Computing Square Roots by Hand
After writing a post about Multiplication, I was going to go off and write a post about the method I’d learned in middle school for computing a square root by hand. It looked a lot like long division, except the digits were considered in pairs, and to be honest, it’s been so long since I used it that I forgot all the details until I looked it up. And that’s when I realized I didn’t have to write it up at all, since Maria of the Homeschool Math Blog had already written it up wonderfully on her main site: both how to do it and why it works (which I was never taught when I originally learned the algorithm).
The how-to post is followed by a great discussion of the relative merits of the long-division-like algorithm versus using the Babylonian method (of iterative approximations) which a commenter presents. You should definitely go read it!
But does computing square roots by hand have any relevance today?
When I was in middle school, you couldn’t buy a calculator at Walmart for $1 that would compute square roots for you. It was useful to actually be able to compute a square root by hand on occasion (though the main instance I remember using it for was to compute the square root of my student ID number, to appease the seniors who would ask for it during freshman initiation at my high school).
But today when you can buy a calculator for $1 that does the four basic operations plus square roots, is there any point in knowing how to do it by hand? Well, most (but by no means all) people still think it’s useful to learn to add, subtract, multiply and divide by hand, even when calculators are readily available that can do all this for you. But square roots?
Maria argues that:
…studying and practicing these algorithms, just like the long division algorithm, will give your child exercise in simple mental math calculations. It is important to let children get lots of mental addition, subtraction, multiplication, and division exercise. Why? Because that helps them to get familiar with numbers and to develop a ‘number sense’.
Now I’m all for number sense. It seems to be sorely lacking in many (most) students today, at all levels. I’m happy to blame most of that on the overuse of calculators in the classroom. Others feel that it is due to the use of “Reform” math curricula such as Everyday Math, that aim to supplement competence with comprehension, but in many cases (particularly when poorly taught) seem to replace competence with, well, not much of anything.
But even if we accept that number sense is important, and critically lacking in so many students, is there something to be gained by teaching an obscure square root formula, in particular? I agree that carrying out the algorithm provides practice in basic operations. But would more practice on those operations be just as good?
Then again, does it do any harm to practice the basic operations in a way that puts another tool in a student’s toolbox?
I find that I don’t have strong feelings on this. The algorithm is highly forgettable, in my opinion, so I’m not convinced that learning it really adds a tool to a student’s toolbox. And why it works is beyond the comprehension of most kids at the upper-elementary and middle school levels, where it seems most likely to be taught. But at an age (maybe 5th or 6th grade) when many students find square roots interesting (dare I say “sexy”) it may be a good motivator for practice that doesn’t seem so mundane.
What do you think? And do you think that the successive approximations method, which is much easier to understand, is a better choice for students, despite being arguably messier to carry out by hand?
Ramblings of a Math Mom 
It seems to me that the square-root-by-hand method isn’t particularly interesting in and of itself. Yeah, I guess it’s handy if you need a square root and all you have is a soroban. But really, it’s fairly useless. Newton-Raphson iteration is faster (given a calculator) and more practical.
But I can’t help thinking that it might be a good example of how you might derive an algorithm. Maybe that’s not so useful, though.
I frequently find I need a square root (and for that matter, exp(x) and ln(x) ) at times when I don’t have a calculator handy.
For square roots, Newton’s method is very handy – take a guess (which is usually pretty easy), and then average your guess with n/guess to get your next guess.
I’ve already discusses the concept of a square root with my kids, but apart from finding it by trial and error (i.e. squaring a guess and then guessing bigger or smaller), I haven’t discussed it further. As soon as they get to the point of needing to find square roots of things, I will show them the “average your guess with n/guess” algorithm.
[Later, I’ll point out why that’s always a bit too big (because arithmetic mean >= geometric mean, which is easily illustrated with a few examples), so if the two numbers you’re averaging aren’t close together, err on the downward side.]
For example, finding square root of 2000. If your first guess is 100, then your next guess will average 100 with 20. But they’re very differen’t, so “60” is going to be substantially too big.
It’s easier to generate a good guess if you start by knocking off an even number of digits until there’s one or two digits left (then you add half the digits taken off back on to the square root), as long as you know the squares of 1-10.
I think it’s essential to be able to calculate a rough guess at almost anything you’d do on a calculator, otherwise how do you know you didn’t make a mistake pressing the buttons?
Knowing how to do stuff like square roots without a calculator is a bit like knowing a bit of spelling in spite of having word processors available – if you don’t know how to spell, how do you know the word the spellchecker gives you is actually the word you wanted? (And what about when you have to write something by hand?)
Thanks Pseudonym and Effrique for your comments.
