[1] Note that we slightly extend the syntax of conditional statements described in section 1.3.2 by allowing another conditional statement in place of the block following else.
[2] We have used 0.001 as a representative small number to indicate a tolerance for the acceptable error in a calculation. The appropriate tolerance for a real calculation depends upon the problem to be solved and the limitations of the computer and the algorithm. This is often a very subtle consideration, requiring help from a numerical analyst or some other kind of magician.
[3] This can be accomplished using error, which takes as argument a string that is printed as error message along with the number of the program line that gave rise to the call of error.
[4] Try this during a boring lecture: Set your calculator to radians mode and then repeatedly press the $\cos$ button until you obtain the fixed point.
[5] $\mapsto$ (pronounced maps to) is the mathematician's way of writing function definitions. $y \mapsto x/y$ means y => x / y, that is, the function whose value at $y$ is $x/y$.
1.3.3 Functions as General Methods