Square Roots as Line Segments

Each of the square roots from √1 to √17
 are shown as the colored segments
Image by Rocky Roer
Made in Geogebra

One thing I wish my students and I had a better grasp of was square roots. We work with integer numbers well enough, but often struggle to make any sense at all of other perfectly valid numbers like square roots. It is unnatural to think of them as numbers, or lengths at all, and perhaps additional understanding could be gained from seeing them like that?

One thing I like to do is realize that each square root really is just a number. Many of them get a bad rep simply because they can't be written in fully written in decimal notation, but they can easily be drawn. Each of the square roots can be constructed quite easily, as illustrated in the picture to the right.  The first 17 square roots are drawn there, simply by drawing a unit length, constructing a perpendicular segment, drawing a unit circle, finding an intersection, and repeating. I stopped at 17 only because if I went further they would have started overlapping and I didn't like the way that looked.

Another thing I have found a little enlightening is a variation in simplest radical form. In PreCalc we've been simplifying expressions like tan(30). That process produces:

That last expression of √3/3 seems devoid of meaning, but is the required "simplified radical form" called for. I have found some enlightenment coming from describing that instead as:

This helps me to have an idea of that length -- it's just a fraction of the √3 length. Other trig segments like √3/2 and √2/2 and such could also be expressed as 1/2*√3  or 1/2*√2 and it's a step towards increasing my understanding of what those segments really are.

In our last unit, sometimes my students found values like sin(15) which is an especially ugly exact value:

Written as 1/4(√6) - 1/4(√2) is not significantly better, but helps me to realize how this number could be constructed. I could take a fraction of the yellow green segment and take away a fraction of the red segment above and make it.  Or alternately, 1/4 (√6 - √2) suggests take the yellow-green segment and chop off a red segment. Save that piece cause you'll use it many times (for secant and cosecants of 15 and 75 for instance) but take a fourth of it for now.

Now if only this understanding would help us avoid the temptation to "simplify" √6 - √2 and make it √4.