Question: Why can't I say just abs() or sin() instead of Math.abs() and Math.sin()?

Answer: The import statement does not bring methods into your local name space. It lets you abbreviate class names, but not get rid of them altogether. That's just the way it works, you'll get used to it. It's really a lot safer this way.
However, there is actually a little trick you can use in some cases that gets you what you want. If your top-level class doesn't need to inherit from anything else, make it inherit from java.lang.Math. That *does* bring all the methods into your local name space. But you can't use this trick in an applet, because you have to inherit from java.awt.Applet. And actually, you can't use it on java.lang.Math at all, because Math is a '??final'?? class which means it can't be extended.