# Extracting square roots by hand

The discussion of “New Math” in GQ (http://boards.straightdope.com/sdmb/showthread.php?t=832870) makes me wonder how many folks know who to extract square roots by hand - I learned one day in 7th or 8th grade when our math teacher had a few extra minutes at the end of class in the late 1970s. I still practice it when meetings get boring (when I want an extra challenge, I do the calculation in a base other than 10).

Let’s see if I know how to add a poll (looks like I can!)

They taught us what I presume was the standard paper-and-pencil algorithm in grade school. My brother still remembers how to do it, but I only learned it well enough to pass the test and then promptly forgot it. Years later I developed my own method (which is probably not original but nobody taught it to me) and can now approximate square roots in my head to a reasonable degree of accuracy.

Didn’t learn it at school. Someone showed me the technique later but I don’t remember the details. I do recall that the best starting point is a power of two.

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

My father taught me how to do it (he was a mechanical engineer).

Every now and then I do it, just to remind myself how it works.

Here’s how, in case you’re wondering:

They taught us the divide-and-average method in high school, but today I’d use the binomial series method (which is more general than just for square roots).

Learned it in high school in the early 80s. Couldn’t do it now but probably could if someone reminded me how to.

The mention of other bases reminds me of the time I wrote an assembly language program to extract square roots in binary. You can, of course, use Newton’s method, but on the original 8088 chip division was very slow. In binary it is extremely easy to extract square roots since all you have to do is see if the next bit is 0 or 1. So you try 1 and half the time (on average) it will be too large so the next bit is 0. Requires only a doubling (shift left by 1) an addition and a comparison. It beat the pants off Newton’s method.

Learned it, never used it, forgot it.

I know at least two methods (and two variants of one of them) well enough that I could do them by hand, without having to look anything up to refresh my memory, or stopping to figure them out. I know of at least two more methods well enough that I could describe how they work in general terms, but would need refreshing or tinkering before I could actually use them. I find puzzling the people who say they remember “the” method, since even of the methods I know, some are better in some contexts than others.

For anyone keeping score at home, the two methods I know well are to divide your original number by your trial number, then find a number between the that and your trial number to use as your next iteration of your trial number, and to do a binary search, squaring each trial number to see whether it’s too low or too high. And the two other methods I know of are Newton’s method, and the power series expansion.

I learned the book method in elementary or middle school, but it’s easier for me to just interpolate between squares.

Example: what’s the square root of 7?

26^2=676, and 27^2=729. So 2.6^2=6.76, and 2.7^2=7.29. So 7 ~ 2.65^2. (Very rough interpolation, I know. But I’m doing this in my head.)

Which is off by only .004.

Yeah, sometimes I mess around with that method too.

I was supposed to learn it, but that was one of my worst math teachers. That year (6th grade) was followed by several years in which practically the only non-integer square root we needed was that of 2 (it crops up all the time in trig). By the time I actually had to use them we had calculators.

I voted yes, but the method I learned was the ol’ trial and error (interpolation) method. Looking at some of the links and posts here, it doesn’t seem like that’s what people are talking about. The Dr. Math link is some method that looks like long division ,but I never learned this. For me, it was

What’s the square root of 95? Well, 9 squared is 81, 10 squared is 100, and 95 is closer to 100, so lets guess 9.7. 9.7^2 is 94.09, so we’re close. Let’s see 9.8. That’s 96.04. OK, so it’s between 9.7 and 9.8. Let’s try 9.75. That’s 95.0625. Let’s do 9.74. That’s 94.8676. So it’s 9.74 something. And you go on and on until you reach your desired precision. Slow and tedious, but it works. It’s also really obvious and doesn’t require memorization of any particular technique.

The method I learned in school was (although I didn’t know it at the time) solving for (A+B)² = A²+2AB+B², with A the digits you’ve solved and B your best guess for the next digit, which gets added to A in the next iteration.

I always wanted to (not too badly, though, as I know I could have learned it myself from the library or asked other adults but I never did).

Now that I think back, it seems like I didn’t learn anything new in math class between long division in the 3rd grade (~1990) and algebra in the 8th (~1995). They could have easily fit square roots into that stretch of time. Okay, I think order of operations was in there, and probably exponents too, but mostly it was just 5 years of practicing arithmetic so the slower folks could catch up before high school.

What do I pick if I don’t know if I learned it or not? I might have, and just not remembered.

I was taught how to do it, but now I couldn’t tell you how if my life depended on it.

Oddly enough, we were also taught how to do it on calculators. By the same public school system that continually reminded us that we needed to learn to do math on paper because we wouldn’t be carrying around a calculator with us all the time (how wrong they were!).

On the subject of Junior High math formulae, I need help figuring out how to compute a sum based on percentages.

For example: 62% of the students, or 446 of them, at Cecil Adams Junior High School are food-insecure. With these known variables, how do we determine the total enrollment at CAJHS?

0.62X=446

Solve for X (which is the total population).