|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 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:
In our last unit, sometimes my students found values like sin(15) which is an especially ugly exact value:
Now if only this understanding would help us avoid the temptation to "simplify" √6 - √2 and make it √4.