How to Design Programs: An Introduction to Computing and Programming
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Section 11
Natural Numbers
The only self-referential data definitions we have seen thus far involved
cons
and lists of arbitrary
length. We needed such data definitions because the classes of lists that we wanted to process were of
arbitrary size. Natural numbers are another class of data whose elements are of arbitrary size; after all,
there is no limit on how large a natural number can be, and, at least in principle, a function should be
able to process them all.
In this section, we study how to describe natural numbers with self-referential data definitions and how
to develop functions that process natural numbers in a systematic fashion. Since such functions come in
many flavors, we study several different flavors of definitions.
11.1
Defining Natural Numbers
People normally introduce natural numbers via enumeration:
0
,
1
,
2
, etc.
34
The abbreviation ``etc.'' at
the end says that the series continues in this manner. Mathematicians and mathematics teachers often use
dots for the same purpose. For us, however, neither the ``etc.'' nor the dots is good enough, if we wish to
design functions on natural numbers systematically. So, the question is what it means to write down
``etc.,'' or put differently, what a complete, self-contained description of the natural numbers is.
The only way to remove the informal ``etc.'' from the enumeration is to describe the collection of
numbers with a self-referential description. Here is a first attempt:
0 is a natural number.
If
n
is a natural number, then one more than
n
is one, too.
While this description is still not quite rigorous,
35
it is a good starting point for a Scheme-style data
description:
A
natural-number
is either
1.
0
or
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