How are square roots of non-perfect squares determined?

Hello there. Random guest who decided to poke her head in after deciding that you guys looked like you had too much fun and wanted in on the action. If I like it, maybe I’ll subscribe. I’ll spare you my life story, save for the bare details: Me = college student, English major, gamer with a soft spot for Nintendo, wannabe fantasy author, insomniac. Anyway.

Nowhere in the above list will you find “math maven,” you will note. I have a terrible track record when it comes to math classes, due to my general dyslexia around numbers and their inability to form any sort of coherent pattern in my brain, but nontheless I have a general knack (when pushed) to discern from what existing math rules a formula stems. I can figure out how formulas are derived usually, and if I don’t know a formula, I can sometimes work one out. That said:

I have no bloody clue how one determines the square root of a non-square number.

I know there must be a formula for it of SOME kind. I know I can plug any unholy-looking number into my calculator, press the square root button, and it will return and equally unholy-looking number without so much as blinking. In ye olde days before TI-85s (or whatever installment they’re up to now) I know people used a table to determine things that ended up in unholy-looking decimals such as unholy sines and cosines and unholy square roots of non-perfect squares, but SOMEBODY had to figure out those numbers to begin with.

I’m a visual person, and I can sometimes work out the math of something by picturing it in my head in little cubes and those awesome colorful wooden shape blocks they gave us to play with in kindergarten (I somehow remember always pretending the big yellow hexagons were gold coins), but whenever I try to picture the square rooting of a non-square I either draw a blank or an ugly, contrived picture that I absolutely cannot translate into “real mathematics.” So would anyone care to explain to me how the heck one goes about this operation?
I’m never gonna calculate /2 just for the heck of it, but it nags at me SO BAD that I don’t know this.

Here’s the paper and pencil method I learned.

Welcome. Perhaps the key to understanding square roots is to recall that they are less tied to “perfect squares” than they are to the idea that a square root is the number that, when multiplied by itself, yields another number, its square.

So a table of perfect squares is where to start “interpolating” the non-square numbers.

By that I mean that since 2-square is 4 and 3-square is 9, you know (or can deduce) that numbers between 2 and 3 are the square roots of 5, 6, 7, and 8.

There are several routines, similar to long division, that let you approximate the answer and then “zoom in” with progressively better answers until you get the number that is the correct square root of a given number.

That may not help a great deal, but it’s the big picture behind what’s just “boring math” to get a more precise description.

There are several ways to do this, which I haven’t time to tell right now. There’s a structured method that looks like long division (that I learned from my father) which works quite well, but it’s not the method calculators use.
I’m sure someone will come by soon with links, but a quick google search ought to net lots of possibilities.

That looks like how I learned it, too, and what I meant by “looking like long division.” I remember learning logarithms along the way and they are a quicker way to get to square roots, but I have forgotten most of what I learned about logarithms. All I can say is that calculators have removed the need to know much “brute force” or “old timey” ways of doing math, and so most kids these days never do learn the fundamentals. Turn the power off at McDonald’s and they can’t make change!

I haven’t seen that in over 35 years (my primary school teacher showed it to us once) and I’d long since forgotten it. Thanks!

You can use Newton-Raphson iteration, thus:

  1. Have a guess at what the square root might be (it doesn’t need to be very accurate).
  2. Divide the number by the guess.
  3. Add the guess to the quotient thus derived, and divide the sum by 2.
  4. This is now your new “guess” so return to step 2
  5. When successive quotients differ by less than an “acceptable” amount, stop.

With minimal programming skills this is easy to implement in the language of your choice. This sequence quickly “converges” - that is, after a few times round the loop you are rapidly closing in on the answer.

Thanks! I wouldn’t say my life is satisfied, but the part of me that latches on to odd questions in order to further my insomnia is. Like I said, I will never, ever use this formula, but knowing random stuff is satisfying.

You know, looking at this, I suspect that my seventh/eighth grade math teacher may have taught us this at some point. But I have almost zero memories of that guy’s lessons. He was very intelligent, yes, but man, he was duller than a perfect circle.

It’s refreshing to learn that your teacher(s) may have at least exposed you to the “older way” of doing this (and hopefully other) exercise(s). My fears about losing basic math skills date back to the “New Math” days when how I learned stuff (obviously not well enough to be able to recall specifics) was discarded in favor of more “useful” ways of acquiring math concepts. That, followed by classroom usage of calculators and even computers for “doing math,” has made me wary of how seriously behind American kids are falling in the “Math & Science” race.

