Class 06: Computing with Symbols and Numbers

Held: Monday, 7 September 2009

Summary: We explore a bit more about data in Scheme, particularly the ways in which our version of Scheme supports numbers.

Related Pages:

• EBoard.
• Lab: Numeric Computation.
• Reading: Numeric Values & Symbolic Values.

Notes:

• EC: Women's Soccer vs. Central, 5:30 p.m., Wednesday.
• I've added instructions for lab writeups to the handouts area.
• Don't forget to fill out the RISC survey at http://www.grinnell.edu/academic/psychology/faculty/dl/risc/. It should take about fifteen minutes.
• The quiz is graded and returned. I do not report average grades or other statistics on grading.
• There was a bit of confusion on 2c on the quiz. Since name is a variable and undefined, the interpreter will respond with an error when it is used.
• A number of you neglected to include the quotation marks in string output. These marks tell you (as user of Scheme) that the result is a string, rather than some other type of value, such as a symbol.
• A question on the first writeup: Some folks wondered what me meant by changing the size of the image. We want you to reflect what happens if we do the same drawing commands in a window of a different size.
• Are there other questions on the lab writeup or assignment?

Overview:

• Types.
• Kinds of Numbers.
• Modulo.
• Lab.

Types

• As you may have noted in your first experiments with Scheme, Scheme assigns types to variables.
• For example, a value might be a number, or a string, or an image identifier, or a procedure, or a drawing, 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.
• As you learn each type, you'll learn a variety of things (that correspond, in some sense, to those two approaches):
  • How to express values in the type. For example, we express string values by surrounding them with double-quotation-marks and we express numbers in much the way we always have.
  • What operations are possible on values in the type. For example, we can use the addition operation (+) on numbers and we can use the string-append operation on strings.

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.
• You can (almost) always use an integer when a real is expected, but you cannot always use
• Scheme also represents some numbers exactly and some numbers inexactly. (That is, it approximates some numbers.)
  • It certainly has to approximate irrational numbers.
  • But it also approximates many other numbers.
  • 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?, integer?, exact?, and inexact?.

Modulo

• The mod (modulo, modulus) operation is one of the trickier operations we use in this class (and we use it a lot).
• Essentially, mod is used to break up the number line into even chunks.
  • If you mod by 7, you break the number line up into chunks of size 7.
  • If you mod by 23, you break the number line up into chunks of size 23.
• For each chunk, we start counting at 0.
• For example

  Number line:  -9 -8|-7 -6 -5 -4 -3 -2 -1| 0  1  2  3  4  5  6| 7  8  9 10 11 
  Modulo 7:      5  6| 0  1  2  3  4  5  6| 0  1  2  3  4  5  6| 0  1  2  3  4 
  

• The (modulo i n) operation allows us to cycle through the numbers between 0 and n-1.

Lab

• Do the lab on numeric values.
• Be prepared to reflect.