en.wikipedia.org/wiki/Newton%27s_method) primarily as an example algorithm to implement in software. It hits a sweet spot between being too trivial to demonstrate the idea of iterative refinement and so complex the details of the algorithm overwhelm the main point of the exercise. [Previous page](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332.md?page=1) [Next page](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332.md?page=3)