Class 08: Numbers

Held: Friday, 31 January 2003

Summary: Today we continue our exploration of numbers in Scheme.

Related Pages:

• EBoard.
• Reading on numbers.
• Lab on numbers.

Assignments

• Read Defining Procedures in Scheme.

Notes:

• The consensus in all three of my classes is that others can visit as long as they are quiet and don't take computers needed by students in the class. So, you can feel free to work in the MathLAN from 10:00-10:50 and from 3:15-4:05 (and beyond).

Overview:

• Types.
• Numbers, Kinds of Numbers, and Numeric Operations.
• Lab on Numbers in Scheme.
• Reflection.

Types

• As you may have noted in your first experiments with Scheme, Scheme assigns types to variables.
• For example, a value might be a list, or a number, or ....
• Computer scientists often think of types in two different ways:
  • Data-driven: A type is a set of values.
  • Purpose-driven: A type provides information on the valid operations that may be applied to a piece of data.
• We will alternate between the two definitions.
• Many languages (particularly the ones you've reported being familiar with) require you to assign a type to a variable when you declare that variable.
• Scheme does not require you to assign types ot variables; it checks the type of each operand when it executes a procedure.
  • Scheme also provides procedures that let you determine the type of a value.
• As the semester progresses, you will learn new types.

Scheme's Numeric Types

• Instead of a general numbers type, Scheme provides a variety of kinds of numbers.
• Integers are numbers without a fractional component.
• Rational numbers can be expressed as the ratio of two integers.
• Real numbers appear somewhere on the number line.
  • In mathematics, real numbers can be rational or irrational.
  • In Scheme, real numbers are all rational.
• Complex numbers may include an imaginary component.
• Scheme also represents some numbers exactly and some numbers inexactly.
• It may surprise you to see which numbers are represented inexactly. (We'll return to this issue later.)
• Some important numeric predicates (procedures that return true or false): number?, real?, rational?, complex?, integer?, exact?, and inexact?.
• Some important numeric procedures: numerator, denominator, quotient, and sqrt.

Lab

• Any questions?
• Do the lab on numbers.
• Reflect.