I believe the method that looked like long division fell out of favor in most curricula some time before calculators were commonplace. After all, if you really needed a square root in real life, you had log tables and slide rules. Rote learning of the sequence of operations in the long division appearing technique didn’t really teach anything fundamental. Far better that the student grasp why dividing the log by 2 gives them the log of the square root.

I can accept that the “old way” was phased out long before calculators and was replaced by more intuitive methods like logs and slide rules and other implements. However, the “old way” was something one could do with pencil and paper (if not in one’s head) even in situations where log tables and slide rules and other external devices weren’t handy. My main point about the “old way’s” utility was that at least the way to get the answer didn’t require ownership of some device or book or the like.

Rote is a difficult and perhaps unnatural way to learn things, especially if algorithms and mental tricks can be used instead. But assuming that we’ll never be in a situation where calculators aren’t accessible is a poor outlook to have on such a fundamental notion as basic arithmetic. I’ll concede that square roots may rise above “basic math” in most people’s progress through math skills, but learning the multiplication tables is something I can’t imagine not being taught.

Wait, you honestly think kids aren’t taught the multiplication tables anymore?

Sure they are. You probably find adults who can’t multiply, but I seriously doubt this number is higher than it was in previous generations. And any adult that can’t multiply certainly isn’t using a calculator, rather this person never does math of any sort except perhaps making change.

If you want to see the square root of two, it’s the length of the diagonal of a unit square (whose sides all have length 1).

If you want a decimal approximation, to however many decimal places, there are various methods, of varying degrees of cleverness and efficiency. But, basically, we know that sqrt(2) is between 1 and 2 (i.e. it’s 1.something) since 1[sup]2[/sup] = 1 and 2[sup]2[/sup] = 4.

Narrowing it down further, we know it’s between 1.4 and 1.5 (i.e.e it’s 1.4something), since 1.4[sup]2[/sup] = 1.96 (too small) and 1.5[sup]2[/sup] = 2.25 (too large).

Similarly, we can show that it’s between 1.41 and 1.42, or that it’s between 1.414 and 1.415, and so on.

Yes but … almost no one will ever be in a situation where they need a square root and don’t have aids available to find it. Is it really worth spending a lot of class time teaching a method that almost zero chance of being useful?

Those who actually use mathematics to solve problems always use the best computation aids available to them. In fact, that’s one main reason logarithms were developed. That is, to reduce the labor and time spent in computation.

To this day, I only remember the multiplications for certain numbers (mostly up in the larger single digits, like 7 x 8 and 8 x 8) thanks to a goofy little book they gave us 3rd-grade or thereabouts with silly little stories about the numbers. To this day, I remember 8^2 is 64 because of the two cold snowmen who needed to build a fire… and the STICKS were FOR the fire! And I think of the firemen in their weird-shaped buildings when thinking about 7 x 8 being 56.

My track record with math teachers has been spotty. Some years, like this past semester, I get awesome charismatic people who explain things and answer questions and who always manage to hold my attention. In fact, I think my professor last semester may have been the best math teacher I’ve ever had. He was brilliant at explaining things. But others, I get these… WEIRD people.

I think my weirdest math teacher ever was the guy with the thick Jamaican accent.

In which case, an iterative trial and error technique is possibly no more tedious than the long division thing. If you ask me what the square root of six is, I can fairly rapidly arrive at 2.4<something> by multiplying a few numbers in my head, which is good enough for a rough estimate.

BTW, ESTIMATING is what I think we are really losing. Back when I was a grad assistant, it used to annoy me no end when students cranked numbers through their calculators, got something that was obviously grossly wrong and seemed unaware of it. If the actual answer is 7.85, and you make a couple arithmetic errors and wind up with 17.2, that’s one thing, but winding up with 988,355,670 should at least make you say “look, this is obviously wrong, but I don’t know where I screwed up”. Not to mention the modern habit of presenting answers to more places than have any relevence to the problem because 8 digits appeared in the calculator window.

Assuming you have access to log tables, you could always look up the log of x, divide it by 2, and take the antilog. BTW, does anyone but me have a CRC Standard Mathematical Tables book? Do they even make them anymore?

Or you could use a slide rule. :slight_smile:

I have one. Yep, they are still around.

I bet at least 99.9999% of the time they are determined electronically on a device like the Intel FPU part of a processor (a Pentium, a Xeon, whatever). There are built in hardware commands, though not very many of them. I don’t think there’s a sqrt() command. But there is a log() command that gives the base 2 log of a float (an 80 bit one), and an exp() command that does the opposite. So maybe they’re usually done that way. The FPU probably uses a variant of the Newton - Raphson method.

I also have one and use it frequently. I’m even thinking about getting a new one.

Far out! Thanks for the info. I’m only 9 editions behind, but somehow I think much is the same.