Effrique, what is it that you do where the need for square roots, natural logs, etc. comes up “frequently” when you don’t have a calculator handy?
Are there good ways of computing ln(x) and exp(x) without a calculator (or slide rule) 😉 ?
I agree with you that Newton’s method is more intuitive. But is it easy for kids to carry out?
My youngest has just become fascinated with square roots. So far he has just gotten to the point of finding integral square roots for numbers where the square root is less than 20 or so. But if given a number without an integral square root, he will tell you which two numbers it must be between. But he doesn’t know long division yet, so finding n/guess would be a challenge for him unless that works out to be a whole number. He can probably do the averaging without long division. I think he could do the “long division style” method more easily at this point (but maybe he’d be better of just learning real long division and sticking with Newton’s method). (But then again, if I teach him the square root by hand method, he won’t be as bored when they learn long division in school…)
Does anyone know the story of how that long-division-style algorithm was “invented”? Wikipedia points out that two advantage of that algorithm are that “[e]very digit of the root found is known to be correct, i.e. it will not have to be changed later” and “[i]f the square root has an expansion which terminates, the algorithm will terminate after the last digit is found.”
I need to learn more about Napier’s Bones. I once saw a video of how they could be used to facilitate carrying out the “long division style” calculation of square roots, and was fascinated, but haven’t taken the time to learn more yet. I think they might make a fun exploration for my middle schoolers.
Hi
If your youngest is fascinated with square roots at the moment then why not just try to teach him the algorithm and see how he gets on? He will either love being able to do it or will find it too much like hard work and move on – either way you won’t have lost too much time and it if it feeds his enthusiasm then its a good thing.
As for Newton’s method – If he struggles with long division then you might choose to let him use a calculator to do the divisions. If he argues that he could just press the square root button then maybe ask “you could – but have you ever wondered how that button works?” (I think that most calculators use a CORDIC algorithm but this might not be the time to mention that) . To me this is a nice way of introducing the idea of algorithms which might lead to an interest in programming.
When I was a child I remember asking my maths teacher how calculators worked out square roots and he told me “Oh they store them in memory.” Even at the age of 10 I was dubious about this and his answer still winds me up today
Mike
Thanks for your thoughts, Mike!
To me this is a nice way of introducing the idea of algorithms which might lead to an interest in programming.
The “interest in programming” is almost inevitable in this house. 😉 I work as a computer scientist, and my husband does mainly technical writing, but also some programming on the side. His older brothers are already writing games in various “toy” programming languages such as KPL, Stagecast Creator, Game Maker, etc. and are also programming Lego Mindstorms.
But you’re right, that “letting” him do the divisions on a calculator would be an interesting idea. (He doesn’t “struggle” with long division as such — it just hasn’t been introduced yet.)
I must be older than you, because I didn’t have a calculator that did square roots at 10yo. I used to work in my dad’s camera store, and he also sold calculators. One of the criteria I’d ask people about was whether or not they needed a square root button on their calculator, if they weren’t buying a scientific. I got a scientific when I started high school (was abou $100 for a Sharp that did way less than the $20 models today!)
I do have a friend who found out how calculators do addition when I was about 11 or 12yo. That was pretty cool. Like you, I wouldn’t have believed your teacher’s answer about storing it in memory. 🙂
I have just hit 30 – which depresses me quite a bit. No longer a young man but anyway….
My father bought me my first scientific when I was 10 (I think) at a time when they were almost unheard of at my school. To this day I believe that that calculator was a major reason for me getting into mathematics (I ended up doing a PhD in theoretical physics).
In a nutshell I wanted to know what all the buttons did and the only person who could tell me was my maths teacher. He was a great guy and missed countless lunch hours explaining things like trigonometry, hexadecimal, logarithms etc etc just so some geeky kid could find out what his calculator could do and why.
I owe my livelihood to that guy and will never forget him.
Yes, Mike, you are quite a bit younger than I.
It’s interesting that back when we were young, a cool scientific calculator could inspire a student to learn more math. Now all they seem to do is substitute for actually learning any math. 😦
I have gotten each of my 2 older boys a scientific calculator for his 11th birthday, as he started middle school. I don’t think the calculator inspired either of them. But my current 11yo recently had a practical problem come up for a project he is building, where he knew the 3 side lengths of a triangle, and one angle, and needed to know the other two angles. What stronger motivation could be possible for learning a little about trig! I helped him solve the problem using his calculator, but now he is quite interested in learning more about this function called “sine”, and the idea of inverse functions.
It is interesting what ends up motivating each of us.
Hi mathmom.
Quoting from Maria: “Because that helps them to get familiar with numbers and to develop a ‘number sense’.“.
