This week I learned something new from one of my learners. This happens quite a bit and I always love it. He had watched a video about a method of estimating square roots. Here’s how it goes.
Of course we start by finding the integer square root of the last perfect square below our number. For 45 our result will be 6 plus a fraction, for 124 it will be 11 plus a fraction. Now we just need to find a fraction to add on to our integer that closely approximates the square root. For the numerator we use the difference between our number and that nearest perfect square below our number (45 – 36 = 9 or 124 – 121 = 3). For the denominator we double the integer square root we arrived at in the first step. So for 45 our final approximation is 6 9/12 and for 124 we get 11 3/22.
This brought up some intriguing questions. Why does this method work? In particular, what’s up with that doubling of the integer to arrive at the denominator of our added fraction? I should have let my learner lead the way on this question, but I let my curiosity get the better of me this time. We worked on the problem together but I had some previous experience to draw on from having proved to myself why the pencil and paper algorithm for taking square roots works. I let that background knowledge I had lead us through it and we pretty quickly arrived at a satisfying explanation. Can you create your own explanation?
Next we wondered: How does the error vary as you increase your starting number? We hypothesized that the error would be smallest for numbers just slightly above a perfect square and largest for numbers just slightly below a perfect square (because those are farther away from the next perfect square below them. We also guessed that the accuracy would get worse as our starting number got bigger.
To test this second hypothesis his first impulse was to try it out with a number near the largest square number. Having satisfied himself that there is no largest perfect square, he settled for numbers near 10,000 (100 squared). This produced a surprising result! Rather than finding a significantly inaccurate estimate for 9800 (one less than 99^2) the algorithm produced an estimate that was very close to exact. So now, we still expected numbers close to the next perfect square to fare worse in the algorithm, but that as the numbers went higher the algorithm would produce better and better accuracy.
Later in the week at home he produced this graphic to show how the error changes as the input number increases.
So cool! Great work on his part. Maybe next we can explain to ourselves exactly why the error falls off with larger inputs!
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