Estimating square roots
(while driving a car)
Some thoughts on mathematics by Dave Coulson, 2014
When I was a kid at school – and I mean primary school, circa 1970 – I was taught a
procedure for estimating the square root of a number. I think I practised it a couple of
times on one school day and then put it aside like all the other things I learned but
never had a use for.
Estimating a square root was a chore, a mathematical version of washing the dishes.
It wasn’t fun because it led me into some pretty intense calculations with fractions.
And what I remember is that the process never ended; we kids just chose a point at
which the working got too hard and we had no choice left but to say “Well the
answer is somewhere near this”.
I’d missed the point entirely. But as a ten year-old (or thereabouts) I wouldn’t likely
have jumped out of my seat with excitement at having become the master of square-
root finding. It had nothing to do with my life then, and to be honest it has little to do
with my life now.
A few years go by and these little electronic boxes enter the classroom and square
roots are never estimated again. Because the square root key on a calculator is a one-
step operation, we never think of estimating square roots anymore, and by rights I
should have no inspiration to write this essay.
AND YET here I am, writing about something that, forty years after the first
calculators, I think is interesting if not relevant.
Think about those people that worked in the technical professions before the arrival
of the calculator: your engineers and pharmacists and draftsmen and so on. All of
these midrange academics would have needed techniques for estimating square
roots and most of them would have become superbly capable of finding them.
I had an uncle who had trained as an aircraft navigator during the second world war.
At his funeral, his son (who is an engineer) made the remark that his father was
extraordinarily good at estimating square roots of numbers and could get an answer
surprisingly fast. I don’t think he was a mathematical genius (although I’m in no
position now to find out). I think he was just well-practised at this art, and through
practise he’d become very good.
So how DO you find the square root of a number. The first step is very important. The
first step is to accept that it is VERY EASY to do and takes essentially NO TIME AT ALL
and can (to a certain extent) be done WITHOUT PAPER.
This is the important thing that has been lost because of the calculator: not that
there are ways of getting square roots, but knowing that the methods are actually
very easy. It’s as if we all forgot how to make a sandwich when the fast food places
I had cause to consider this question while I was driving my car and listening to a
report on the radio. This fellow said that he’d started up an underground farm in an
abandoned tube station under London and was growing veges under artificial lighting
and heating, and that all he had to do to sell his produce was carry the goods upstairs
to the most densely populated streets in the country.
What a brilliant idea! I listened on. The interviewer asked how big the space was and
the farmer said he had 6000 square metres of farmable land down there.
6000 square metres... I started to visualise this (not a safe thing to do while driving a
car). But for a few moments it was important to me to understand just how big a plot
of land 6000 square metres is. So I started estimating, and that eventually led to this
You see, 6000 square metres is a chunk of land. It has a length and a width, and if it’s
a rectangle, it might be 60 metres by 100 metres. Okay, that gives me a really good
idea already of the size of the farm. It’s kind of like the piece of land in the middle of
a sports track. It’s way bigger than my front yard.
Okay but let’s go further. What if it was a square field and not a rectangle?
Abandon your mathematical senses for a while and go back to being a preschooler in
a playpen. If someone gives you a rectangle and asks you to make a square out of
it, you squash the top down and watch the sides bulge out, right?
So squash the number 100 and watch the 60 bulge out. Now you arrive at the
number 80; a square of side length 80.
Is this the square root of 6000? Check it out. 8x8 is 64 so 80x80 is 6400.
Oh, that’s not right. But oh again, it’s pretty close to the right answer.
How can that be? I’m off by 400! That’s where we go wrong. We see 400 and say that
that’s a big difference when what we should be doing is seeing how big 400 is up
against 6400. It’s less than ten percent out....
... Which means that the answer of 80 is also out by less than ten percent, and we
know it’s too big by that amount. So the answer is somewhere between 72 and 80.
Probably 74 or 75.
Now think about that for a while and see if this answer is ‘good enough’ or not. I’m
driving a car in Christchurch New Zealand and wondering about a farm in London. Do
you think I need to have the answer to the nearest decimal place? My life isn’t going
to change because of this investigation. No, I’m doing this for pleasure. I am just
asking a what-if kind of question because I want to imagine the life of a person that
owns a 6000 square metre farm.
So I arrive at the first traffic light with that number in my head. Now I’m bored with
the traffic and I begin to wonder if I can make the answer more accurate, just for
something to take my mind off the business of watching cars. It’s a game now, a
personal challenge to see how accurate I can make that answer before the light turns
There’s a procedure for finding the square of a number that ends in zero point five.
Most mathematicians know it. For example, 2.5 squared is 2 x 3 + 0.25.
3.5 squared is 3 x 4 + 0.25.
Whatever our number is, get the two whole numbers on either side of it and multiply
them together and add 0.25.
9.5 squared is 90.25. Get it?
Now that gives me a few more numbers I can estimate with. I know that the square
root of 72 is a scratch less than 8.5 because 8 x 9 = 72.
You see? You can draw up a chart of squared whole
numbers and intersperse it with squares of half
numbers and have a chart that lets you estimate
the square roots of numbers that lie between.
So now I know that the square root of 6000 is
somewhere between 75 and 80, and I haven’t
needed a piece of paper, and I’m sitting in a car at
an intersection. This is not hard maths! All I’m
doing is referring to the times table in my head.
7 x 8 = 56
8 x 8 = 64
Okay, this is getting really interesting for me because I’m so close to the answer. Can I
get any closer without making this a mathematical chore? Well, think of the sizes of
those numbers. 75 is too small and 80 is too big. 75 squared gave me about 7200 and
80 squared gave me 6400. I want a number that’s about halfway between. Okay, so
choose 77 or 78.
Now I’m at a roadblock – no not on the road but in my head. How do I square a
number like 77 or 78 to see which one is right? Then it dawns on me that I’m talking
about the difference between two adjacent numbers, a distance in metres that is
about the length of the seat I’m sitting on, compared to a field that’s seventy or
eighty times as long! Honestly, by this stage, who cares to know the answer more
precisely than that?
From here there are numerous pen and paper methods that will all work to improve
upon my answer. But I said at the outset that estimating square roots is not hard and
doesn’t take long and generally doesn’t need pen and paper, and shouldn’t feel like
washing the dishes. And that means knowing how close an estimate is good enough.
77? 78? I’m done!
For those who’d like to see the pen-and-paper methods, consider this:
‘a’ is a small number, a decimal number, so ‘a2’ is even smaller. Throw it away.
‘a’ is 0.46, and so a very accurate estimate of the square root now is 7.46
You can repeat the process to further refine the estimate:
‘a’ is REALLY small now, so throwing it away is even less of a crime than before.
‘a’ is -0.00033 The square root of 6000 is about 7.46 – 0.00033 = 7.45967
The problem here is that I needed a calculator to perform those calculations, in which
case I may as well have simply hit the square root key.
You see, we’re now living in an age where estimating numbers not only has to
work, it has to work faster and less painfully than reaching for a calculator, which is to
say not painful at all.
So it seems that there is an event horizon like black holes have which we don’t cross
for fear that we’ll get sucked into a mathematical vortex that takes us nowhere.
Outside of that vortex, estimation is simple and we do it frequently because it’s easy.
The whole point of estimating is that it’s supposed to be easy. So keep it easy! Know
when to stop!
That answer of 77.459 takes me down to the level of millimetres. Remember we’re
talking about a farm measured probably to the nearest metre. Even the farmer
himself doesn’t care about the millimetres! I have obtained an answer more accurate
than it had any right to be, as if I was trying to wash the discolouration off a dish that
was never white to begin with. Know when to stop!