And coincidentally, Pseudonym mentions the soroban and this is the subject of my Friday Math movie this week: The Amazing Abacus.
I think it is worth dropping calculators every now and then (maybe often?) so that students get a better “number sense”. It certainly will help them when it comes to algebra.
As for “It is interesting what ends up motivating each of us.”… It’s always what we are passionate about. In your 11 yo’s case, it was a practical and real problem – not some textbook question.
Zac, I agree that practical problems are great motivators when they arise naturally. I think that a lot of teachers and text books try to include “real and practical” problems but if it doesn’t happen to be a problem that a student is currently interested in, it is nowhere near as motivating.
I just saw your Abacus video — it is truly incredible (especially the kids who can use an “imaginary” abacus with such speed and accuracy!)
I also showed my 11up your Trig Graphs Movie. He wasn’t interested in all the details at this point, but he thought it was cool how the circle generated the graph.
Oh, on approximations to exp and ln:
The usual approach here is to find a value that’s close to what you want, then use a few terms of the Taylor expansion.
ln(10), for example, is close to ln(e^2), so set:
dx = (10 – e^2) / e^2
Then:
ln(10) ~= ln (e^2) + dx – dx^2/2 ~= 2.291
The correct value is approximately 2.303.
You can also use the arithmetic-geometric mean approach to get a better initial estimate. Essentially, you use the property:
ln(sqrt(ab)) = (ln a + ln b)/2
So if you know two logarithms that bracket the desired one, you can take the arithmetic and geometric means to get a better estimate. Using our example, 10 is between exp 2 and exp 3. So:
ln(sqrt(e^(2+3))) = (ln (e^2) + ln (e^3))/2
or:
ln(approx 12.182) = 2.5
Plugging this into the Taylor expansion above gives 2.305 as the estimate.
You can also use the arithmetic-geometric mean method to binary search. Incidentally, binary arithmetic was invented to compute logarithms using this process. This might be easier to visualise computing the common logarithm rather than the natural one.
Similarly, 10^x can be simplified as 10^(a+x) = 10^a 10^x. If you let a be the closest integer to x, you can approximate the 10^x part as:
10^x =~ 1 + 2.303x + (2.303x)^2/2
(where 2.303 is 1 / log e)
So, for example, 10^2.2 = 100 * 10^0.2 =~ 157. The correct value is approximately 158.5.
It might also be worth rounding to the nearest half-integer. 10^2.6 = 100 * sqrt(10) * 10^0.1.
The Bemer algorithm is a generalisation of this that uses nine precomputed values. See http://www.myreckonings.com/Dead_Reckoning/Chapter_4/Materials/Bemer_Exponentials.pdf for details.
need help trying to help my daughter. square roots: square root of 60? like the 56’s square root is 2*14. please help me
I think you must have meant that sqrt(56) = 2*sqrt(14).
The way you get this is to factor out squares. Sqrt(56) = sqrt(4*14) = sqrt(4)*sqrt(14) = 2*sqrt(14). You just keep working until you’ve taken out all the square factors that you can.
Hope that helps!
Hey…
Im not a math genius, and throughout all of my schooling, we have been required (even in college) to use a calculator for math and physics. Lo and behold, it is MCAT season, and we CANNOT use calculators on the exam. The exam covers a lot of mathematics in chemistry and physics. We have about a minute or so to answer questions without calculators. These questions are including square roots, exponents, fractions etc. which do requires calculators, but I have now relaized that it is mainly taking me a longer time to study techniques to help me solve the math problems by hand (and no calculator) than it is for me to study the topics of the exam (physics, chemistry, verbal, biology and organic chem)
Would you guys know of a site that may help me improve my math skills? The exam is coming up very soon. Thanks everyone!!!!!!
I would have LOVED this algorithm when I was about 9 or 10!!! I used to wonder how to calculate square roots. Of course, I figured out the guess and check way, but despite spending ages trying to figure our a better way in primary school, I was too stupid to get this algorithm.
Now I just use the Taylor series expansion of the square root around the closest perfect square.
I certainly wouldn’t call not coming up with this algorithm as a kid “stupid”!
Historically the way to compute square roots is to draw up a table of squares, and then surprise surprise you have a table of square roots. This also applies to other roots. Archimedes probably calculated cube roots in this way, that is by first drawing up a table of cubes.
Archimedes’s method of expressing the square root of 3 as an approximate fraction was to multiply one of the figures in the square column by 3, and then look further down the column to find this product. The corresponding figure in the root column would be the numerator of the required fraction with the initial figure in the root column as the denominator. This procedure can be followed for all roots, but I agree a calculator would be quicker and more efficient, but without the mathematical thought processes.