# 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:** 12\ **Page:** 3
**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 13, 2017, 12:23am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/41 "2017-08-13T00:23:20Z")
> [@ftg](#): > > The convergence rate is great so it is much more efficient than one might think. Yeah–it _doubles_ the number of correct digits on each iteration. If you have a lot of digits to compute, even if you’re pretty good with the long-root method you’re better off switching to Newton-Raphson past a certain number of digits. ---
**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 13, 2017, 12:38am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/42 "2017-08-13T00:38:49Z")
> [@Dr.Strangelove](#): > > 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 Yeah, I learned that method from an article in Popular Science I think. It works very well. Somewhere I have an example I worked out that shows even if you truncate your calculations. ---
**Author:** ![kopek](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kopek/32/337_2.png) [@kopek](https://boards.straightdope.com/u/kopek)\ **Post date:** [August 13, 2017, 1:02am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/43 "2017-08-13T01:02:18Z")
> [@What\_Exit](#): > > I learned but have since forgotten it. Same with Logs. Same here. Not having used it at all I am surprised what advanced math I’ve retained and what I’ve lost. Working in various factors and different bases; yes. But a lot of my trig and calc are gone. ---
**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 14, 2017, 7:05am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/44 "2017-08-14T07:05:19Z")
> [@Andy\_L](#): > > Yeah, I learned that method from an article in Popular Science I think. It works very well. Somewhere I have an example I worked out that shows even if you truncate your calculations. Another nifty thing is that it’s easy to do with fractions. Take sqrt(2), for instance. Start with 1: (1/1 + 2/1)/2 = 3/2 (3/2 + 4/3)/2 = 17/12 (17/12 + 24/17)/2 = 577/408 (already good to 6 decimal digits) ---
**Author:** ![Novelty\_Bobble](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)\ **Post date:** [August 14, 2017, 10:05am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/45 "2017-08-14T10:05:51Z")
> [@Andy\_L](#): > > I should have provided a link to the method I learned - this is very close to that [http://mathforum.org/library/drmath/view/52610.html](http://mathforum.org/library/drmath/view/52610.html) Never did maths to any great level and never learned a manual method for doing it and I always find explanations for any mathematics to be confusing. In the example above, I follow the method up to the point where the “20+?” in the left hand column turns into “200+?” for the next step and then “2120+?” etc. Where do those numbers come from?, the instructions don’t seem to mention how they are constructed. The 20 becomes 200 without any explanation as to why edit - scratch that, I just realised that when it talks about 20 times the “number above the radical” it means the full number, not the single digit you just entered. Makes sense now…carry on, I’m going to play around with this. ---
**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 14, 2017, 5:09pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/46 "2017-08-14T17:09:17Z")
I don’t see why Newton’s method is somehow easier or less tedious. I just randomly picked sqrt(5421). Just following Newton’s method, I’m already having to square a 5 digit number in cycle 2:10321. From there I have to subtract 5421, which requires me to multiply that by 140 to get an exact answer. Then I have to divide that by 10321/70. Instead, I could have just seen how many times 140 goes into 521. then multiplied 143 \* 3, got 429, subtracted that from 521, then done the same process with 9200 and 1460. I get why it’s a great method for computers, since you get close really quickly, but, for humans, it gets unwieldy very quickly. ---
**Author:** ![UncleRojelio](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/unclerojelio/32/3160_2.png) [@UncleRojelio](https://boards.straightdope.com/u/UncleRojelio)\ **Post date:** [August 14, 2017, 6:42pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/47 "2017-08-14T18:42:36Z")
I learnt Newton’s Method in high school. Ever since then it’s been one of the standard algorithms I’ve implemented when playing with a new computer language. Right after the Fibonacci sequence and right before Shortest Path. ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [August 14, 2017, 6:55pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/48 "2017-08-14T18:55:04Z")
I learned Newton’s Method for deriving the roots of many different types of equations by hand. I don’t recall that I learned anything special for square roots. ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [August 14, 2017, 7:07pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/49 "2017-08-14T19:07:03Z")
Been a long time, and I would need to refresh my memory, but I did do it in high school. ---
**Author:** ![Civil\_Guy](https://avatars.discourse-cdn.com/v4/letter/c/d78d45/32.png) [@Civil\_Guy](https://boards.straightdope.com/u/Civil_Guy)\ **Post date:** [August 14, 2017, 10:16pm UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/50 "2017-08-14T22:16:20Z")
Learned how, long ago - the sort-of long division method. Forgot it soon afterwards. Would now use some kind of interpolation method, if I had to. Probably won’t see the day. ---
**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 15, 2017, 6:06am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/51 "2017-08-15T06:06:22Z")
> [@BigT](#): > > I get why it’s a great method for computers, since you get close really quickly, but, for humans, it gets unwieldy very quickly. It really all depends on how you use it and what you want to get out of it. To use your example, I first need a guess for the initial number. It’s easy to see that it’s somewhere between 70 and 80, so I’ll guess 75. Average 75 and 5421/75 to get 5523/75. Now it’s a matter of normal long division: that comes to 73.64. I can see already that my original guess had maybe 1.5 digits of precision, and so my new answer is at least 3 digits. I can leave it at 73.6 if that’s good enough. If not, I can do another iteration, but this time starting at 74, _not_ 73.64. The reason is that it’s easier to do (especially considering that 74_74 is easy if I already know 75_75), and I know I’ll still get at least 4 digits out of it. Since the original 5421 was just 4 digits, there’s no need for more assuming this came from some measurement. The answer comes to 73.628, which is close enough to get \<0.1 away from the true square. The other advantage is that I can always rederive it, whereas I always forget the long method. I can rederive the fundamental Newton-Raphson technique, too (I always forget whether it’s f(x)/f’(x) or f’(x)/f(x), and also whether it’s added or subtracted, but it’s obvious when you draw a picture). ---
**Author:** ![GuanoLad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/guanolad/32/42_2.png) [@GuanoLad](https://boards.straightdope.com/u/GuanoLad)\ **Post date:** [August 15, 2017, 7:11am UTC](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332/52 "2017-08-15T07:11:35Z")
Probably was taught it. Cannot remember it if I did. I was pretty decent at maths through school, until it reached a complexity I found hard to follow, somewhere after quadratic equations, and I have never found the need to pursue it since. I remember most of the basics fairly well, I am not as quick at it as I used to be, and some of it is still a bit hard for me to grasp, but the good news is in real life calculators are allowed. [Previous page](https://boards.straightdope.com/t/extracting-square-roots-by-hand/793332.md?page=2)