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!

On 6/28 this year I introduced quite a few people to the circle constant tau, as I usually do.  Surfing around on the web there were quite a few articles explaining why tau is better than pi, and there’s always the Tau Manifesto.  But none of them cut to the chase in quite the way I wanted.  There is one reason that stands out to me as by far the most important reason to start using tau as a circle constant, and that is that it makes basic trigonometry way more intuitive to learn.  Here’s why.

If you think back to advanced algebra in high school you might remember having to master radian measures of angles.  Instead of 30 degrees, you had to use pi/6.  Instead of 90 degrees, pi/2.  Your teacher may have given you a list of common angles to memorize in pi radians so that you could convert these angles easily from one to the other.  Very likely, you found this confusing and frustrating.  As you finally mastered it, you may have been able to visualize these fractions of pi as fractions of a half a circle.  Stop and ponder that for a moment… fractions of a half of a circle?

Radians measure angles by tracing out a fraction of the circumference on the unit circle.  3.14159… radians gets you half way around the unit circle.  6.2831… radians on the other hand, gets you all the way around the circle!  This is tau, equal to 2 times pi.  If you use tau as your circle constant then angles measured in radians suddenly become easy.  Now instead of thinking of fractions of a half of a circle, you just have to think of fractions of a circle!

I have a fond memory of one of the first times I introduced tau to a student I was tutoring.  We met in the evening and she told me that she had come away from her math class that day feeling confused.  They were beginning to study radians.  I asked her if she wanted to know a better way of writing these radian angles even if it was going to be different than the way her teacher was writing it.  She agreed so I showed her tau radians and how they related to pi radians.  After a very short explanation she pointedly picked up her pencil and proceeded to cross out her entire two pages of notes from class that day, in order to re-do them using tau.  Ah, the delights of elegant and intuitive notation!