And now, a brief break from non-blogging:

Today, I’d like to start with a specific example and move on to a general point. The specific example is a way to approximate the squares of numbers and then refine those approximations to get exact answers, and the general point concerns the place such techniques should have in mathematics education.

My last calculator broke years ago, so when I have to do a spot of ciphering, I have to work the answer out in my head or push a pencil. (If the calculation involves more numbers than can fit on the back of an envelope, then it’s probably a data-analysis job which is being done on a computer anyway.) Every once in a while, the numbers teach you a lesson, in their own sneaky way.

It’s easy to square a smallish multiple of 10. We all learned our times tables, so squaring a number from 1 to 9 is a doddle, and the two factors of 10 just shift the decimal point over twice. Thus, 50^{2} is 2500, no thinking needed.

Now, what if we want to square an integer which is *near* 50? We have a trick for this, a stunt which first yields an answer “close enough for government work,” and upon refinement gives the exact value. (I use the “close enough for government” line advisedly, as this was a trick Richard Feynman learned from Hans Bethe while they were calculating the explosive power of the first atomic bomb, at Los Alamos.) To get your first approximation, find the difference between your number and 50, and add that many hundreds to 2500. The correction, if you need it, is to add the difference squared. Thus, 48^{2} is roughly 2300 and exactly 2304, while 53^{2} is roughly 2800 and exactly 2809.

I wouldn’t advise teaching this as “the way to multiply,” first because its applicability is limited and second because it’s, well, arcane. What a goofy sequence of steps! Surely, if we’re drilling our children on an algorithm, it should be one which works on any numbers you give it. The situation changes, though, after you’ve seen a little algebra, and you realize where this trick comes from. It’s just squaring a binomial:

[tex](a + b)^2 = a^2 + 2ab + b^2,[/tex]

which when [tex]a = 50[/tex] becomes

[tex](50 + b)^2 = 2500 + 100b + b^2.[/tex]

And once you’ve seen it like *this,* you can generalize it and cultivate your own tricks from this seed.

We were taught to multiply binomials in eighth grade, if my memory does not fail me completely. Our lessons involved multiplying a pageful of binomials with the good ol’ FOIL method, squaring another pageful, multiplying a few dozen more for review. . . eventually moving on to the exciting operation of *factoring* a *trinomial!*

I do not remember middle-school math with any particular fondness. And, let’s face it, I was a nerd (he says as he reaches for the diet soda bottle perched precariously atop the DVD box sets of *Firefly* and *The Prisoner*). I don’t know the best way to fold stuff like the squaring numbers near fifty trick into the curriculum, but I suspect that items of this sort might help at least some fraction of the students stay awake. If we were shown how the algebraic formulas could be helpful in doing arithmetic, then the lesson would have had more purpose and been more memorable, which I think would be beneficial. Budding scientists need to learn about taking successive approximations, too. And if we stick such lessons in a coursebook *somewhere,* then maybe we can boost the number of teachers at all levels who are wise in these ways, and who are thus capable of teaching them at other grade levels when the circumstances require it.

(Dealing with a complicated machine requires flexibility, and you can’t be flexible when you only know how to do one thing.)

Education is not a straight-line race. I see it more as a great big scavenger hunt (in a dark forest belurked with goblins, naturally). There are treasures which everybody needs to bring back, like how to do long division, and there is a path through the forest where the terrain has been pounded flat and much of the undergrowth trimmed back. The path works for a lot of people — maybe a majority, maybe just a plurality — but some have no luck following it, and others can scamper over the rocks to reach many of the prizes faster.

Great post, this would have helped me enjoy binomials back in 8th grade algebra. My memory of all that early algebra is of extreme boredom from having no idea what I’d ever use it for. I don’t remember being shown any practical uses for most of algebra I or II. If teachers could work in at least some application examples I think it could help a lot of students appreciate the subject. But then again maybe it’s only a nerd thing (as I set my cup of water down on my Firefly box set).

This sort of stuff is great. Thanks!

It’s exactly the sort of thing I want my students to see – not to use it necessarily, but to get a feel for playing with numbers.

