Extracting square roots by hand

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.

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.

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.

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)

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.

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.

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.

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.

Been a long time, and I would need to refresh my memory, but I did do it in high school.

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.

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 7474 is easy if I already know 7575), 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).

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.