I can’t imagine trying to get through life without the dozens of numerical tricks I carry around. One thing that amuses me no end (when I occasion to teach undergrads) is the frequency with which I am able to compute the first three or four digits of a numerical answer before any of them have even punched the first number into their calculator.

For me mental calculation like this is essential even when you do have a calculator or computer to hand, because you need to be sure you didn’t make an error using the calculator; if you can get a good approximation to the answer in your head, you can be pretty confident when you have a very similar answer on your calculator that it’s actually the right answer.

Squaring numbers is just the start – suffer the little children to extract square roots and – abracadabra – make the acquaintance of continued fractions

It looks like I made at least a couple people happy with this one. Count this as my good deed for the year!

A while back, some of us were talking about extracting square roots by hand, and it seems that schools stopped teaching that in the early 1980s or thereabouts. It’s really too bad.

They stopped teaching extracting square roots even before that in schools around here (certainly by around 1977), if you’re talking about the old somewhat-long-division-like method. But we did learn to do it with log-tables, and I also figured out binary search for myself about that time.

These days, if I need the square root of some number (which I often do since I calculate standard deviations a lot) with no calculator handy, I just use Newton-Raphson (these days usually just called ‘Newton’s method’); in the case of square roots, you just average your previous guess and (the number divided by the guess). Good enough first guesses are easy – you just need to know the squares of the first 10 integers to have 1 significant figure accuracy, and NR will approximately double the significant figures of accuracy each step after that. I can usually get roughly two figure accuracy on a first guess anyway. There are few problems for which 3 figure accuracy is not sufficient (if you need more than that, you probably have no business doing it in your head), so one or maybe two steps of Newton-Raphson is usually enough (I sometimes prefer to do the calculation approximately and run it for an extra step rather than do fewer steps more accurately, since an extra step will give more accuracy anyway; NR is tolerant of approximations along the way).

I have a way of extracting the first few digits of large roots of large numbers in my head (25th root of a billion, 40th root of a trillion, that kind of thing). It doesn’t work so good when I am tired (I get very slow and tend to make mistakes), but otherwise it, it’s pretty quick and really impresses people who don’t know how to compute a square root.

And yet all it consists of is a few nifty facts (e.g. “tenth root of 10 ~= cube root of 2” – the resulting tricks from that fact are well known to anyone who has done calculations with decibels; plus the rule of 72 – sometimes called the ‘rule of 70’ – the ‘doubling time’ rule for investments). That, and the fact that the percentage errors in approximations of numbers go down as you take fractional powers…

So, for example, the 25th root of a billion: 10^9~=2^30 (decibel trick – yes, that’s very rough, about 8% error, but no matter).

So the 25th root of that is 2^(30/25) = 2×2^(1/5).

The fifth root of 2 is roughly (1+ 72/500) (rule of 72 – it helps to know when to use 72 and when to use 70, though if you pick the wrong one it is still pretty decent) or about 1.144.

Double that (for the factor of 2 from before) and round off to 2.29. Which is close enough to make people look at you like you just did something impossible.

When you’re not explaining it, the whole thing takes only a few seconds.

Oh, and with a bit of practice, you can start saying the answer before you finish calculating it… it’s obvious at step 1 of the example calculation that it’s a bit bigger than 2, so you start with “two point …” as you go. In practice, I’d usually say something like “hmm… that’s roughly … two point … two eight something… say 2.29” – it sounds like you got it almost instantly, but in fact you’re using the speaking time to finish the calculation; people tend to assume that when you start saying the number you must have already finished.

I’ll often use the binomial equation for quick estimates when running around. Just rewrite

A * B = (a*10 + c) * (b*10 + d) = ab*100 + (ad+bc)*10 + cd

if you remember the magical -subtraction- operation you can usually keep c and d under 5 so the second term is easy to get and the third term is in the noise. E.g., 28 * 23 = (30-2)*(20+3) = 600 + (9-4)10 – ? or a little under 650. (644).

This is just a little more effort than just rounding (30 * 20, in that case), but good enough for an extra digit or two of accuracy.

(Obviously you can do the same thing with hundreds and thousands.)

Nice trick — I’ve pulled that one out from time to time, as well. Again, it makes best sense when you’re familiar with algebra, and it points out the importance of approximations.